Professor S P Gul'ko

Professor E A Tarasov

The goal of this course is to study the main tools of the renewal theory and their applications to some problems of the actuarial analysis for insurance companies in the framework of the Crem er -Lundberg models. We consider such important problems in the renewal theory as limit theorems for the renewal processes and the ruin problems for the insurance companies with investments in the stochastic nancial markets.

1 Poisson processes

Denition and main properties

In this section we study the principal properties of the Poisson process. Let (τ j ) ≥1 be i.i.d. exponential random variables with some parameter λ > 0. We set T n = ∑ n j=1 τ j for n ≥ 1 and T 0 = 0.

Denition 1.1. The random R + → N function

N t = ∞ ∑ n=1 1 (T n ≤t) (1.1)
is called the homogeneous Poisson process of the intensity λ > 0.

Proposition 1.1. If (τ j ) j≥1 is i.i.d. exponential random variables with a parameter λ > 0, then the vector (T 1 , . . . , T n ) has the distribution density with respect to the Lebesgue measure in R n dened as

f n (x 1 , . . . x n ) = λ n e -λx n 1 {0<x 1 <...<x n } . (1.2)
Proposition 1.2. If (N t ) t≥0 is a homogeneous Poisson process with an intensity λ > 0, then for any t > 0 the random variable N t has the Poisson distribution, i.e. for any integer n ≥ 0

P(N t = n) = (λt) n n! e -λt . (1.3)
Now we study the main properties of the Poisson processes.

Proposition 1.3. Let (N t ) t≥0 be a homogeneous Poisson process of an intensity λ > 0. Then 1. almost sure the function (N t ) t≥0 is increasing, with integer values and right continuous;

2. conditionally with respect to N t = n, the vector (T 1 , . . . , T n )

has the same distribution as n order statistics uniformly distributed on the interval [0, t];

3. the Poisson process (N t ) t≥0 has homogeneous increments, i.e. for all 0 < s < t and any integer n ≥ 0

P(N t -N s = n) = P(N t-s = n);
4. the Poisson process (N t ) t≥0 has independent increments, i.e.

for any time moments 0 = t 0 < t 1 < . . . < t m and any integer numbers n 1 , . . . , n m P

( N t 1 = n 1 , N t 2 -N t 1 = n 2 , . . . , N t m -N t m-1 = n m ) = m ∏ j=1 P ( N t j -N t j-1 = n j ) ;
5. the Poisson process (N t ) t≥0 is a process of rare events, i.e. for any t ≥ 0 and ∆ > 0

P ( N t+∆ -N t = 1 ) = λ ∆ + o(∆) , P ( N t+∆ -N t > 1 ) = o(∆) , (1.4)
as ∆ → 0.

Remark 1.1. As we will see later all these properties are very use- ful in the actuarial mathematics for the constructing the principal insurance models. Indeed, the Poisson process is used to model the number of claims on the time interval [0, t]. Especially, the independent increments and rare events properties are very natural for the insurance models.

Principal features

Proposition 1.4. Let (N t ) t≥0 be a stochastic process that satises the following conditions:

• for almost every ω, the trajectory (N t (ω)) t≥0 is zero in 0, increasing, right continuous and with integer values;

• the process (N t ) t≥0 has independent and homogeneous increments;

• (N t ) t≥0 is a process of rare events, i.e. there exists λ > 0, for which the asymptotic properties (1.4) hold.

Then (N t ) t≥0 is the Poisson process of the intensity λ > 0.

Proof. Firstly, we show that P(N t = 0) = e -λt .

(1.5)

We denote by f (t) = P(N t = 0). Indeed, due to the independence N t and N t+s -N t we obtain f (t + s) = P(N t+s = 0) = P(N t+s = 0 , N t = 0)

= P(N t+s -N t = 0 , N t = 0) = f (t)f (s) .

Using here the rare events property, we get (1.5). Now we nd the distribution of N t for arbitrary xed t > 0. To this end, we set

G(t) = Ez N t for 0 < z < 1.
Taking into account that the increments are independence and homogeneous, we can represent the function

G(t + s) as G(t + s) = E z N t+s -N t z N t = G(t)G(s) .
Moreover, for all t > 0

G(t) ≥ Ez N t 1 {N t =0} = e λt > 0 .
Therefore, G(t) = e tg (z) and

g(z) = lim t→0 G(t) -1 t .
Then the rare events property directly implies that for t → 0 G(t) = P(N t = 0)+zP(N t = 1)+Ez N t 1 {N t ≥2} = e -λt +zλt+o(t) .

Therefore, g(z) = λ(t -1) and

G(t) = e -λt e λzt = ∞ ∑ n=0 z n ( e -λt (λt) n n!
) .

This directly implies that for all t > 0

P(N t = n) = (λt) n n! e -λt .
Next, note that we can represent the process (N t ) t≥0 as

N t = ∞ ∑ n=1 1 {T n ≤t} ,
where T n = inf{t ≥ 0 : N t ≥ n}. From here it follows that

P(T n > t) = P(N t ≤ n -1) = n-1 ∑ j=0 P(N t = j) = n-1 ∑ j=0 e -λt (λt) j j! = λ n ∫ ∞ t ( v n-1 e λ v ) dv .
This implies that the distribution of T n coincides with the distribution of a sum i.i.d. exponential random variables of the parameter λ > 0. Thus, in view of the denition (1.1), the random function

(N t ) t≥0 is a homogeneous Poisson process.

The last jump of the Poisson process

Let's study the properties of the delay between the present time moment t > 0 and the last jumping moment T Nt . Putting T 0 = 0, we get

T Nt = ∞ ∑ k=0 T k 1 {Nt=k} .
Therefore, T Nt is a random variable. We will study the properties of the two random variables V t = T Nt+1 -t and V * t = t -T Nt .

Proposition 1.5. The random variable V t is independent of the σeld, generated by the variables {N s , s ≤ t} and has the exponential distribution with the parameter λ > 0.

Proof. We note that for u > 0

{V t ≤ u} = {T Nt+1 -t ≤ u} = ∪ ∞ n=0 {T n+1 -t ≤ u , N t = n} = ∪ ∞ n=0 {N t+u ≥ n + 1 , N t = n} = ∪ ∞ n=0 {N t+u -N t ≥ 1 , N t = n} = {N t+u -N t ≥ 1 } .
This immediately implies Proposition 1.5.

Proposition 1.6. For any Borelian sets A ⊆ R and for any t > 0

P(V * t ∈ A) = e -λt 1 {t∈A} + λ ∫ A∩[0,t]
e -λv dv .

(1.6)

Proof. It is clear that for this proposition it suces to show (1.6) for the sets of the form A = [0, u[ with u > 0. We note that V * t ≤ t a.s., i.e. for u ≥ t the equation (1.6) is true. For u < t we get that

P(V * t ∈ A) = P(V * t < u) = ∞ ∑ n=0 P(t -T n < u , N t = n) = ∞ ∑ n=0 P(N t-u < n , N t = n) = ∞ ∑ n=0 P(N t -N t-u > 0 , N t = n) = P(N t -N t-u > 0) = P(N u > 0) .
Therefore,

P(V * t ∈ A) = 1 -e -λu = λ ∫ A∩[0,t]
e -λv dv .

To nish this proof we note that

P(V * t = t) = P(N t = 0) = e -λt .
Hence Proposition 1.6.

Propositions 1.5 and 1.6 imply that is twice as long on average as an interval (c) Show that for all A ∈ B(R n )

E (T Nt+1 -T Nt ) = 2 λ (1 -e -λt
[T n , T n+1 ] since E(T n+1 - T n ) = 1/λ.
P ((T 1 , • • • , T n ) ∈ A | N t = n) = n! t n ∫ A 1 {0<s 1 <s 2 <•••<sn≤t} ds 1 • • • ds n . (1.8) (d) Let X 1 , .
P ((Z 1 , • • • , Z n ) ∈ A) = n! t n ∫ A 1 {0<s 1 <s 2 <•••<sn≤t} ds 1 • • • ds n . (1.9)
Deduce that conditionally with respect to {N t = n} the random variables (T 1 , . . . , T n ) has same distribution as the order statistics of n uniform independent random variables on the interval [0, t].

(e) Show that for 0 < s < t

P (N s = k|N t ) =    N t k    ( s t ) k ( 1 - s t ) N t -k 1 {k>N t } .
(f) Show that (N t ) t≥0 is a process with homogeneous increments in the sense that for all 0 < s < t the increment N t -N s has the same distribution as N t-s .

(g) Show that (N t ) t≥0 has independent increments, i.e. for any increasing time moments

0 = t 0 < t 1 < • • • < t k the random variables N t 1 = N t 1 -N t 0 , N t 2 -N t 1 , . . . , N t k -N t k-1
are independent.

(h) Show that (N t ) t≥0 is a rare events process, i.e. for any t ≥ 0 and ∆ > 0

P(N t+∆ -N t = 1) = λ∆+o(∆) , P(N t+∆ -N t > 1) = o(∆)
as ∆ → 0.

Let (N

N t = ∞ ∑ j=1 1 {S j ≤t} , ( 2.1)
where S 0 = 0 and S j = ∑ j l=1 η l for j ≥ 1. Note that if the distribution G is exponential, then (N t ) t≥0 is the Poisson process. Using the large numbers law (Theorem A.1), one can establish that

lim t→∞ N t t = 1 Eη 1 a.s. (2.2)
Denition 2.1. We say that a random variable ξ is arithmetic if there exists d > 0 such that

P(ξ ∈ Γ d ) = 1 ,
where

Γ d = {(kd) -∞<k<∞ } is the grid of size d > 0. A random variable ξ is called non-arithmetic if P(ξ ∈ Γ d ) < 1 for any d > 0.
In this section we need the Blackwell Renewal Theorem (see, for example, in [3]):

Theorem 2.1. Assume that η 1 is non-arithmetic and

0 < E η 1 < ∞.
Then the expectation of the counting function has the following asymptotic properties:

lim t→∞ E N t t = 1 E η 1
and for any h > 0

lim t→∞ E (N t+h -N t ) = h E η 1 .
We will use this theorem to study the renewal function

Q(t) = E ∞ ∑ j=0 V (t -S j )1 {S j ≤t} , ( 2.3)
where V : R + → R is bounded over all the nite intervals function.

One can check directly that this function satises the following renewal equation

Q(u) = V (u) + ∫ u 0 Q(u -z) dG(z) . ( 2
sup 0≤u≤t |Q(u)| ≤ sup 0≤u≤t |V (u)| (E N t + 1) < ∞ on each nite interval [0, t].
Moreover, let B(R + ) be a linear space of R + → R bounded on each nite interval functions. We will introduce the following linear

B(R + ) → B(R + ) operator T (f )(u) = ∫ t 0 f (u -z) dG(z) .
In this case we can rewrite the renewal equation as

f = V + T (f ) .
This implies that for all n ≥ 1

f = n ∑ j=0 T j (V ) + T n+1 (f ) . (2.5)
To study this equation one needs to know how to calculate the n-th power of T . Let's show by induction that for each n ≥ 1

T n (f )(u) = E f (u -S n ) 1 {S n ≤u} . (2.6)
For n = 1 this is the denition. Assume now that this equality holds for some xed n > 1. We set

f (u) = T n (f )(u) = E f (u -S n ) 1 {S n ≤u} = ∫ +∞ 0 f (u -y)1 {y≤u} dF S n (y) ,
where F S n (y) = P(S n ≤ y). Using this function, we can represent the (n + 1)-th power as

T n+1 (f )(u) = T ( f )(u) = ∫ u 0 E f (u -z -S n ) 1 {S n ≤u-z} dG(z) = E f (u -η n+1 -S n ) 1 {S n ≤u-η n+1 } = E f (u -S n+1 ) 1 {S n+1 ≤u} .
It means that equality (2.6) is true for any n ≥ 1. Using it in (2.5), we get that

f (u) = n ∑ j=0 E V (u -S j ) 1 {S j ≤u} + E f (u -S n+1 ) 1 {S n+1 ≤u} . (2.7)
According to our condition, we try to solve the equation (2.4) among the functions which are bounded on each nite interval. So, the last term in (2.7) is bounded by

| E f (u -S n+1 ) 1 {S n+1 ≤u} | ≤ sup 0≤s≤u |f (s)| P(S n+1 ≤ u)
and, by the large numbers law (Theorem A.1), for any xed u > 0 this term tends to zero as n → ∞. So, taking the limit in (2.7) as n → ∞, we obtain that any solution of the equation (2.4) which is bounded on every nite interval is equal to the renewal function

(2.3).

Smith theorem

Now we study the asymptotic properties of the function (2.3).

To this end one needs the following denition.

Denition 2.2. We say that a

R + → R function V is directly integrable by Riemann on [0, ∞[ if ∞ ∑ k=1 sup k-1≤x≤k |V (x)| < ∞ . ( 2
lim u→∞ Q(u) = 1 E η 1 ∫ ∞ 0 V (z) dz . (2.9)
Proof. First, we show this theorem for linear combinations of indicator functions, i.e. we assume that

V (x) = α 1 1 [t 0 ,t 1 ] (x) + m ∑ k=2 α k 1 (t k-1 ,t k ] (x) , ( 2.10)
where

0 = t 0 < t 1 < . . . < t m < ∞. It's easy to see that this function for u ≥ t m Q(u) = α 1 E ∞ ∑ j=0 1 {u-t 1 ≤S j ≤u} + m ∑ k=2 α k E ∞ ∑ j=0 1 {u-t k ≤S j <u-t k-1 } = m ∑ k=1 α k E (N u-t k-1 -N u-t k ) -α 1 E ∆N u-t 1 - m ∑ k=2 α k E (∆N u-t k-1 -∆N u-t k ) ,
where

∆N t = ∑ ∞ j=1 1 {S j =t} . Note that for any h > 0 ∆N t ≤ N t+h -N t-h
and, by the Blackwell theorem,

lim sup t→∞ E ∆N t ≤ 2h Eη 1 .
Therefore,

lim t→∞ E ∆N t = 0 and lim u→∞ Q(u) = 1 E η 1 m ∑ k=1 α k (t k -t k-1 ) = 1 Eη 1 ∫ ∞ 0 V (z) dz .
Let now V be a function that satises the conditions of this theorem, i.e. it is directly integrable by Riemann and has a nite number of the jumps on all nite intervals. In this case, for each L > 0 we can nd a sequence of functions (V m ) m≥1 of the form (2.10) such that

lim m→∞ sup 0≤x≤L |V (x) -V m (x)| = 0 .
So, we can represent the function Q as

Q(u) = I 1 (u) + I 2 (u) + I 3 (u) , ( 2.11)
where

I 1 (u) = E ∑ ∞ j=0 V m (u -S j ) 1 {u-L≤S j ≤u} , I 2 (u) = E ∞ ∑ j=0 (V (u -S j ) -V m (u -S j )) 1 {u-L≤S j ≤u}
and

I 3 (u) = E ∞ ∑ j=0 V (u -S j ) 1 {S j ≤u-L} .
Taking into account that V m (z) = 0 for z > L, we nd

I 1 (u) = E ∞ ∑ j=0 V m (u -S j 1 S j ≤u} ,
and, therefore,

lim u→∞ I 1 (u) = 1 E η 1 ∫ ∞ 0 V m (z) dz = 1 E η 1 ∫ L 0 V m (z) dz . (2.12) Moreover, |I 2 (u)| ≤ sup 0≤z≤L |V (z) -V m (z)| ( E (N u -N u-L ) + E ∆ N u-L
) .

And we get that

lim sup u→∞ |I 2 (u)| ≤ sup 0≤z≤L |V (z) -V m (z)| L E η 1 .
This implies that for any L > 0

lim m→∞ lim sup u→∞ |I 2 (u)| = 0 . (2.13)
Now we consider the last term in (2.11). Setting

v * k = sup k-1≤x≤k |V (x)|,
we can estimate it from above as

|I 3 (u)| ≤ E ∞ ∑ j=0 ∞ ∑ k=L+1 |V (u -S j )|1 {u-k≤S j ≤u-k+1} ≤ E ∞ ∑ j=0 ∞ ∑ k=L+1 v * k 1 {u-k≤S j ≤u-k+1} ≤ ∞ ∑ k=L+1 v * k ( 1 + E(N (u-k) + +1 -N (u-k) + ) + E ∆N u-k) + ) ≤ sup x≥0 ( 1 + E(N x+1 -N x ) + E ∆N x ) ∞ ∑ k=L+1 v * k .
Thus,

lim L→∞ lim sup u→∞ |I 3 (u)| = 0 . (2.14)
From here, taking into account (2.11), we have

Q(u) - 1 Eη 1 ∫ ∞ 0 V (y)dy ≤ I 1 (u) - 1 Eη 1 ∫ L 0 V m (y)dy + 1 Eη 1 ∫ L 0 |V m (y) -V (y)|dy + 1 Eη 1 ∫ ∞ L |V (y)|dy + |I 2 (u)| + |I 3 (u)| .
Taking in this inequality the limit as

lim sup L→∞ lim sup m→∞ lim sup u→∞ ,
we get (2.11). Hence Theorem 2.3.

Exercises II

1. Let (N t ) t≥0 be counting function, that is

N t = ∑ n≥1 1 {η 1 +...+η n ≤t} ,
where (η j ) j≥1 are i.i.d. random variables uniformly distributed on the interval [0, z] with a xed z > 0. Calculate the following limits (a)

lim t→∞ EN t 1 + 2t ; (b) lim t→∞ E ( N 3t -N 3t+4 ) ; (c) lim t→∞ EN 2t √ 1 + t 2 ; (d) lim t→∞ E ( N t -N t-1/3
) ;

(e)

lim t→∞ sin(1/t) E N 10t ; (f) lim t→∞ ( 1 -e 1/t ) E N 4t .
2. Are the following functions directly integrable by Riemann

1 1 + x 2 , e -x , sin(x) 1 + x 4 ? 3. Calculate the limit lim t→∞   1 1 + t 2 + E ∞ ∑ j=1 1 1 + (t -T j ) 2 1 {T j ≤t}   ,
where T j = ∑ j i=1 ξ 2 i and (ξ j ) j≥1 are i.i.d. Gaussian random variables with the parameters (0, 1).

3 Cram er -Lundberg models

Main denitions and results

In this section we consider non-life insurance models in which the claim sizes are dened by i.i.d. positive random variables (Y j ) j≥1 and the inter-arrival times

with µ = E Y 1 < ∞ . ( 3
τ 1 = T 1 , τ k = T k -T k-1 , k ≥ 2 , ( 3.2)
are i.i.d. exponentially distributed random variables with Eτ 1 = 1/λ. We dene the total claim amount process as

X t = Nt ∑ j=1 Y j (3.3)
and X t = 0 for N t = 0. In the theory of stochastic processes such process is called a compound Poisson process. Moreover we assume that a continuous stream of revenue brings in c t during the time interval [0, t], where c > 0 is the premium income rate. In this case the risk process is dened as

U t = u + c t -X t , ( 3.4)
where u > 0 is the initial endowment of the insurance company.

Denition 3.1. The event

A -= { ∃t > 0 such that U t < 0} = ∪ t>0 { U t < 0} (3.5)
is called the ruin.

The denition of the risk process (3.4) immediately implies that

A -= ∪ k≥1 { U T k < 0} . (3.6)
This means that this set is measurable. The moment τ u when the risk process goes below zero is called the ruin time:

τ u = inf{ t > 0 : U t < 0} . (3.7)
The ruin probability or ruin function is given by

ψ(u) = P(A -| U 0 = u) = P( τ u < ∞) . ( 3.8)
Setting

σ u = inf{k ≥ 1 : U T k < 0} (3.9)
and taking into account the denition (3.8), we obtain

ψ(u) = P( σ u < ∞) . (3.10)
Firstly, we study the properties of the total claim amount process

(3.3).
Theorem 3.1. For the process (3.3) the following law of large num- bers holds

lim t→∞ 1 t X t = λ µ a.s. (3.11)
Moreover, if E Y 2 1 < ∞, then for the process (3.3) the limit theorem holds also, i.e.

X t -λ µt √ t =⇒ N (0, λ E Y 2 1 ) as t → ∞ . (3.12)
Proof. To show (3.11) we note that, in view of the denition of the Poisson process in (1.1), for any t > 0

T N t ≤ t < T N t +1 . (3.13)
Therefore, taking into account that N t → ∞ a.s. as t → ∞, we obtain through the large numbers law that

lim t→∞ T N t N t = Eτ 1 = 1 λ a.s.
Therefore, from the inequalities (3.13) it follows that

lim t→∞ N t t = λ a.s.
and, using again the large numbers law given in Theorem A.1, we come to the limit (3.11). As to the second equality, note that the deviation X t -λµt can be represented as

X t -λµt = S N t + λµ(T N t -t) , (3.14)
where

S n = n ∑ j=1 η j and η j = Y j -µ + µ(1 -λτ j ) .
Note that

E η j = 0 and E η 2 j = E Y 2 1 ,
and, in view of (3.13),

0 ≤ t -T N t ≤ τ N t +1 .
Moreover, we have

E τ N t +1 = ∞ ∑ k=0 E τ k+1 1 {N t =k} ≤ Eτ 1 + λ ∫ +∞ 0 z Υ(t, z) e -λz dz , (3.15)
where

Υ(t, z) = ∞ ∑ k=1 P(T k ≤ t < T k + z) = λ(t -(t -z) + ) ,
and (x) + = max(0, x). Therefore, the bound (3.15) yields

E τ N t +1 ≤ 1 λ + λ 2 ∫ t 0 z 2 e -λz dz + λ 2 t ∫ ∞ t z 2 e -λz dz i.e. sup t≥0 E τ N t +1 < ∞
and, therefore,

P -lim t→∞ T N t -t √ t = 0 .
Using this equality in (3.14), we obtain the asymptotic representa-tion

X t -λµt √ t = S N t √ t + o P (1) , (3.16)
where o P (1) is a term going to zero in probability as t → ∞. Moreover, let now m = [λt] and [x] be the integer part of the number x.

Then

E ( S N t -S m ) 2 = E S 2 N t -2E S N t S m + E S 2 m = E η 2 1 E|N t -m| , i.e. E ( S N t -S m ) 2 t ≤ 1 t + √ E(N t -E N t ) 2 t = 1 t + √ λ √ t .
Using this in (3.16), we get

X t -λµt √ t = S m √ t + o P (1) .
Now, applying to the sequence (S n ) n≥1 the central limit Theorem

A.4, we come to the limit property (3.12). Hence Theorem 3.1. for all u > 0.

Proof. Let c < λµ. We can represent the sequence (U T k ) k≥1 as

U T k = u - k ∑ j=1 ξ j , ( 3.17)
where ξ j = Y j -cτ j . In this case, by applying the strong large numbers law (Theorem A.1) for S k = ∑ k j=1 ξ j in the equality (3.10), we nd that

lim n→∞ U Tn n = -lim n→∞ S n n = -E ξ 1 = c λ -µ < 0 a.s.
So, taking into account (3.9) and (3.10), we obtain that ψ(u) = 1 for all u ≥ 0. Let now c = λµ, i.e. E ξ 1 = 0. In this case note that for any k ≥ 1 and ϵ > 0

P(|ξ k | > ϵ) = P(|ξ 1 | > ϵ) > 0 .
Using Kolmogorov three-series theorem and Kolmogorov zero-one law (Theorems A.2 -A.3), we obtain that

lim sup k→∞ S k = +∞ a.s.
From the equalities (3.9) and (3.10) it follows that

1 = P(lim sup k→∞ S k = +∞) ≤ P(σ u < ∞) .
Thus, ψ(u) = 1. Hence Proposition 3.1.

Remark 3.1.

E ξ 1 = E (Y 1 -c τ 1 ) = µ -c 1 λ < 0 . (3.18)
In the sequel we will assume that the premium rate is equal to

c = (1 + ρ) λ µ , (3.19)
where ρ is a positive constant, which provides the net prot condition.

Exercises III

Let (Y j ) j≥1 be i.i.d. random variables with values in R + and with the nite on a neighborhood around 0 generator function dened as

m Y (h) = E e hY j .
Let (N t ) t≥0 be a homogeneous Poisson process of an intensity λ > 0 independent of (Y j ) j≥1 . For any t ≥ 0 we set

X t = N t ∑ j=1 Y j and U t = u + ct -X t
with u > 0 and c > 0.

1. Calculate expectation and variance of U t .

2. Calculate the generator function for X t .

3. Let α > 0. Show that there is only one solution c α for the equation E e -α(ct-X t ) = 1 , for any t > 0.

4. Show that E U t > u for c = c α . What is the limit of c α as α → 0?

Lundberg inequality

In this section we will study the behavior of the function ψ(u)

under the condition (3.18). Moreover, we assume that the sequence of claims amounts (Y j ) j≥1 satises the following condition, called the Lundberg condition,

H 1 )
There exists δ > 0 such that Proof. First, we note that the function L is convex. Indeed, by Holder's inequality for 0 < α < 1 and for 0 ≤ x, y ≤ δ we obtain that

E e δY 1 < ∞ . ( 3
L(αx + (1 -α)y) = ln (E e αxξ 1 e (1-α)yξ 1 )
≤ ln

( (E e xξ 1 ) α (E e yξ 1 ) 1-α ) = ln (E e xξ 1 ) α + ln (E e yξ 1 ) 1-α = α L(x) + (1 -α) L(y) .
We assume that there is 0 < r 1 < r 2 such that L(r 1 ) = L(r 2 ) = 0.

Then for all z ∈ [r 1 , r 2 ] we obtain

L(z) = L(αr 1 + (1 -α)r 2 ) ≤ αL(r 1 ) + (1 -α)L(r 2 ) = 0 ,
where α = (r 2 -z)/(r 2 -r 1 ). If L(z) = 0 (i.e. E e zξ 1 = 1) for all r 1 ≤ z ≤ r 2 , then we would have E ξ 2 1 e zξ 1 = 0 and, so ξ 1 = Y 1 -cτ 1 = 0 a.s. But this is not possible since the random variables Y 1 and τ 1 are independent. Therefore, it exists 0 < r 1 < z 1 < r 2 such that L(z 1 ) < 0. Similar, as L(0) = 0, we get that it exists

z 0 ∈ [0, r 1 ] such that L(z 0 ) < 0. Setting α = (z 1 -r 1 )/(z 1 -z 0 ), we nd that 0 = L(r 1 ) = L(αz 0 + (1 -α)z 1 ) ≤ αL(z 0 ) + (1 -α)L(z 1 ) < 0 .
This implies the uniqueness of the positive root. Hence Proposition 3.2. Denition 3.3. If the equation L(x) = 0 admits a root r > 0, then this root is called the Lundberg coecient.

We will assume the following condition.

H 2 )
The equation L(x) = 0 admits a root r > 0.

Remark 3.2. It is easy to see that the assumptions H 1 )H 2 ) imply the net prot condition (3.18). Indeed, if E ξ 1 ≥ 0, then by Jensen inequality we obtain that Proof. First, one notes that according to (3.10), we can represent the ruin probability as the distribution tail of the extreme value for a sequence of sums of i.i.d. random variables:

ψ(u) = P(inf k≥1 U T k < 0) = P(max k≥1 S k > u) ,
where S k = ∑ k j=1 ξ j and ξ j = Y j -cτ j . Let now

ψ n (u) = P( max 1≤k≤n S k > u) .
It's obvious that

ψ(u) = lim n→∞ ψ n (u) .
So, for this theorem it suces to show the inequality (3.22) for the functions ψ n (u) for all n ≥ 1. We will do it by the induction. We start with n = 1. In this case S 1 = ξ 1 and by the Markov inequality

ψ 1 (u) = P(ξ 1 > u) ≤ E e rξ 1 e -ru = e -ru .
Moreover, if the inequality (3.22) holds for some xed n ≥ 1, then for n + 1 we get

ψ n+1 (u) = P( max 1≤k≤n+1 S k > u , ξ 1 > u) + P( max 1≤k≤n+1 S k > u , ξ 1 ≤ u) = P(ξ 1 > u) + P ( max 2≤k≤n+1 S k > u , ξ 1 ≤ u ) . (3.23)
We estimate now the rst term in (3.23) more precisely, i.e.

P(ξ

1 > u) ≤ e -ru E e rξ 1 1 {ξ 1 >u} . ( 3.24)
Taking into account that S k is the sum of i.i.d. random variables and using the inequality (3.22) for ψ n (•), we can estimate the second term in (3.23) as

P ( max 2≤k≤n+1 S k > u , ξ 1 ≤ u ) = P   max 2≤k≤n+1 n ∑ j=1 ξ j+1 > u -ξ 1 , ξ 1 ≤ u   = E 1 {ξ 1 ≤u} ψ n (u -ξ 1 ) ≤ e -ru E 1 {ξ 1 ≤u} e rξ 1 .
Using this inequality and the upper bound (3.24) in (3.23), we ob-tain that

ψ n+1 (u) ≤ e -ru (E e rξ 1 1 {ξ 1 >u} + E e rξ 1 1 {ξ 1 ≤u} ) = e -ru E e rξ 1 = e -ru .
So, for all n ≥ 1 the functions ψ n (u) ≤ e -ru . Taking here the limit as n → ∞, we get the bound (3.22). Hence Theorem 3.2.

Example 3.1. We consider the Cram er-Lundberg model in which the random variables (Y j ) j≥1 are exponential with a parameter γ > 0. In this case the net prot condition (3.19) takes the form

c = (1 + ρ) λ /γ ,
where ρ is a positive constant. Note, that the condition H 1 ) holds for δ < γ. Moreover, it is easy to see that the Lundberg coecient in this case is

r = γ - λ c = γ ρ 1 + ρ .
So, in view of the Lundberg inequality, we get for all u ≥ 0

ψ(u) ≤ e -γ ρ 1+ρ u .
(3.25)

Exercises IV

We consider the risk process U t = u + ct -X t for a reinsurance company, where X t = ∑ N t i=j (Y j -K) + with K > 0, (N t ) t≥0 is a homogeneous Poisson process of intensity λ > 0 independent of the i.i.d. sequence (Y j ) j≥1 random exponential variables of parameter γ > 0. We choose the premium rate as

c = (1 + ρ)λE (Y 1 -K) + with ρ > 0 . 1. Calculate c.

Show that

E e it(Y 1 -K) + = 1 + it γ -it e -Kγ , t ∈ R.
3. Show that X t has the same distribution as Xt = ∑ Ñt i=1 Y j , where ( Ñt ) t≥0 is a homogeneous Poisson process of the intensity λ = λe -Kγ independent of (Y j ) j≥1 .

Fundamental equation for the non-ruin probability

Denote by ϕ(u) = 1 -ψ(u) the non-ruin probability. Theorem 3.3. We assume that the Cram er-Lundberg model satises the net prot condition (3.18) and the distribution function (•) of the random amounts (Y j ) has a density f Y . Then the nonruin probability ϕ(u) satises the following integral equation

F Y
ϕ(u) = ρ 1 + ρ + 1 1 + ρ ∫ u 0 ϕ(u -y) dF Y,I (y) , ( 3.26)
where

F Y,I (y) = 1 µ ∫ y 0 F Y (z) dz and F Y (y) = 1 -F Y (y) = P(Y 1 > y) .
Proof. Taking into account that S n = ∑ n j=1 ξ j and (ξ j ) j≥1 are i.i.d. random variables, one has

ϕ(u) = P(sup n≥1 S n ≤ u) = P(ξ 1 ≤ u , sup n≥2 S n ≤ u) = P(ξ 1 ≤ u, sup n≥2 n ∑ j=2 ξ j ≤ u -ξ 1 ) = E1 {ξ 1 ≤u} ϕ(u -ξ 1 ) = E 1 {Y 1 -cτ 1 ≤u} ϕ(u -Y 1 + cτ 1 ) , i.e. ϕ(u) = λ ∫ ∞ 0 ∫ u+cv 0 ϕ(u -y + cv) dF Y (y) e -λv dv = λ c e uλ/c ∫ ∞ u e -λz/c ∫ z 0 ϕ(z -y) dF Y (y) dz .
Taking the derivatives in this equality, we nd that

ϕ ′ (u) = λ c ϕ(u) - λ c ∫ u 0 ϕ(u -y) dF Y (y)
and, therefore,

ϕ(t)-ϕ(0) = λ c ∫ t 0 ϕ(u) du- λ c ∫ t 0 ∫ u 0 ϕ(u-y) dF Y (y) du . (3.27)
Moreover, the integration by parts yields

∫ t 0 ∫ u 0 ϕ(u -y) dF Y (y) du = ∫ t 0 ( ϕ(0) F Y (u) + ∫ u 0 F Y (y) ϕ ′ (u -y) dy ) du = ϕ(0) ∫ t 0 F Y (u)du + ∫ t 0 F Y (y) ( ∫ t y ϕ ′ (u -y)du ) dy = ∫ t 0 F Y (y)ϕ(t -y)dy .
Using now the condition (3.19), we obtain from (3.27) that

ϕ(t) -ϕ(0) = 1 (1 + ρ)µ ∫ t 0 ϕ(t -y) F Y (y) dy = 1 1 + ρ ∫ t 0 ϕ(t -y) dF Y,I (y) .
(3.28)

It should be noted now that ϕ(∞) = 1. Therefore, the passing here to the limit as t → ∞ yields

ϕ(0) = ρ 1 + ρ
and we obtain from (3.28) the equality (3.26). Hence Theorem 3.3.

Note that (3.26) immediately implies the equation for the ruin prob-

ability ψ(u) = 1 -ϕ(u): ψ(u) = F Y,I (u) 1 + ρ + 1 1 + ρ ∫ u 0 ψ(u -y) dF Y,I (y) , (3.29)
where

F Y,I (y) = 1 -F Y,I (y).
Example 3.2. In the case, when distribution of (Y j ) j≥1 is exponen- tial, as in the example 3.1, i.e. F Y (y) = 1 -e -γy , this equation has the following form

ψ(u) = e -γu 1 + ρ + γ 1 + ρ ∫ u 0 ψ(u -y) e -γy dy . (3.30)
We can resolve this equation directly and get that the solution is

ψ(u) = 1 1 + ρ e -γ ρ 1+ρ u . ( 3

Exercises V

Let (Y j ) j≥1 be i.i.d. random variables with values in N and N a random variable with values in N independent of (Y j ) j whose distribution is of the form

q n := P(N = n) = ( a + b n ) q n-1 , n = 1, 2, . . . ,
where q 0 = P(N = 0), for a < 1 and b ∈ R are xed constants.

Moreover let

X = N ∑ j=1
Y j and p k := P(X = k) .

1. Show that the Poisson and binomial distributions verify the previous hypotheses on N .

Let S

n = ∑ n j=1 Y j . Show that for i ≥ 1 E ( Y 1 S i S i ) = 1 i .

Show that

E ( a + b Y 1 n S i = n ) = n ∑ k=0 ( a + k n ) P(Y 1 = k)P(S i-1 = n -k) P(S i = n) .

Show that

p 0 =      q 0 , if P(Y 1 = 0) = 0 ; E (P(Y 1 = 0)) N , else .
5. Show that for n ≥ 1

p n = ∞ ∑ i=1 P(S i = n)q i .
6. Show that the probabilities p k can be calculated recursively (Panjer's algorithm):

p k = 1 1 -aP(Y 1 = 0) k ∑ i=1 ( a + bi k ) P(Y 1 = i)p k-i , k ≥ 1.

Cram er bound

In this section we will study the limit of ψ(u) when u → ∞ for small claims, i.e. for claims that verify the condition H 1 ).

Theorem 3.4. We assume that the conditions H 1 )H 2 ) hold with 0 < r < δ and the random variable Y 1 has a density f Y . Then

lim u→∞ e ru ψ(u) = ψ * > 0 , (3.32)
where

ψ * = ρ µ r ∫ ∞ Therefore, G(+∞) = - q rµ + q rµ E e rY 1 .
Note here that the denition of r and (

V V (u) = q µ ∫ ∞ u e rz P(Y 1 > z) dz ≤ q δµ E e δY 1 e -(δ-r)u ,
i.e. the function V satises the Riemann direct integrability condition. This means that we can apply Smith theorem to the function

Q, i.e. lim u→∞ Q(u) = 1 Eη 1 ∫ ∞ 0 V (z) dz ,
where

Eη 1 = q µ ∫ ∞ 0 z e rz P(Y 1 > z) dz and ∫ ∞ 0 V (z) dz = q ∫ ∞ 0 e rz F Y,I (z)dz = q r e rz F Y,I (z) | ∞ 0 + q µr ∫ ∞ 0 e rz P(Y 1 > z) dz = - q r + 1 r G(+∞) = 1 -q r .
This directly implies (3.32). Hence Theorem 3.32.

Exercises VI

1. We consider a Cram er-Lundberg model with the risk process

U t = u + ct -X t ,
where the total claim amount process

X t = ∑ N t j=1 Y j .
(a) Show that the random variables X t -X s and X s are independent for 0 < s < t.

(b) Show that the random variables X t -X s and X t-s have the same distribution for 0 < s < t.

(c) Calculate

E(e -hU t |X s ) .

(d) Assuming that the Lundberg coecient r > 0 exists,

show that E(e -rU t |X s ) = e -rU s .

(e) Show that Ee -rU t independent of t.

2. Assume that in a Cram er-Lundberg model the distribution of the claim amounts Y j is given by the density Find this function for n = 1.

f n (x) = α n 1 Γ(n) x n-1 e -αx for x > 0 (α > 0 , n ≥ 1). (a)

Large claims

In this section we study the problem of ruin for the claims (Y j ) j≥1 which do not hold the condition H 1 ), i.e., E e δY 1 = +∞ for all δ > 0. We replace the condition H 1 ) by a weaker condition, i.e. we assume that the distribution of (Y j ) j≥1 is subexponential.

Denition 3.4. We say that a random variable Y is subexponential if i.i.d. random variables (Y j ) j≥1 having the same distribution as Y for all n ≥ 1 satisfy the following condition

lim z→∞ P( ∑ n j=1 Y j > z) P(Y 1 > z) = n . (3.36)
Example 3.3. Let Y a positive random variable such that for any

z ≥ 0 P(Y > z) = 1 (1 + z) α and α > 0 .
Let's show, by the induction, that Y satises the condition (3.36).

Assuming that the property (3.36) holds for n -1, we will check this condition for n. To this end we set

F n (z) = P(S n > z) ,
where S n = ∑ n j=1 Y j . We have

F n (z) = P( n-1 ∑ j=1 Y j > z -Y n ) = E F n-1 (z -Y n ) 1 {Yn≤z} + P(Y n > z) = ∫ z 0 F n-1 (z -t) dF (t) + F (z) ,
where F is the distribution function of Y and F (z) = F 1 (z). So, we obtain that

F n (z) F (z) = 1 + ∫ z 0 F n-1 (z -t) F (z) dF (t) . (3.37)
Then we can represent the last term in this equality as

∫ rz 0 F n-1 (z -t) F (z) dF (t) + ∫ z rz F n-1 (z -t) F (z) dF (t) = I 1,r (z) + I 2,r (z) ,
where 0 < r < 1. Note that for the function

F (z) = 1 -(1 + z) -α
for all t ≥ 0 we have

lim z→∞ F (z -t) F (z) = 1 .
Thus, in view of the induction hypothesis for all t > 0 lim z→∞

F n-1 (z -t) F (z) = n -1 .
Moreover, for t ≤ rz with 0 < r < 1 we obtain the following upper bound

lim sup z→∞ F n-1 (z -t) F (z) ≤ lim sup z→∞ F n-1 ((1 -r)z) F (z) ≤ n -1 (1 -r) α .
Therefore, by the dominated convergence theorem,

lim z→∞ I 1,r (z) = n -1
for all 0 < r < 1. As to the function I 2,r (z), we obtain that for any

z > 0 I 2,r (z) ≤ 1 F (z) (F (z) -F (rz)) ≤ ( 1 (1 + rz) α - 1 (1 + z) α ) (1 + z) α .
This means that for any

0 < r < 1 lim sup z→∞ I 2,r (z) ≤ r -α -1
and, passing here to the limit as r → 1, we nd

lim sup r→1 lim sup z→∞ I 2,r (z) = 0 .
Therefore, the equality (3.37) implies directly (3.36).

Proposition 3.3. Let Y be a subexponential random variable. Then 57 for any ε > 0 it exists K = K(ε) > 0 such that for any n ≥ 1

sup z≥0 F n (z) F (z) ≤ K (1 + ε) n , ( 3.38)
where

F n (z) = P(S n > z), F (z) = F 1 (z), S n = ∑ n j=1 Y j and (Y j ) j≥1 are i.i.d. random variables of the same distribution as Y .
Proof. We set

α n = sup z≥0 F n (z) F (z) ,
then we get

F n+1 (z) F (z) = 1 + ∫ z 0 P(S n > z -t) dF (t) F (z) ≤ 1 + α n P(Y 1 + Y 2 > z , Y 2 ≤ z) F (z) = 1 + α n ( P(S 2 > z) F (z) -1
) .

Note here that for any ε > 0 there exists

T = T (ε) > 0 such that sup z≥T ( P(S 2 > z) F (z) -1 ) ≤ (1 + ε) .
Therefore,

α n+1 ≤ sup 0≤z≤T F n (z) F (z) + sup z≥T F n (z) F (z) ≤ α n (1 + ε) + K 0 ,
where

K 0 = 1 + 1 P(Y > T )
.

This inequality means that for n ≥ 1

α n+1 = (1 + ε) α n + β n+1 (3.39)
with α 1 = 1 and β n+1 = α n+1 -(1 + ε)α n ≤ K 0 . We can resolve this equation and nd that

α n = (1 + ε) n-1 α 1 + n ∑ j=2 (1 + ε) j β n-j ≤ (1 + ε) n-1 + K 0 n ∑ j=2 (1 + ε) j ≤ K (1 + ε) n ,
where K = 1 + K 0 /ε. From here we obtain (3.38).

In Section 3. Proof. Let us denote the right part in equality (3.40) by g, i.e.

g(u) = p ∞ ∑ j=0 q j P( Sj ≤ u) .
It is clear that this function is bounded, i.e.

g(u) ≤ p ∞ ∑ j=0 q j = p 1 -q = 1 .
Moreover, one can see that this function satises the equation (3.26).

Indeed,

g(u) = p + pq P( Ỹ1 ≤ u) + p ∞ ∑ j=2 q j P ( j ∑ l=2 Ỹl ≤ u -Ỹ1 ) = p + pq P( Ỹ1 ≤ u) + p ∞ ∑ j=2 q j ∫ u 0 P ( j ∑ l=2 Ỹl ≤ u -t ) dF Y,I (t) = p + q ∫ u 0 g(u -t) dF Y,I (t) .
We show now that g(u) = ϕ(u). To this end we set δ(u) = g(u) -

M u = |δ(t 0 )| = q | ∫ u 0 δ(u 0 -t) dF Y,I (t)| ≤ q ∫ u 0 |δ(u 0 -t)| dF Y,I (t) ≤ q M u P( Ỹ1 ≤ u 0 ) ≤ qM u .
Taking into account that q < 1, we get that M u = 0 for all u > 0.

Therefore, ϕ(u) = g(u) for any u ≥ 0. Proof. The equality (3.41) implies directly that

ψ(u) = 1 -ϕ(u) = p ∞ ∑ j=1 q j P( Sj ≥ u) .
Thus, taking into account that

p ∞ ∑ j=1 j q j = ρ -1 ,
we have

∆(u) = ψ(u) F Y,I (u) -ρ -1 = p ∞ ∑ j=1 q j σ j (u) ,
where

σ j (u) = P( Sj ≥ u) F Y,I (u) -j .
Now we xe ε > 0 such that θ = q(1 + ε) < 1. Then, by Proposition 3.3, we obtain that there is a positive constant K such that for any j ≥ 1 and for all u > 0 q j |σ j (u)| ≤ K θ j + q j j .

Moreover, for any j ≥ 1 lim u→∞ σ j (u) = 0 .

Then the dominated convergence theorem directly implies that

lim u→∞ ∆(u) = p ∞ ∑ j=1 q j lim u→∞ σ j (u) = 0 .
Hence Theorem 3.5.

F (z) = P(Y ≤ z) = 1 - 1 (1 + z) 1+α
for z ≥ 0. In this case µ = 1/α and, therefore, for all z ≥ 0

F Y,I (z) = 1 - 1 (1 + z) α .
We have seen already that this distribution function is subexponential. Therefore, in this case, in view of Theorem 3.5, we obtain that

u α ψ(u) → ρ -1 as u → ∞.

Exercises VII

Let F be some distribution function for the claim Y > 0. We denote F (x) = 1 -F (x). We say that F is light tailed if there exist a and b > 0 such that F (x) ≤ ae -bx for all x (or that F is heavy-tailed). Let x l = inf{x|F (x) > 0} and x r = sup{x|F (x) < 1} .

We set e F (u) = E(Y -u|Y > u) for u ∈ (x l , x r ) .

1. Show that e F (u) can be written as

e F (u) = 1 F (u) ∫ +∞ u F (x)dx .
2. Show that if F (x) > 0 for all x > 0 and that F is continuous, then we have for any x > 0

F (x) = e F (0) e F (x) exp { - ∫ x 0 1 e F (y) dy } .
3. Show that if lim u→+∞ e F (u) = +∞, then F is heavy tailed. 4. Show that if F is heavy tailed, then the generator function of Y is innite for any z > 0.

5. Calculate e F (u) when Y is an exponential random variable of a parameter λ > 0.

6. Calculate e F (u) when Y has the Pareto distribution

F (x) = ( κ κ + x ) α
of parameters κ > 0 and α > 1.

7. Show that if Y has the Gamma distribution of order m ≥ 1 and a parameter γ > 0, i.e.

F (x) = γ m x m-1 m! e -γx ,
then F is light-tailed.

8. Show that if Y has the Weibull distribution with parameters c > 0 and τ > 0, i.e.

F (x) = exp{-cx τ } ,
then F is light or heavy tailed depending on the value of τ .

Ruin problem with investment

In this section we consider an insurance company that invests its capital in a Black-Scholes market with the two assets B = (B t ) t≥0

and S = (S t ) t≥0 dened as

     dB t = r B t dt , B 0 = 1 ; dS t = aS t dt + σS t dw t , S 0 > 0 , ( 3.42)
where (w t ) t≥0 is a Brownian motion, r, a and σ are non-negative constants. Let (F t ) t≥0 be ltration on this model dened as

F t = σ{w s , X s , s ≤ t} , ( 3.43)
where (X t ) t≥0 is the total claim amount process dened in (3.3).

We assume that in each time moment t ≥ 0 the insurance company has β t of the assets (B) and γ t of the assets (S). So, the wealth (the risk process) at the instant t > 0 is equal to (3.45) where u > 0 is an initial endowment and Z t = ct -X t .

U t = β t B t + γ t S t . ( 3
U t = β t B t + γ t S t = u + ∫ t 0 β s dB s + ∫ t 0 γ v dS v + Z t ,
Proposition 3.5. Let u > 0 and γ = (γ t ) t≥0 be a square integrated process, i.e. for all t > 0

∫ t 0 γ 2 v dv < ∞ a.s.
We set We denote by ς = (ς t ) t≥0 proportional strategy, i.e.

β t = u + ∫ t 0 γ v d S v -γ t S t + Z t , ( 3
ς t = γ t S t β t B t + γ t S t = γ t S t β t + γ t S t = γ t S t u + ∫ t 0 γ v d S v + Z t .
For this strategy we can rewrite the equation (3.45) as

U t = u + ∫ t 0 (r + (a -r)ς s ) U s ds + σ ∫ t 0 ς s U s dw s + ct -X t .
In this section we assume that

ς t ≡ δ , ( 3.47)
where δ ≥ 0 is a xed nonrandom constant. In view of

d S t = (a -r) S t dt + σ S t dw t ,
we obtain for V t = γ t S t the following stochastic dierential equation

dV t = δ(a -r)V t dt + δσV t dw t + δd Z t , V 0 = δu .
Through Ito formula we can represent the process V t in the following form

V t = e ξ t ( δ u + δ c ∫ t 0 e -ξ s B -1 s dZ s ) ,
where ξ t = a 1 t + σ δ w t , a 1 = δ(a -r) -σ 2 δ /2 and σ δ = δσ. So, to get the property (3.47) we set

γ t = e ξ t ( δ u + δ c ∫ t 0 e -ξ s B -1 s dZ s ) S t , 0 ≤ t ≤ T .
For this strategy the risk process can be written as

U t = u + a δ ∫ t 0 U s ds + σ δ ∫ t 0 U s dw s + Z t , (3.48)
where u > 0 is the initial capital, a δ = r + δ(a -r) and σ δ = δσ.

The ruin probability is

ψ(u) = P(inf t≥0 U t < 0) . (3.49)
We start to study this function for any u ≥ 0.

Proposition 3.6. The function ϕ(u) = 1 -ψ(u) satises for all u ≥ 0 the following dierential equation Proof. We denote by (η t ) t≥0 the solution of the following stochastic equation dη t = (a δ η t + c) dt + σ δ η t dw t , η 0 = u .

By Ito formula, we can resolve it, i.e. where a = a δ -σ 2 δ /2. Now we xe some h > 0 and then, in view of the denition of ϕ, we can represent this function as

η
ϕ(u) = E ( 1 {inf t≥0 U t ≥0} ) = E ( 1 {inf 0≤t≤h U t ≥0} E ( 1 {inf t≥h U t ≥0} | F h )) .
It should be noted that the process (3.48) is Markovian, i.e.

E

(

1 {inf t≥h U t ≥0} | F h ) = ϕ(U h )
and, therefore,

ϕ(u) = E ( 1 {inf 0≤t≤h U t ≥0} ϕ(U h )
) .

Moreover, we can represent this function in the form

ϕ(u) = A 1 (u) + A 2 (u) + A 3 (u) (3.53)
where

A 1 (u) = E ( 1 {N h =0} ϕ(η h ) ) , A 2 (u) = E ( 1 {inf 0≤t≤h U t ≥0} 1 {N h =1} ϕ(U h )
)

and

A 3 (u) = E ( 1 {inf 0≤t≤h U t ≥0} 1 {N h ≥2} ϕ(U h )
) .

We rewrite the equation (3.53) as

A 1 (u) -ϕ(u) h + A 2 (u) h + A 3 (u) h = 0 . (3.54)
It is clear that A 1 (u) -ϕ(u) = e -λh E (ϕ(η h ) -ϕ(u)) .

In addition, it is well known (see, for example, in [START_REF] Paulsen | Stochastic Calculus with Applications to Risk Theory[END_REF]) that the function ϕ is two times continuously dierentiable. So, by the Ito formula A 1 (u) -ϕ(u) = (e -λh -1) ϕ(u)

+ ∫ h 0 E ( (a δ η v + c) ϕ ′ (η v ) + σ 2 δ η 2 v 2 ϕ ′′ (η v )
) dv and, therefore, Then we can represent the term A 2 (u) as

A 2 (u) = E ( 1 {η T 1 ≥Y 1 } 1 {N h =1} ϕ(U h )
) .

Note here that on the set {N h = 1} Finally, for the last term in (3.54) it's easy to see that

U h = ( η T 1 -Y 1 ) e ζ h -ζ T 1 + c ∫ h T 1 e ζ h -
A 3 (u) h ≤ P (N h ≥ 2) h → 0 , as h → 0 .
So, taking the limit in (3.54) as h → 0 we get the equation (3.50).

Hence Proposition 3.6.

To study the asymptotic properties of the ruin probability (3.49) sary that for any c > 0 the following series are convergent

∑ n≥1 E ξ c j , ∑ n≥1 E (ξ c j -E ξ c j ) 2 , ∑ n≥1 P ( |ξ j | > c ) ,
and suciently that these series are convergent for some xed c > 0,

where ξ c = ξ1 {|ξ j |≤c} .

A.4 Central limit theorem

First, we recall the weak convergence for random variables.

Denition A. Theorem A.4. Let (ξ j ) j≥1 be i.i.d. random variables with Eξ 1 = 0 and Eξ 2 1 = σ 2 . Then

∑ n j=1 ξ j √ n =⇒ ξ as n → ∞ ,
where ξ is a Gaussian random variable with the parameters (0, σ 2 ).

The publication was prepared in the author's edition Printed in the area of digital printing Publishing House of Tomsk State University Order number from " " in February 2020, 50 copies.

  (a) Calculate EN t and Var(N t ) for t > 0. (b) Calculate the distribution of T n for n ≥ 1.

  Now we come back to the ruin problem, i.e. we study the properties for the ruin probability (3.10). Proposition 3.1. (Almost sure ruin) If c ≤ µλ, then ψ(u) = 1

  .20) Also we dene the Lundberg function asL(x) = ln E e xξ 1 .(3.21)The condition H 1 ) implies that the function L(x) is nite in absolute value for any 0 ≤ x ≤ δ.Proposition 3.2. We assume that the condition H 1 ) holds. If the equation L(x) = 0 has a strictly positive root, then this root is unique.

LH 2 )

 2 (x) = ln E e xξ 1 > ln e xEξ 1 ≥ 0 for any x > 0. So, the function L has no strictly positive root. Theorem 3.2. (Lundberg inequality) Under the conditions H 1 ) for all u ≥ 0 the ruin function admits the exponential upper bound ψ(u) ≤ e -ru . (3.22)

  .31) Remark 3.3. Note that, if we compare the form (3.31) with the upper bound (3.25), then one can see that the Lundberg inequality gives sharp upper bound for the coecient (1 + ρ) -1 .

  Calculate the generator function Ee hY 1 . For which values h > 0 this function is well dened? Calculate EY 1 . (b) Find the net prot condition for this model. (c) Calculate the Lundberg coecient for n = 1 and n = 2. (d) Write the integral equation for the ruin function ψ n (u).

Proposition 3 . 4 .

 34 5 we have shown that the probability of non-ruin ϕ(u) satises the equation (3.26). Now one needs to resolve this equation. We assume that in the Cram er-Lundberg model the distribution function F Y (•) for the random amounts (Y j ) j≥1 has a density f Y and µ = E Y 1 < ∞. Moreover, we assume that this model satises the net prot condition (3.18), i.e. c = λµ(1+ρ) with ρ > 0. Then the solution of the equation (3.26) has the following form ϕ(u) = p ∞ ∑ j=0 q j P( Sj ≤ u) , (3.40)where p = ρ/(1 + ρ), q = 1/(1 + ρ), S0 = 0, Sj = ∑ j i=1 Ỹi and ( Ỹ ) j≥1 are i.i.d. random variables with the distribution function

F

  Y,I(•) dened in(3.26).

  ϕ(u). We have already seen that |δ(u)| ≤ 2 for all u ≥ 0. Let now u be a xed positive number. Denote by M u = sup 0≤t≤u |δ(t)|. Then there exists 0 ≤ t 0 ≤ u such that M u = |δ(u 0 )| because the function δ(•) is continuous on the interval [0, u]. Therefore,

Theorem 3 . 5 .

 35 We assume that all the conditions in Proposition 3.4 hold and the distribution function F Y,I (•) is subexponential. Thenlim u→∞ ψ(u) F Y,I (u) = ρ -1 .(3.41)

Example 3 . 4 .

 34 We consider the Cram er-Lundberg model in which the positive random variables (Y j ) j≥1 are distributed according to the Pareto distribution function F (•) dened as

  )(a δ u + c) -λϕ(u) + λ ∫ u 0 ϕ(u -y)dF (y) = 0 (3.50) with the boundary conditions cϕ ′ (0) = λϕ(0) and ϕ(+∞) = 1 . (3.51)

lim h→0 A 1 σ 2 δ u 2 2 ϕ

 h→012 (u) -ϕ(u) h = (a δ u + c) ϕ ′ (u) + ′′ (u) -λϕ(u) .

  Remark 1.2. The equation(1.7) is called the "bus paradox" . If we associate the jump moments of a Poisson process with the time moments of passages of a bus through a station, then according to(1.7) for suciently large t the bus waiting time interval [T Nt , T Nt+1 ]

	) .	(1.7)

  Proposition 3.1 means that insurance companies have to choose the premium rate c > 0 such that Eξ 1 < 0. This is the only possibility to avoid being bankrupt almost sure in the framework of the Cram er -Lundberg model. So, if Eξ 1 < 0, then we can hope that the ruin function ψ(u) will be less then 1.

Denition 3.2. The Cram er-Lundberg model satises "net prot condition" if

  .44)We denote by π t = (β t , γ t ) and assume that the process π = (π t ) t≥0 is adapted to the ltration (F t ) t≥0 . In this case π is said a nancial strategy. Denition 3.5. Financial strategy π = (π t ) t≥0 with π t = (β t , γ t ) is said to be admissible if for any t ≥ 0

	∫ t	(	|β s | + γ 2
	0

s

)

ds < ∞ a.s. and, for anyy t ≥ 0,

  .46) where S t = S t /B t and Z t = ∫ t 0 B -1 v dZ v . Then the nancial strategy (β t , γ t ) t≥0 is admissible. Proof. First of all note that the process (3.46) is integrable, i.e. Taking into account here that dZ t = B t d Z t and dS t = d S t B t = S t dB t + B t d S t , we get the equality (3.45).

	for any t > 0	∫ t
		|β v | dv < ∞ a.s.
		0
	The denition (3.46) implies that the discounted wealth process
		U t =	U t B

t = β t + γ t S t admits the following stochastic dierential:

d U t = γ t d S t + d Z t .

Therefore, Ito formula implies that

dU t = dB t U t = B t d U t + U t dB t = β t dB t + γ t ( S t dB t + B t d S t ) + B t d Z t .

  ζ s ds , where the process (ζ t ) t≥0 is given in(3.52). So, on the set{ N h = 1} ∩ {Y 1 ≤ η T 1 u|e x -1| + c h e 2x ) |e 2x -1| + 2c h e 2x + u|e x -1| .Taking into account that the process (ζ t ) t≥0 is continuous, we obtainIn view of the independence of the processes (ζ t ) t≥0 and (N t ) t≥0 ,

	} h -1 + c h e 2ζ * h ) e 2ζ * |U h -u + Y 1 | ≤ (u + η * h + η * h , η * h = sup h (x) = we get where where B * ( lim h→0 ζ * h = 0 a.s. Moreover, using the properties of a Brownian motion, one can check (3.55) directly that for all γ > 0 and 0 < t < ∞ E e γζ * t < ∞ . So, by the dominated convergence theorem lim h→0 EB * h (ζ * h ) = 0 . we obtain that u + that lim h→0 ∫ u A 2 (u) 0 h = λ ϕ(u -y) dF (y) . (3.56)

0≤s≤h |η s -u| and ζ * h = sup 0≤s≤h |ζ s | .

From the denition of (η t ) t≥0 it follows that

η * h ≤ u|e ζ * h -1| + c h e 2ζ * h .

Therefore, on the set (3.55)

|U h -u + Y 1 | ≤ B * h (ζ * h ) ,

  1. The sequence of random variable is called conver- gent weakly to a random variable ξ, i.e. ξ n =⇒ ξ as n → ∞, if for any bounded continuous R → R function g lim

n→∞ E g(ξ n ) = E g(ξ) .

z e rz P(Y

> z) dz and the parameter ρ > 0 is given in the net prot condition(3.19).

The notes are intended for students of the Mathematics This work was supported by the Ministry of Education and Science of the Russian Federation (project No 1.472.2016/1.4).

Proof. First, note that Remark 3.2 implies the condition (3.19).

Moreover, by the Theorem 3.3, we can write the equation for ψ ψ(u) = q F Y,I (u) + q ∫ u 0 ψ(u -y) dF Y,I (y) , (3.33) where q = (1 + ρ) -1 . From here we directly get the equation for the function Q(u) = e ru ψ(u), i.e.

where

Now let us to show that G is a distribution function, i.e. G(+∞) = 1. Indeed, the integrating by parts yields

Taking into account that 0 < r < δ, we obtain

The following theorem gives us the asymptotic behavior of the ruin probability in the depending of this parameter (see, for example, [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF] and [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments // Stoch[END_REF]).

Theorem 3.6. For the proportional strategy (3.47) the ruin probability (3.49) has the following asymptotic (as

Now one needs to choose the proportional investment coecient δ > 0 in the strategy (3.47) such that the power parameter ν dened in (3.57) will be positive, i.e. 0 < δ < δ * , where

Therefore, if there exists 0 < δ < δ * for which E Y

A Appendix

In this section we announce main limit results of the probability theory which can be found, for example, in [10].

A. Then for any A ∈ X the probability P(A) = 0 or P(A) = 1.

A.3 Three series theorem

Theorem A.


Amélie Slembrouck-Brec

Amélie Rodrigues

Oriane Rabesandratana

Giuliana Gagliardi

Céline Nanteau

Stéphane Fouquet

Gilles Thuret

Sacha Reichman

Gael Orieux

Olivier Goureau
email: olivier.goureau@inserm.fr

Reprogramming of adult retinal Müller glial cells into human induced pluripotent stem cells as an efficient source of retinal cells

The reprogramming of human somatic cells to induced pluripotent stem cells (iPSCs) has broad applications in regenerative medicine. The generation of self-organized retinal structures from these iPSCs offers the opportunity to study retinal development and model specific retinal disease with patient-specific iPSCs, and provides the basis for cell replacement strategies. In this study we demonstrated that the major type of glial cells of the human retina, Müller cells, can be reprogrammed into iPSCs that acquire classical signature of pluripotent stem cells. These Müller glial cell-derived iPSCs were able to differentiate toward retinal fate and generate concomitantly retinal pigmented epithelial cells and selfforming retinal organoids structures containing retinal progenitor cells. Retinal organoids recapitulated retinal neurogenesis with differentiation of retinal progenitor cells into all retinal cell types in a sequential overlapping order. With a modified retinal maturation protocol characterized by the presence of serum and high glucose levels, our study revealed that the retinal organoids contained pseudo-laminated neural retina with important features reminiscent of mature photoreceptors, both rod and cone subtypes. This advanced maturation of photoreceptors supports the possibility to use 3D retinal organoids for studying photoreceptor development, but also offers novel opportunity for disease modeling, particularly for inherited retinal diseases.

Introduction

Human pluripotent stem cells (PSCs) represent a valuable tool to study human neuronal development and neurodegenerative diseases and to develop future stem-cell based therapies [1]. As concerns the retina, both human embryonic stem cells (ESCs) and induced pluripotent stem cells (iPSCs) have been shown to be able to produce retinal cells [START_REF] Wiley | Patient-specific induced pluripotent stem cells (iPSCs) for the study and treatment of retinal degenerative diseases[END_REF][START_REF] Leach | Concise Review: Making Stem Cells Retinal: Methods for Deriving Retinal Pigment Epithelium and Implications for Patients With Ocular Disease[END_REF][START_REF] Zhao | Stem cell therapies for retinal diseases: recapitulating development to replace degenerated cells[END_REF][START_REF] Rabesandratana | Pluripotent Stem Cell-Based Approaches to Explore and Treat Optic Neuropathies[END_REF][START_REF] Gao | Stemming retinal regeneration with pluripotent stem cells[END_REF][7], including retinal ganglion cells, photoreceptors and retinal pigmented epithelial (RPE) cells, corresponding to the major cell types affected in the most common retinal degenerative diseases. The differentiation of human PSCs toward the retinal lineage has evolved considerably in the last few years and several innovative protocols allowing the self-formation of 3D retinal organoids have been reported [8][9][START_REF] Tucker | Patient-specific iPSC-derived photoreceptor precursor cells as a means to investigate retinitis pigmentosa[END_REF][START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF][START_REF] Kuwahara | Generation of a ciliary margin-like stem cell niche from self-organizing human retinal tissue[END_REF][START_REF] Mellough | IGF-1 Signaling Plays an Important Role in the Formation of Three-Dimensional Laminated Neural Retina and Other Ocular Structures From Human Embryonic Stem Cells[END_REF][START_REF] Wahlin | Photoreceptor Outer Segment-like Structures in Long-Term 3D Retinas from Human Pluripotent Stem Cells[END_REF][START_REF] Gonzalez-Cordero | Recapitulation of Human Retinal Development from Human Pluripotent Stem Cells Generates Transplantable Populations of Cone Photoreceptors[END_REF][START_REF] Llonch | Organoid technology for retinal repair[END_REF].

Many types of somatic cells, such as skin fibroblasts, blood cells, keratinocytes or urinederived cells, have been successfully used for reprogramming and the production of human iPSCs [1,[START_REF] González | Methods for making induced pluripotent stem cells: reprogramming à la carte[END_REF]. Due to their high availability through noninvasive and routine sampling in clinical settings, blood and urine-derived cells have been considered as a preferred source for reprogramming. Nevertheless, independently of the initial somatic identity of reprogrammed cells, human iPSCs can be guided to differentiate into retinal organoids with relatively similar efficiency using different retinal differentiation protocols [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF][START_REF] Zhong | Generation of three-dimensional retinal tissue with functional photoreceptors from human iPSCs[END_REF][START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF][START_REF] Li | Generation of Retinal Organoids with Mature Rods and Cones from Urine-Derived Human Induced Pluripotent Stem Cells[END_REF][START_REF] Capowski | Reproducibility and staging of 3D human retinal organoids across multiple pluripotent stem cell lines[END_REF].

Recently, Wang et al. reprogrammed four different neuronal cell types (rod photoreceptors, cone photoreceptors, bipolar cells and amacrine/horizontal cells) from early and late postnatal mouse retina into iPSCs and further differentiated them into retinal organoids [START_REF] Wang | Retinal Cell Type DNA Methylation and Histone Modifications Predict Reprogramming Efficiency and Retinogenesis in 3D Organoid Cultures[END_REF].

Mouse Müller glial cells (MGCs), which originate from the same pool of retinal progenitors as retinal neurons, can also be reprogrammed to pluripotency with high efficiency and can be guided to differentiate into retinal organoids [START_REF] Wang | Retinal Cell Type DNA Methylation and Histone Modifications Predict Reprogramming Efficiency and Retinogenesis in 3D Organoid Cultures[END_REF]. Even though no data regarding the reprogramming of human retinal cells has been reported, human glial cells such as astrocytes isolated from cerebellar tissue can be efficiently reprogrammed into iPSCs and further differentiated into different neural cell types [START_REF] Ruiz | High-Efficient Generation of Induced Pluripotent Stem Cells from Human Astrocytes[END_REF][START_REF] Tian | Reprogrammed mouse astrocytes retain a "memory" of tissue origin and possess more tendencies for neuronal differentiation than reprogrammed mouse embryonic fibroblasts[END_REF].

Here, we report the generation of iPSCs from human MGCs. By applying our multistep retinal differentiation protocol [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF], we differentiated human MGC-derived iPSCs into pseudolaminated retinal organoids that contained all major retinal cell types.

Materials and Methods

Human postmortem tissue and Müller glial cell cultures

Postmortem eye tissues were collected within 24h after death from bodies donated to science (Laboratory of Anatomy, Faculty of Medicine of St-Etienne, France) in accordance with the French bioethics law. Handling of donor tissues adhered to the tenets of the Declaration of Helsinki of 1975 and its 1983 revision in protecting donor confidentiality.

Retina was dissected from globes by circumferential hemisection behind the ora serrata to gently remove the anterior segment including lens. The retina was carefully separated from the vitreous by transection of the papillary head and around the peripheral regions and transferred into CO 2 -independent medium (Thermo Fisher Scientific). After removing the major blood vessels, the retina was chopped into small fragments (<2mm 2 ) and cells dissociated as previously described [START_REF] Hicks | The growth and behaviour of rat retinal Müller cells in vitro. 1. An improved method for isolation and culture[END_REF][START_REF] Gaudin | Survival and regeneration of adult human and other mammalian photoreceptors in culture[END_REF] with some modifications. After 3 washes in Ringer solution (NaCl 155 mM; KCl 5 mM; CaCl 2 2 mM; MgCl 2 1 mM; NaH 2 PO 4 2 mM; HEPES 10 mM and Glucose 10 mM), retinal fragments were dissociated with pre-activated papain at 31.8 U/mg (Worthington) in Ringer solution during 30 min at 37°C. Digestion was arrested by the addition of 1 ml of DMEM (Thermo Fisher Scientific) containing 10% of fetal bovine serum (FBS) (Thermo Fisher Scientific ) and 25µg/ml of DNase1 (Sigma-Aldrich). The cells were homogeneously suspended with gentle up and down pipetting in pre-warmed DMEM with 10% FBS and 10 µg/ml gentamycin (MGC-medium) and seeded in 6-cm dishes previously coated with Poly-D-Lysin at 2 µg/cm² and Laminin at 1 µg/cm² (Sigma-Aldrich). Dissociated cells were incubated at 37°C in a standard 5% CO 2 / 95% air incubator and medium was left unchanged for 3 to 5 days and then renewed every 2 to 3 days. By the end of the first week in culture, large flattened glial cells were observed with few neuron-like cells scattered on the glial surface. Mitotic glial cells became confluent within 10 to 14 days after plating. At this time cells were passaged by a brief incubation in TrypLE Express (Thermo Fisher Scientific) and banked (cryopreservation in FBS-10%DMSO) or seeded in 6 wellplates previously coated with Poly-D-Lysin and Laminin for reprogramming.

Human Müller glial cell reprogramming and validation

Human MGCs at passage 1 were transduced using the CytoTune Sendai reprogramming vectors Oct4, Klf4, Sox2 and c-Myc (Thermo Fisher Scientific) and cultured for 6 days in MGC-medium before plating on mitomycined human foreskin (MHF)-seeded dishes. The day after, the MGC-medium was replaced with the iPS medium, corresponding to ReproStem medium (ReproCELL, Ozyme) supplemented with 10 ng/ml of human recombinant FGF2 (PreproTech France). The emergent human iPSC colonies were picked under a stereomicroscope according to their human ESC-like colony morphology and expanded on MHF feeder layers. After generation of a frozen stock, human iPSCs were cultured on MHF feeder layers and subsequently adapted to feeder-free conditions on truncated recombinant human vitronectin-coated dishes with Essential 8 TM medium (both from Thermo Fisher Scientific) as previously described [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF]. Cells were routinely cultured at 37°C in a standard 5% CO 2 / 95% air incubator with a daily medium change and passaged with the enzyme-free Gentle cell dissociation reagent (Thermo Fisher Scientific) every week. The undifferentiated state of iPSC colonies was characterized by alkaline phosphatase expression as previously described [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF]. Between twelve and sixteen passages, the clearance of the exogenous reprogramming factors and Sendai virus genome was confirmed by qPCR following the manufacturer's instructions (Thermo Fisher Scientific). At the same time, conventional cytogenetic analysis and in vivo pluripotency analysis by teratoma formation assay were performed as described previously [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF]. The genomic integrity was confirmed by SNP genotyping (Integragen Genomics).

Retinal differentiation and human iPSC-derived retinal cell cultures

Retinal cell differentiation was based on our previously established protocol with adherent human iPSC [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF][START_REF] Slembrouck-Brec | Defined Xeno-free and Feeder-free Culture Conditions for the Generation of Human iPSC-derived Retinal Cell Models[END_REF]. iPSCs derived from human MGCs cultured in Essential 8 TM medium were switched in chemical defined Essential 6 TM medium (Thermo Fisher Scientific). After 2 days, iPSCs were cultured in E6N2 medium composed of Essential 6 TM medium, 1% N2 supplement (Thermo Fisher Scientific), 10 units/ml Penicillin and 10 µg/ml Streptomycin (Thermo Fisher Scientific). The medium was changed every 2-3 days. Around Day 28 (D28), identified self-formed retinal organoids were isolated, using a needle, with the surrounding cells and cultured as floating structures in the ProB27 medium supplemented with 10 ng/ml of recombinant human FGF2 (Peprotech) and half of the medium was changed every 2-3 days. ProB27 medium is composed of chemical defined DMEM:Nutrient Mixture F-12 (DMEM/F12, 1:1, L-Glutamine), 1% MEM non-essential amino acids, 2% B27 supplement (Thermo Fisher Scientific), 10 units/ml Penicillin and 10 µg/ml Streptomycin. At D35, retinal organoids were cultured in absence of FGF2 in ProB27 medium with 10% FBS and 2mM of Glutamax for the next several weeks. To evaluate maturation of photoreceptors, at D84, the retinal organoids were cultured in ProB27 medium with 2% B27 supplement without vitamin A (Thermo Fisher Scientific) until D200.

For human iPSC-derived RPE cell cultures, identified pigmented patches were cut around D42 and transferred, noted as passage 0 (P0), onto plates coated with Geltrex (Thermo Fisher Scientific). Human iPSC-derived RPE cells were expanded in the ProN2 medium composed of DMEM/F12, 1% MEM non-essential amino acids, 1% N2 supplement, 10 units/ml Penicillin and 10 µg/ml Streptomycin and passaged or banked as previously described [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF][START_REF] Slembrouck-Brec | Defined Xeno-free and Feeder-free Culture Conditions for the Generation of Human iPSC-derived Retinal Cell Models[END_REF].

Cryosection

For cryosectioning, retinal organoids were fixed for 15 min in 4% paraformaldehyde (PAF) at 4°C and washed in Phosphate-buffered saline (PBS). Structures were incubated at 4°C in PBS / 30% Sucrose (Sigma-Aldrich) solution during at least 2 hrs and embedded in a solution of PBS, 7.5% gelatin (Sigma-Aldrich), 10% sucrose and frozen in isopentane at -50°C. 10 µm-thick cryosections were collected in two perpendicular planes.

Immunostaining and imaging on retinal sections and dissociated cells

Dissociated cells were fixed with 4% PAF in PBS for 10 min before immunostaining. Sections and fixed dissociated cells were washed with PBS, nonspecific binding sites were blocked for 1 hr at room temperature with a PBS solution containing 0.2% gelatin and 0.1% Triton X-100 (blocking buffer) and then overnight at 4°C with the primary antibody (Table S1) diluted in blocking buffer. Slides were washed three times in PBS with 0.1% Tween and then incubated for 1 hr at room temperature with appropriate secondary antibodies conjugated with either Cy3, Alexa Fluor 488, 594 or 647 (Interchim) diluted at 1:600 in blocking buffer with 4',6diamidino-2-phenylindole (DAPI) diluted at 1:1000 to counterstain nuclei. Fluorescent staining signals were captured with an Olympus FV1000 confocal microscope.

Retinal organoid immunostaining-clearing and imaging

Retinal organoids were fixed with 4% PAF in PBS for 30 to 60 min before immunostaining with a PBS solution containing 0.2% gelatin, 0.5% Triton X-100 and 0.01% thimerosal (Sigma-Aldrich) as previously described [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF]. Samples were next submitted to the 3D imaging of solvent-cleared organ (3DISCO) clearing procedure [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF]. 3D imaging was performed with an inverted confocal microscope (Olympus FV1200) with numerical objectives for high-resolution imaging. Images, 3D volume and movies were generated with Imaris x64 software (version 7.6.1, Bitplane) using the "snapshot" and "animation" tools.

RNA extraction and Taqman Assay

Total RNAs were extracted using Nucleospin RNA II kit (Macherey-Nagel) and cDNA synthesized using the QuantiTect reverse transcription kit (Qiagen) following manufacturer's recommendations. qPCR analysis was performed with custom TaqMan® Array 96-Well Fast plates (Thermo Fisher Scientific) according to the manufacturer's protocol. All primers and MGB probes labeled with FAM for amplification were purchased from Thermo Fisher Scientific (Table S2). Results were normalized against 18S and quantification of gene expression was based on the DeltaCt Method in minimum three independent biological experiments.

Phagocytosis assay.

Photoreceptor outer segments (POS) phagocytosis assay was performed as previously described [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF]. Briefly, four weeks after plating confluent human iPSC-derived RPE cells were challenged for 3 hrs with 1×10 6 FITC-labeled POS before detection of surface-bound and internalized FITC-POS particles. For exclusive detection of internalized particles, fluorescence of surface-bound FITC-POS was selectively quenched by incubation in 0.2% trypan blue before cell fixation in ice cold methanol. Following rehydration, cells were incubated with DAPI for nuclei counterstaining. Fluorescent signals were quantified with the Infinite M1000 Pro (Tecan). The immortalized rat RPE cell line RPE-J was used as a positive control for phagocytic activity.

Results

Derivation of iPSCs from human Muller cells

Retinal cells were dissociated from human postmortem retina and were seeded into tissue culture plates and left to develop as monolayer culture for one week in culture conditions previously described to favor glial cell culture while eliminating other cell types [START_REF] Hicks | The growth and behaviour of rat retinal Müller cells in vitro. 1. An improved method for isolation and culture[END_REF]. After one passage, immunostaining of the cells with anti-Vimentin and anti-Glutamine Synthase (GS) antibodies revealed homogenous labeling throughout the cytoplasm of the cells, consistent with a MGC phenotype (Figures 1A-1B). In contrast, markers of microglia and monocytes/macrophages such as Ionized calcium binding adaptor molecule 1 (Iba1) and CD18 were absent (Figures 1A-1B). We investigated the potential of human MGCs to be reprogrammed into PSCs using the four reprogramming factors OCT3/4, SOX2, CMYC and KLF4 delivered using non-integrative Sendai viruses previously used for dermal fibroblast reprogramming [START_REF] Terray | Establishment of an induced pluripotent stem (iPS) cell line from dermal fibroblasts of an asymptomatic patient with dominant PRPF31 mutation[END_REF]. Emerging iPSC colonies were detectable between 10 and 15 days after transduction and around 10 colonies were picked up and expanded for at least 5 passages before being adapted in feeder-free conditions (vitronectin coating). Expanded human iPSC colonies displayed alkaline phosphatase activity as shown in Figure 1C for one clone (iPSC line 5f). Immunofluorescence analysis of human iPSC line-5f revealed co-expression of transcription factors OCT4 and SOX2, and surface markers SSEA4 and TRA1-81 (Figures 1D-1E), characteristic of pluripotent stem cells. RT-qPCR revealed that the expression of pluripotency genes markedly increased over the respective human MGC population and was similar with that seen in human ESCs (Figure 1F).

Human iPSC line-5f could be differentiated in vivo into derivatives of all three germ layers, as shown by teratoma formation in NSG mouse (Figure S1A-S1B), and exhibited a normal karyotype after 15 passages (Figure S1C). The clearance of the vectors and the exogenous reprogramming factor genes was confirmed by qPCR after 15 passages (Figure S1D). Furthermore, genomic integrity of the iPSC line-5f was confirmed by SNP genotyping (Figure S1E).

Induction of human MGC-derived iPSCs towards retina cell fates

Based on our retinal differentiation protocol in xeno-free / feeder-free conditions [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF][START_REF] Slembrouck-Brec | Defined Xeno-free and Feeder-free Culture Conditions for the Generation of Human iPSC-derived Retinal Cell Models[END_REF], we first evaluated the ability of overgrowing human MGC-derived iPSCs to give rise to neuroepithelial-like structures that could acquire an eye field (EF) fate. As previously reported for iPSCs derived from dermal fibroblasts, self-forming neuroepithelial-like structures can be observed about 4 weeks after the initiation of differentiation (Figure 2A).

RT-qPCR analysis demonstrated that cells of 28-day-old (D28) structures expressed EF transcription factors, such as LHX2, MITF, PAX6, RAX, SIX3 and VSX2, while losing the expression of the pluripotency marker POU5F1, and showed low or no expression of forebrain and midbrain markers, such as EN1 and NKX2-1 (Figure 2B). Interestingly, the expression of transcription factors involved in the photoreceptor lineage, such as CRX, NRL, and NEUROD1, was also detected (Figure 2B). In previously published 3D induction protocols, inhibition of Wnt and BMP/TGFβ pathways contributed to directing human PSCs to a retinal identity [7,[START_REF] Llonch | Organoid technology for retinal repair[END_REF]. In our protocol, RT-qPCR analysis demonstrated that differentiating human MGC-derived iPSCs expressed DKK1 and NOGGIN, endogenous antagonists of Wnt and BMP respectively (Figure 2C). In these conditions, inhibition of BMP/TGF and Wnt functions could occur in absence of exogenous inhibitors, enabling the formation of neuroepithelial structures after 14 days. Activation of Insulin/Insulin-like growth factor-1 signaling pathway has been also reported to contribute to directing pluripotent stem cells toward a retinal identity [START_REF] Mellough | IGF-1 Signaling Plays an Important Role in the Formation of Three-Dimensional Laminated Neural Retina and Other Ocular Structures From Human Embryonic Stem Cells[END_REF][START_REF] Llonch | Organoid technology for retinal repair[END_REF]. To test the possible role of this signaling pathway, we added the pharmacological inhibitor (OSI-906) of Insulin and Insulin-like growth factor-1 receptor signaling to the E6N2 medium at day 2. In this condition, very rare neuroepithelial-like structures were observed after 28 days compared to the control conditions (Figure 2A). All these results suggested that the activation of Insulin/IGF-1 pathway and endogenous production Wnt and BMP antagonists could explain the self-formation of these structures.

After mechanical detachment at D28, structures were collected for further culture in suspension in the presence of FGF2 for one week to favor differentiation of the neural retina (Figure 2D). Co-expression of RAX and PAX6 detected by immunostaining of sections from D35 organoids (after one week in floating culture), confirmed the EF identity of the organoids (Figure 2E). PAX6-positive cells located in the outer part of the developing neuroepithelium also expressed the retinal progenitor cell marker VSX2 (Figure 2F). At D35, many VSX2positive cells were still mitotic as revealed by the Ki67 proliferation marker staining (Figure 2G), confirming the retinal progenitor cell phenotype of these cells.

Formation of retinal organoids from human MGC-derived iPSCs

During retinal development, multipotent retinal progenitors can give rise to all retinal cell types in a defined overlapping chronological order with early-born cell types like retinal ganglion cells (RGCs), horizontal/amacrine cells and late-born cell types including bipolar cells and MGCs. Regarding photoreceptors, cones are mostly produced during early stages of retinal development, whereas rods are generated later [START_REF] Brzezinski | Photoreceptor cell fate specification in vertebrates[END_REF][START_REF] Wang | Photoreceptor Fate Determination in the Vertebrate Retina[END_REF]. To assess the identity of differentiating retinal cells within organoids, immunohistochemistry was performed on sections after different periods in floating culture conditions, based on our previous retinal differentiation protocol (see methods 2.3 for details). At D56, BRN3A-positive RGCs were scattered throughout the organoid with a higher concentration at the inner-most part of the forming neural retina (Figure 3A), while cells committed toward photoreceptor lineage expressing CRX and Recoverin (RCVRN) were found on the opposite side of the structure, in the external part of the retinal organoids (Figure 3A-C). Sparse horizontal cells expressing transcription factor LHX1 can also be detected at the same time (Figure 3B). Immunostaining of D56 retinal organoid sections with the exclusive markers CRX for photoreceptors and PAX6 for early-born non-photoreceptor cells (RGCs, amacrine/horizontal cells) showed that PAX6-positive neurons and CRX-positive photoreceptors started to form two distinct layers (Figure 3C). Interestingly, cells displaying the highest level of PAX6 were found in the most central part of the structure, while in the external part of the retinal organoid, Pax6-positive cells displayed low level of expression of this transcription factor. As time progressed, these two layers can be more distinctly visualized, as shown by the exclusive location of CRXexpressing photoreceptors and PAX6-positive cells at D100 (Figure 3D) and D150 (Figure 3G). In addition to displaying a pseudo lamination with both presumptive outer nuclear layer (ONL) and inner nuclear layer (INL), retinal organoids also showed organized PAX6-positive neurons presented towards the basal surface, where RGCs are located, and also localized in the presumptive INL (Figure 3D-E). Amacrine cells identified by PAX6 and AP-2 co-staining formed a clear nuclear layer at D100, while sparsely distributed PAX6-positive / AP-2negative cells in the apical part of the presumptive INL could correspond to the horizontal cells (Figure 3E). The population of CRX/Recoverin-positive photoreceptors congregated at the outer part of the retinal organoids (Figure 3F), corresponding to the presumptive ONL.

Consistent with in vivo retinogenesis, late-born bipolar cells can be identified by co-staining with PKC and VSX2 antibodies (Figure 3H), demonstrating that our culture conditions allowed the generation of all five types of retinal neurons in organoids. Furthermore, RPCs were also able to differentiate in MGCs, as shown by the presence of cells co-expressing Glutamine Synthase (GS) and the transcription factor SOX9 in D175 retinal organoids (Figure 3I).

When retinal organoids from human MGC-derived iPSCs were subjected to differentiation in ProB27 medium in absence of FBS and Glutamax, efficient retinal differentiation was still observed with the presence of RGCs expressing BRN3A in D56 organoids and photoreceptors identified by Recoverin and CRX immunostaining in D56 and D100 organoids (Figure 4). However, organoids failed to display a continuous laminar organization with the neuro-epithelial layer mostly developing into small rosettes, where the innermost cells expressed photoreceptor markers, surrounded by PAX6-positive non-photoreceptor cells (Figure 4). This phenotype was similar to previously characterized iPSC line (hiPSC-2 clone) derived from human fibroblasts [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF]. Interestingly, pseudo-laminated retinal organoids can also be obtained from this hiPSC-2 line in presence of 10%FBS and Glutamax, with similar formation of presumptive ONL, bearing CRX and Recoverin-positive photoreceptors, and presumptive INL containing PAX6-positive cells, including amacrine cells expressing AP-2 (Figure S2). Cone arrestin (CAR) immunostaining was also restricted to the presumptive ONL (Figure S2), confirming that the generation of pseudo-laminated organoids was independent on the iPSC line used as a source for differentiation and was rather due to the refinement of our original retinal differentiation protocol [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF][START_REF] Slembrouck-Brec | Defined Xeno-free and Feeder-free Culture Conditions for the Generation of Human iPSC-derived Retinal Cell Models[END_REF].

Maturation of rods and cones in retinal organoids

Next, we sought to study the maturation of photoreceptor precursors, previously identified with CRX staining (Figure 3), in presence of FBS and Glutamax for long-term culture.

Constant application of Retinoic Acid (RA) has been reported to inhibit cone maturation in zebrafish [START_REF] Stevens | Plasticity of photoreceptor-generating retinal progenitors revealed by prolonged retinoic acid exposure[END_REF] and recently in mouse PSC-derived retinal organoids [START_REF] Kruczek | Differentiation and Transplantation of Embryonic Stem Cell-Derived Cone Photoreceptors into a Mouse Model of End-Stage Retinal Degeneration[END_REF]. Exogenous RA was never present in our culture conditions. Retinal organoids first cultured with normal B27 supplement containing vitamin A were switched at D84 in ProB27 medium with B27 supplement without vitamin A to prevent the formation of endogenous RA, one active metabolite of Vitamin A. Rods and cones can be clearly identified either by Rhodopsin or Cone arrestin immunostaining restricted to the apical presumptive ONL in D175 retinal organoids (Figure 5). Rhodopsin staining was distributed throughout the soma of rods and intensively stained the peripheral region of the presumptive ONL, reflecting the formation of outer segment (OS)-like structures (Figure 5B and5C). The ciliary protein ARL13B presented an expected punctuated labeling pattern closed to the presumptive ONL, indicating the formation of correctly positioned IS and OS-like structures (Figure 5B). Both short wavelength Opsin (Blue Opsin) and long/medium wavelength Opsin (R/G Opsin) cones were observed with the presence of specific cone pedicle-like structures, apical to the presumptive ONL (Figure 5D and5E). To visualize photoreceptors in 3D, we performed highresolution confocal imaging of immunolabelled D175 whole organoids subjected to 3DISCO clearing procedure [START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF]. Spatial arrangement of cones characterized by the expression of CRX and Cone arrestin was observed on 3D-reconstructed images with unambiguous morphological stubby bud-like cone features (Figure 5F and 5G; Video S1).

Generation of RPE cells from human MGC-derived iPSCs

After removal of retinal organoids and further differentiation of iPSC cultures in proneural medium, pigmented patches of cells appeared around D42 and could be mechanically isolated and spread onto new plates for expansion and amplification. After 4 weeks, pigmented cells formed a confluent monolayer with a cobblestone-shaped morphology (Figure 6A) and expressed the RPE-specific transcription factor MITF and the tight junction marker ZO-1 (Figure 6B). Z-stack analysis of confocal images showed the expression of Ezrin at the apical side of the cells and basolateral localization of Bestrophin (Figure 6C and 6D), which is consistent with a mature polarized RPE. RT-qPCR studies demonstrated that iPSC-derived RPE cells expressed classic markers of RPE such as PEDF, MERTK, RPE65 and BEST1 after long-term cultures (Figure 6E). We also evaluated the functionality of the iPSC-derived RPE cells by measuring phagocytosis of fluorescent-labeled photoreceptor outer segments (POS). As shown in Figure 5F, iPSC-derived RPE cells after one passage were able to phagocyte with an average of 37.3  0.07 % (mean  SEM; n=3) internalized POS within 3 hours, similar to the control rat RPE-J cell line (49.6  0.02; mean  SEM; n=3).

Discussion

In this study we generated iPSCs from a differentiated cell type isolated from normal human retina and showed that these human iPSCs derived from MGCs can be efficiently differentiated toward retinal lineage with simultaneous formation of RPE cells and retinal organoids. Specification and maturation of retinal cells generated from MGC-derived iPSCs follow the same spatial-temporal pattern observed with retinal cells derived from other human somatic-derived iPSCs and consistent with in vivo development.

Since all body cells seem to have the potential to become iPSCs, though at different yields, it is not surprising that glial cells from the retina, such as MGCs can be reprogrammed into iPSCs. Furthermore, MGCs represent the most plastic cell type found in the retina. In coldblood vertebrate, MGC population constitutes an adult retinal stem cell niche able to dedifferentiate, proliferate and generate new retinal cells, mainly after activation of the Ascl1/Lin28 pathway following injury [START_REF] Hamon | Müller glial cell-dependent regeneration of the neural retina: An overview across vertebrate model systems[END_REF][START_REF] Goldman | Müller glial cell reprogramming and retina regeneration[END_REF]. This physiologic response is absent in mammals but in vivo ectopic expression of a specific combination of factors targeting mouse MGCs enabled MGCs to generate functional retinal neurons in different conditions [START_REF] Jorstad | Stimulation of functional neuronal regeneration from Müller glia in adult mice[END_REF][START_REF] Yao | Restoration of vision after de novo genesis of rod photoreceptors in mammalian retinas[END_REF],

confirming the latent stem cell potential of MGCs even in mammals.

Detailed examination of a variety of iPSCs has shown that these cells can retain some epigenetic memory of the cell of origin that bias their differentiation tendency toward the original cell type [START_REF] Polo | Cell type of origin influences the molecular and functional properties of mouse induced pluripotent stem cells[END_REF][START_REF] Kim | Donor cell type can influence the epigenome and differentiation potential of human induced pluripotent stem cells[END_REF]. While this phenomenon was obvious in early-passage iPSCs, the differences in epigenetics and differentiation capacities tended to be attenuated after prolonged passages [START_REF] Polo | Cell type of origin influences the molecular and functional properties of mouse induced pluripotent stem cells[END_REF][START_REF] Kim | Donor cell type can influence the epigenome and differentiation potential of human induced pluripotent stem cells[END_REF]. Other studies have shown that the cell type of origin contributed minimally to iPSC variability and that the variations in differentiation were largely attributable to donor differences rather than to the original cell type [START_REF] Kajiwara | Donor-dependent variations in hepatic differentiation from human-induced pluripotent stem cells[END_REF][START_REF] Rouhani | Genetic Background Drives Transcriptional Variation in Human Induced Pluripotent Stem Cells[END_REF][START_REF] Kyttälä | Genetic Variability Overrides the Impact of Parental Cell Type and Determines iPSC Differentiation Potential[END_REF]. Even though no epigenetic characterizations have been done on the iPSCs used in this study, either derived from human MGCs or from dermal fibroblasts [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF][START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF], no major difference in the characteristics of retinal organoids has been observed. In agreement with our observation, Capowski et al.

recently reported the ability of human iPSCs derived from fibroblasts or blood cells to generate retinal organoids, demonstrating the reproducibility of retinal organoid generation independently of the starting cell type [START_REF] Capowski | Reproducibility and staging of 3D human retinal organoids across multiple pluripotent stem cell lines[END_REF]. A different observation has been made using mouse iPSCs derived from different retinal cell types [START_REF] Hiler | Quantification of Retinogenesis in 3D Cultures Reveals Epigenetic Memory and Higher Efficiency in iPSCs Derived from Rod Photoreceptors[END_REF]. These authors showed that rodderived iPSCs were more efficient to differentiate into retinal structures containing all retinal cell types compared to structures obtained from iPSCs derived from fibroblasts. The authors hypothesized that epigenetic memory and especially the methylation status of Lhx9 gene could explain these differences. Epigenetic differences could be lower between human fibroblasts and human MGCs than between mouse fibroblasts and mouse photoreceptors.

However, retinal organoids generated from mouse ESCs or iPSCs by other groups with relatively similar protocols of differentiation never reported a significant impact of the source of cells on the formation of retinal organoids [START_REF] Eiraku | Self-organizing optic-cup morphogenesis in three-dimensional culture[END_REF][START_REF] Gonzalez-Cordero | Photoreceptor precursors derived from three-dimensional embryonic stem cell cultures integrate and mature within adult degenerate retina[END_REF][START_REF] Assawachananont | Transplantation of Embryonic and Induced Pluripotent Stem Cell-Derived 3D Retinal Sheets into Retinal Degenerative Mice[END_REF][START_REF] Völkner | Retinal Organoids from Pluripotent Stem Cells Efficiently Recapitulate Retinogenesis[END_REF].

The number of methods for generating 3D retinal cultures from human PSCs has greatly increased in the past five years [START_REF] Reichman | From confluent human iPS cells to selfforming neural retina and retinal pigmented epithelium[END_REF][START_REF] Kuwahara | Generation of a ciliary margin-like stem cell niche from self-organizing human retinal tissue[END_REF][START_REF] Mellough | IGF-1 Signaling Plays an Important Role in the Formation of Three-Dimensional Laminated Neural Retina and Other Ocular Structures From Human Embryonic Stem Cells[END_REF][START_REF] Wahlin | Photoreceptor Outer Segment-like Structures in Long-Term 3D Retinas from Human Pluripotent Stem Cells[END_REF][START_REF] Gonzalez-Cordero | Recapitulation of Human Retinal Development from Human Pluripotent Stem Cells Generates Transplantable Populations of Cone Photoreceptors[END_REF][START_REF] Llonch | Organoid technology for retinal repair[END_REF][START_REF] Zhong | Generation of three-dimensional retinal tissue with functional photoreceptors from human iPSCs[END_REF][START_REF] Reichman | Generation of Storable Retinal Organoids and Retinal Pigmented Epithelium from Adherent Human iPS Cells in Xeno-Free and Feeder-Free Conditions[END_REF][START_REF] Li | Generation of Retinal Organoids with Mature Rods and Cones from Urine-Derived Human Induced Pluripotent Stem Cells[END_REF][START_REF] Capowski | Reproducibility and staging of 3D human retinal organoids across multiple pluripotent stem cell lines[END_REF], all seeking at reproducing, despites few differences, the key steps of retinal development that are highly conserved throughout the vertebrate evolution. Morphological, cellular and molecular analysis allow to determine the identity and the distribution of retinal cell types at different stages of organoid formation. As in many 3D protocols of differentiation, we can distinguish three successive stages in cultures of differentiating organoids. Early organoids around D35 correspond essentially to expansion of RPCs within the neuroepithelial cell layer. At intermediate stages (between D50-D100), a clear separation between presumptive ONL and INL -corresponding to differentiation of photoreceptors and neuron populations of inner retina-could be observed in the majority of organoids. Finally, in late retinal organoids (D150-D175), an advanced stage of photoreceptor development can be obtained, revealed by the formation of IS and OS, within an expected time range regarding human development during which OS formed and elongated between fetal weeks 23 and 30 [START_REF] Hendrickson | Rod photoreceptor differentiation in fetal and infant human retina[END_REF][START_REF] Hoshino | Molecular Anatomy of the Developing Human Retina[END_REF]. However, a progressive disorganization of the inner region of the structures was observed concurrently with the RGC loss that is expected in this type of cultures in absence of central projections targets. The requirement of RGCs for proper retinal lamination has been demonstrated in mouse and zebrafish retina, where the developmental ablation of a RGC subpopulation or interfering with the RGC migration and positioning led to a lamination defect [START_REF] Icha | Independent modes of ganglion cell translocation ensure correct lamination of the zebrafish retina[END_REF][START_REF] Tufford | Melanopsin Retinal Ganglion Cells Regulate Cone Photoreceptor Lamination in the Mouse Retina[END_REF]. Neuronal migration and lamination play central role in the building of the retinal laminar architecture [START_REF] Amini | Neuronal Migration and Lamination in the Vertebrate Retina[END_REF]. It is likely that human iPSC-derived retinal organoids do not entirely recapitulate this highly controlled process of neuronal layering that takes place during retinogenesis. It is also important to consider that in vivo the inner retina is vascularized in contrast to the outer retina; thus, the absence of vasculature in retinal organoids could limit the metabolic support of the inner retina structures and explain the disorganization for long-term culture. Absence of RPE could also contribute to impaired organoid lamination. Even though our protocol allowed the generation of RPE cells from iPSCs, this monolayer of epithelial cells facing the photoreceptors in vivo is not present in retinal organoids as a monolayer. As reported in mice, ablation of RPE during early retinal development is accompanied by disruption of retinal lamination with formation of rosette [START_REF] Raymond | The retinal pigmented epithelium is required for development and maintenance of the mouse neural retina[END_REF].

In this study we used FBS and Glutamax for long-term maintenance of cell survival [START_REF] Zhong | Generation of three-dimensional retinal tissue with functional photoreceptors from human iPSCs[END_REF] without addition of RA to the media throughout the whole differentiation process in order to try optimizing cone differentiation. Indeed, RA signaling pathway plays an important role in photoreceptor development by promoting rod differentiation but inhibiting cone maturation in chicken and zebrafish [START_REF] Stevens | Plasticity of photoreceptor-generating retinal progenitors revealed by prolonged retinoic acid exposure[END_REF][START_REF] Hyatt | Retinoic acid alters photoreceptor development in vivo[END_REF][START_REF] Alfano | Vax2 regulates retinoic acid distribution and cone opsin expression in the vertebrate eye[END_REF][START_REF] Da Silva | Fgf8 Expression and Degradation of Retinoic Acid Are Required for Patterning a High-Acuity Area in the Retina[END_REF]. Recently, different groups reported that continued supplementation of RA hindered or prevented the expression of cone markers in retinal organoids derived from mouse or human iPSCs [START_REF] Zhong | Generation of three-dimensional retinal tissue with functional photoreceptors from human iPSCs[END_REF][START_REF] Kruczek | Differentiation and Transplantation of Embryonic Stem Cell-Derived Cone Photoreceptors into a Mouse Model of End-Stage Retinal Degeneration[END_REF] and that exogenous RA addition was not required to generate retinal organoids with both mature rod and cone photoreceptors [START_REF] Li | Generation of Retinal Organoids with Mature Rods and Cones from Urine-Derived Human Induced Pluripotent Stem Cells[END_REF]. Most of the recent protocols aiming at generating retinal organoids applied a relatively high concentration of RA in the culture medium in a specific time window, generally before photoreceptor maturation (around day 85-120 depending of the protocol used, yielding rodenriched organoids [START_REF] Wahlin | Photoreceptor Outer Segment-like Structures in Long-Term 3D Retinas from Human Pluripotent Stem Cells[END_REF][START_REF] Gonzalez-Cordero | Recapitulation of Human Retinal Development from Human Pluripotent Stem Cells Generates Transplantable Populations of Cone Photoreceptors[END_REF][START_REF] Capowski | Reproducibility and staging of 3D human retinal organoids across multiple pluripotent stem cell lines[END_REF]. In our culture conditions, the precursor of RA, Vitamin A, is present in B27 supplement only at early time of organoid floating cultures but not for long-

Figure 1 :

 1 Figure 1: Derivation and characterization of iPSCs from human MGCs (A-B) Characterization of human MGCs by immunostaining with glial (GS and Vimentin) and microglial (CD18 and Iba1) markers. Nuclei were counterstained with DAPI (blue). Note the absence of staining with microglial markers as expected. (C) Positive alkaline phosphatase staining of human iPSC-5f derived from human MGCs at P10. (D-E) Immunofluorescence of pluripotency markers (SSEA4, OCT4, TRA1-81 and SOX2) for iPSC-5f at P15. (F) qRT-PCR analysis of pluripotency and self-renewal markers in human ESCs, iPSC-5f and human MGCs. Data are normalized to human ESCs. (Scale bars: A, B, D-E, 60μm; C, 200μm).

Figure 2 :

 2 Figure 2: Commitment of human MGC-derived iPSCs into retinal lineage (A) Examples of self-forming neuroepithelial-like structures derived from iPSC-5f at D28 and involvement of IGF-1/insulin signaling in neuroepithelium formation during iPSC differentiation. (B) qRT-PCR analysis of eye-field transcription factors (EFTF), photoreceptorrestricted lineage markers and non-retinal markers in neuroepithelial-like structures at D28. Data are normalized to iPSC-5f at D0. (C) qRT-PCR analysis of NOGGIN and DKK1 in differentiating iPSC-5f at D14 and D28. Data are normalized to iPSC-5f at D0. (E-G) Immunofluorescence co-staining of cryosections from neuroepithelial-like structures at D35 for RAX and PAX6 (E), PAX6 and VSX2 (F), or VSX2 and Ki67 (G). Nuclei were counterstained with DAPI (gray). (Scale bars: A and D, 200μm; E, F, G, 100μm).

Figure 3 :

 3 Figure 3: Generation of pseudo-laminated retinal organoids containing all retinal cell types from human MGC-derived iPSCs (A-F) Immunofluorescence staining of cryosections from retinal organoids at D56 (A-C) and D100 (D-F) using markers for retinal ganglion cells (BRN3A, PAX6), horizontal cells (LHX1, PAX6), amacrine cells (AP2, PAX6) and photoreceptors (CRX, RCVRN). (G-I) Immunofluorescence staining of cryosections from retinal organoids at D150 (G) and D175 (H, I) using markers for photoreceptors (CRX), bipolar cells (VSX2 and PKC) and MGCs (GS and SOX9). Nuclei were counterstained with DAPI (blue). (Scale bars: A, 100μm; B-I, 50μm).

Figure 4 :

 4 Figure 4: Improvement of retinal organoid lamination in presence of FCS and Glutamax. Immunofluorescence staining of cryosectioned retinal organoids from human MGC-derived iPSCs using markers for retinal ganglion cells (BRN3A, PAX6), amacrine/horizontal cells (PAX6) and photoreceptors (CRX) after 56 and 100 days in floating cultures in absence (left panels) or in presence (right panels) of 10%FCS and 2mM Glutamax in ProB27 medium. Nuclei were counterstained with DAPI (blue). (Scale bars: 100μm).

term cultures (after 12 weeks), in attempt to control the endogenous RA production during the culture. It would be interesting to investigate the expression of RA-synthesizing enzymes (Aldh1a1 and Aldh1a3) and catabolizing enzymes (Cyp26a1 and Cyp26a3) in human retinal organoids at different stages of maturation, since the transient and highly localized expression pattern of these enzymes in human fetal retina could explain the foveal patterning [START_REF] Da Silva | Fgf8 Expression and Degradation of Retinoic Acid Are Required for Patterning a High-Acuity Area in the Retina[END_REF]. Recent data in mouse PSC-derived retinal organoids reported a drastic down-regulation of RA metabolism enzymes (Raldh1, Raldh3 and Cyp26a1) at late differentiation stages, when cone precursors are present [START_REF] Kruczek | Differentiation and Transplantation of Embryonic Stem Cell-Derived Cone Photoreceptors into a Mouse Model of End-Stage Retinal Degeneration[END_REF]. As reported by other groups [START_REF] Zhong | Generation of three-dimensional retinal tissue with functional photoreceptors from human iPSCs[END_REF][START_REF] Li | Generation of Retinal Organoids with Mature Rods and Cones from Urine-Derived Human Induced Pluripotent Stem Cells[END_REF][START_REF] Capowski | Reproducibility and staging of 3D human retinal organoids across multiple pluripotent stem cell lines[END_REF][START_REF] Eldred | Thyroid hormone signaling specifies cone subtypes in human retinal organoids[END_REF], we found that human organoids have similar distribution of cone subtypes, with less blue cones than red/green cones, as to the in vivo situation of human retina. The control of cone subtype specification by thyroid hormone signaling, largely described in mice [START_REF] Brzezinski | Photoreceptor cell fate specification in vertebrates[END_REF][START_REF] Wang | Photoreceptor Fate Determination in the Vertebrate Retina[END_REF], was recently confirmed in human PSC-derived retinal organoids by elegant CRISPR/Cas9-mediated gene editing [START_REF] Eldred | Thyroid hormone signaling specifies cone subtypes in human retinal organoids[END_REF]. In our study, the presence of triiodothyronine (T3), the most active form of thyroid hormone in the B27 supplement, makes possible to control the cone subtype specification in retinal organoids via the thyroid hormone receptor beta.

In summary, stem cell-derived retinal organoid technology represents a useful tool for studying the development of human retina and for modeling human retinal diseases, particularly those targeting the outer retina. With the generation of relevant cells, this innovative technology is expected to facilitate the development of cell therapies for retinal dystrophies.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of interest

The authors declare that they have no conflicts of interest. Table S1: List of antibodies used for immunohistochemistry analysis Table S2: List of TaqMan® Gene Expression ID Assays used for RT-qPCR Movie S1: Movie of a D175 retinal organoid labelled with anti-CRX (white) and Cone arrestin (red). Scale bar = 300µm.

Supplementary Material