Th Ème Num

Mireille Bossy

Awa Diop

apport de recherche

Keywords: Euler scheme, weak error, CIR process, Hull-White process, SABR processes Schéma d'Euler, erreur faible, processus de CIR, processus d'Hull-White, processus SABR

An efficient discretisation scheme for one dimensional SDEs with a diffusion coefficient

function of the form |x| α , α

Introduction

We consider (X t , t ≥ 0), the R-valued process solution to the following one-dimensional Itô stochastic differential equation

X t = x 0 + t 0 b(X s )ds + σ t 0 |X s | α dW s , (1.1)
where x 0 and σ are given constants, σ > 0 and (W t , t ≥ 0) is a one-dimensional Brownian motion defined on a given probability space (Ω, F, P). We denote by (F t , t ≥ 0) the Brownian filtration. To ensure the existence of such process, we state the following hypotheses: Under hypotheses (H0) and (H1), strong existence and uniqueness holds for equation (1.1). Moreover, when x 0 ≥ 0 and b(0) > 0, the process (X t , t ≥ 0) is valued in [0, +∞) (see e.g. [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]). Then (X) is the unique strong solution to

X t = x 0 + t 0 b(X s )ds + σ t 0 X α s dW s . (1.2)
Simulation schemes for Equation (1.1) are motivated by some applications in Finance: in [START_REF] Cox | A theory of the term structure of the interest rates[END_REF], Cox, Ingersoll and Ross (CIR) proposed to model the dynamics of the short term interest rate as the solution of (1.1) with α = 1/2 and b(x) = a -bx. Still to model the short term interest rate, Hull and White [START_REF] Hull | Pricing interest-rate derivative securities[END_REF] proposed the following mean-reverting diffusion process dr t = (a(t) -b(t)r t )dt + σ(t)r α t dW t with 0 ≤ α ≤ 1. More recently, the stochastic-αβρ model or SABR-model have been proposed as a stochastic correlated volatility model for the asset price (see [START_REF] Hagan | Managing smile risk[END_REF]):

dX t = α t X β t dW 1 t dα t = να t dB t
where B t = ρW 1 t + (1 -ρ 2 )W 2 t and (W 1 , W 2 ) is a 2d-Brownian motion. CIR-like models arise also in fluid mechanics: in the stochastic Lagrangian modeling of turbulent flow, characteristic quantities like the instantaneous turbulent frequency (ω t ) are modeled by (see [START_REF] Dreeben | Wall-function treatment in pdf methods for turbulent flows[END_REF])

dω t = -C 3 ω t (ω t -ω t ) dt -S( ω t )ω t dt + C 4 ω t 2 ω t dW t
where the ensemble average ω t denotes here the conditional expectation with respect to the position of the underlying portion of fluid and S(ω) is a given function.

In the examples above, the solution processes are all positive. In the practice, this could be an important feature of the model that simulation procedures have to preserve. By using the classical Euler scheme, one cannot define a positive approximation process. Similar situations occur when one consider discretisation scheme of a reflected stochastic differential equation. To maintain the approximation process in a given domain, an efficient strategy consists in symmetrizing the value obtained by the Euler scheme with respect to the boundary of the domain (see e.g. [START_REF] Bossy | A symmetrized Euler scheme for an efficient approximation of reflected diffusions[END_REF]). Here, our preoccupation is quite similar. We want to maintain the positive value of the approximation. In addition, we have to deal with a just locally Lipschitz diffusion coefficient.

In [START_REF] Deelstra | Convergence of discretized stochastic (interest rate) processes with stochastic drift term[END_REF], Deelstra and Delbaen prove the strong convergence of the Euler scheme apply to dX t = κ(γ -X t )dt + g(X t )dW t where g : R → R + vanishes at zero and satisfies the Hölder condition |g(x) -g(y)| ≤ b |x -y|. The Euler scheme is applied to the modified equation dX t = κ(γ -X t )dt + g(X t l l {Xt≥0} )dW t . This corresponds to a projection scheme. For reflected SDEs, this procedure convergences weakly with a rate 1 2 (see [START_REF] Costantini | Numerical approximation for functionals of reflecting diffusion processes[END_REF]). Moreover, the positivity of the simulated process is not guaranteed. In the particular case of the CIR processes, Alfonsi [START_REF] Alfonsi | On the discretization schemes for the CIR (and Bessel squared) processes[END_REF] proposes some implicit schemes, which admit analytical solutions, and derives from them a family of explicit schemes. He analyses their rate of convergence (in both strong and weak sense) and proves a weak rate of convergence of order 1 and an error expansion in the power of the time-step for the explicit family. Moreover, Alfonsi provides an interesting numerical comparison between the Deelstra and Delbaen scheme, his schemes and the present one discussed in this paper, in the special case of CIR processes.

In section 2, we construct our time discretisation scheme for (X t , t ∈ [0, T ]), based on the symmetrized Euler scheme and which can be simulated easily. We prove a theoretical rate of convergence of order one for the weak approximation error. We analyze separately the cases α = 1/2 and 1/2 < α < 1. The convergence results are given in the next section in Theorems 2.3 and 2.5 respectively. The sections 3 and 4 are devoted to the proofs in this two respective situations. We denote (X t , t ∈ [0, T ]) the approximation process. To study the weak error Ef (X T ) -Ef (X T ), we will use the Feynman-Kac representation Ef (X x T -t ) = u(t, x) where u(t, x) solves the associated Kolmogorov PDE. The two main ingredients of the rate of convergence analysis consist in, first obtaining the upper-bounds on the spatial derivatives of u(t, x) up to the order four. To our knowledge, for this kind of Cauchy problem, there is no generic result. The second point consists in studying the behavior of the approximation process at the origin.

Let us emphasis the difference between the situations α = 1/2 and 1/2 < α < 1. The case α = 1/2 could seem intuitively easier as the associated infinitesimal generator has unbounded but smooth coefficients. In fact, studying the spatial derivative of u(t, x) with probabilistic tools, we need to impose the condition b(0) > σ 2 , in order to define the derivative of X x t with respect to x. In addition, the analysis of the approximation process (X) at the origin shows that the expectation of its local time is in ∆t b(0)/σ 2 . In the case 1/2 < α < 1, the derivatives of the diffusion coefficient of the associated infinitesimal generator are degenerated functions at point zero. As we cannot hope to obtain uniform upper-bounds in x for the derivatives of u(t, x), we prove that the approximation process goes to a neighborhood of the origin with an exponentially small probability and we give upper bounds for the negative moments of the approximation process (X).

2 The symmetrized Euler scheme for (1.1) For x 0 ≥ 0, let (X t , t ≥ 0) given by (1.1) or (1.2). For a fixed time T > 0, we define a discretisation scheme (X t k , k = 0, . . . , N ) by

X 0 = x 0 ≥ 0, X t k+1 = X t k + b(X t k )∆t + σX α t k (W t k+1 -W t k ) , (2.1) k = 0, . . . , N -1,
where N denotes the number of discretisation times t k = k∆t and ∆t > 0 is a constant time step such that N ∆t = T .

In the sequel we will use the time continuous version (X t , 0 ≤ t ≤ T ) of the discrete time process, which consists in freezing the coefficients on each interval [t k , t k+1 ):

X t = X η(t) + (t -η(t))b(X η(t) ) + σX α η(t) (W t -W η(t) ) , (2.2)
where

η(s) = sup k∈{1,...,N } {t k ; t k ≤ s}. The process (X t , 0 ≤ t ≤ T ) is valued in [0, +∞).
By induction on each subinterval [t k , t k+1 ), for k = 0 to N -1, by using the Tanaka's formula, we can easily show that (X t ) is a continuous semi-martingale with a continuous local time (L 0 t (X)) at point 0. Indeed, for any t ∈ [0, T ], if we set

Z t = X η(t) + b(X η(t) )(t -η(t)) + σX α η(t) (W t -W η(t) ), (2.3)
then X t = |Z t | and

X t = x 0 + t 0 sgn(Z s )b(X η(s) )ds + σ t 0 sgn(Z s )X α η(s) dW s + 1 2 L 0 t (X), (2.4)
where sgn(x) := 1 -2 l l (x≤0) . The following lemma ensures the existence of the positive moments of (X t ), starting at x 0 at time 0, and of (X t ), its associated discrete time process: Lemma 2.1. Assume (H0) and (H1). For any x 0 ≥ 0, for any p ≥ 1, there exists a positive constant C, depending on p, but also on the parameters b(0), K, σ, α and T , such that

E sup t∈[0,T ] X 2p t + E sup t∈[0,T ] X 2p t ≤ C(1 + x 2p 0 ). (2.5)
In the following proof, as well as in the rest of the paper, C will denote a constant that can change from line to line. C could depend on the parameters of the model, but it is always independent of ∆t.

Proof. We prove (2.5) for (X t , 0 ≤ t ≤ T ) only, the case of (X t , 0 ≤ t ≤ T ) could be deduced by similar arguments. By the Itô's formula, and noting that for any t ∈ [0, T ] t 0 X 2p-1 s dL 0 s (X) = 0, we have

X 2p t = x 2p 0 + 2p t 0 X 2p-1 s sgn(Z s )b(X η(s) )ds +2pσ t 0 X 2p-1 s sgn(Z s )X α η(s) dW s + σ 2 p(2p -1) t 0 X 2p-2 s X 2α η(s) ds.
(2.6)

To prove (2.5), let's start by showing that sup

t∈[0,T ] E X 2p t ≤ C(1 + x 2p 0 ). (2.7)
(2.5) will follow from (2.7), (2.6) and the Burkholder-Davis-Gundy Inequality. Let τ n be the stopping time defined by τ n = inf{0 < s < T ; X s ≥ n}, with inf{∅} = 0. Then,

EX 2p t∧τn ≤ x 2p 0 + 2pE t∧τn 0 X 2p-1 s b(X η(s) )ds + σ 2 p(2p -1)E t∧τn 0 X 2p-2 s X 2α η(s) ds .
By using (H0), (H1) and the Young Inequality, we get

EX 2p t∧τn ≤x 2p 0 + T b(0) 2p + (2p -1)E t∧τn 0 X 2p s ds + 2pKE t∧τn 0 X 2p-1 s X η(s) ds + σ 2 p(2p -1)E t∧τn 0 X 2p-2 s X 2α η(s) ds .
Replacing X s by (2.2) in the integrals above, by using another time (H1) and the Young Inequality, we easily obtain that for any t ∈ [0, T ],

EX 2p η(t)∧τn ≤ x 2p 0 + C 1 + η(t) 0 E X 2p η(s)∧τn ds ,
where C > 0 depends on p, b(0), K, σ, α and T . A discrete version of the Gronwall Lemma allows us to conclude that sup k=0,...,N

E X 2p t k ∧τn ≤ C(1 + x 2p 0 ),
for another constant C, which does not depend on n. Taking the limit n → +∞, we get that sup k=0,...,N E(X

2p t k ) ≤ C(1 + x 2p 0 )
, from which we easily deduce (2.7) using (2.2).

Main results

In addition of hypotheses (H0) and (H1), we will analyze the convergence rate of (2.1) under the following hypothesis: Under (H1), (X t , 0 ≤ t ≤ T ) satisfies

X t = x 0 + t 0 b(X s )ds + σ t 0 X s dW s , 0 ≤ t ≤ T. (2.8)
When b(x) is of the form a -βx, with a > 0, (X t ) is the classical CIR process used in Financial mathematics to model the short interest rate. When b(x) = a > 0, (X t ) is the square of a Bessel process. Here we consider a generic drift function b(x), with the following restriction :

(H3) b(0) > σ 2 .
Remark 2.2. When x 0 > 0 and b(0) ≥ σ 2 /2, by using the Feller's test, one can show that P(τ 0 = ∞) = 1 where τ 0 = inf{t ≥ 0; X t = 0}. We need the stronger Hypothesis (H3) to prove that the derivative (in the sense of the quadratic mean) of X x t with respect to x is well defined (see Proposition 3.4 and its proof in Appendix B). In particular, we need to use the Lemma 3.1 which controls the inverse moments and the exponential inverse moment of the CIR-like process (X t ), for some values of the parameter ν = 2b(0) σ 2 -1 > 1. Section 3 is devoted to the proof of the following Theorem 2.3. Let f be a R-valued C 4 bounded function, with bounded spatial derivatives up to the order 4. Let α = 1 2 and x 0 > 0. Assume (H1), (H2) and (H3). Choose ∆t sufficiently small in (2.1), i.e. ∆t ≤ 1/(2K) ∧ x 0 . Then there exists a positive constant C depending on f , b, T and x 0 such that

Ef (X T ) -Ef (X T ) ≤ C   ∆t + ∆t x 0 b(0) σ 2   .
Under (H3), the global theoretical rate of convergence is of order one. When b(0) < σ 2 , numerical tests for the CIR process show that the rate of convergence becomes under-linear (see [START_REF] Diop | Sur la discrétisation et le comportement à petit bruit d'EDS unidimensionnelles dont les coefficients sont à dérivées singulières[END_REF] and the comparison of numerical schemes for the CIR process performed by Alfonsi in [START_REF] Alfonsi | On the discretization schemes for the CIR (and Bessel squared) processes[END_REF]).

Convergence rate when

1/2 < α < 1 Under (H1), (X t , 0 ≤ t ≤ T ) satisfies X t = x 0 + t 0 b(X s )ds + σ t 0 X α s dW s , 0 ≤ t ≤ T.
(2.9)

We restrict ourselves to the case

(H3') x 0 > b(0) √ 2 ∆t.
Remark 2.4. When x 0 > 0, the Feller's test on process (X t ) shows that it is enough to suppose b(0) > 0, as in (H1), to ensure that P(τ 0 = ∞) = 1, for τ 0 = inf{t ≥ 0; X t = 0}.

In Section 4, we prove the following Theorem 2.5. Let f be a R-valued bounded C 4 function, with bounded spatial derivatives up to the order 4. Let 1 2 < α < 1. Assume (H1), (H2) and (H3'). Choose ∆t sufficiently small in (2.1), i.e. ∆t ≤ 1/(4K). Then there exists a positive constant C depending on f , α, σ, b, T and x 0 such that

|Ef (X T ) -Ef (X T )| ≤ C 1 + 1 x q(α) 0 ∆t,
where q(α) is a positive constant depending only on α.

3 The case of processes with α = 1/2

Preliminary results

In this section (X t ) denotes the solution of (2.8) starting at the deterministic point x 0 at time 0. When we need to vary the deterministic initial position, we mention it explicitly by using the notation (X x t ) corresponding to the unique strong solution of the equation

X x t = x + t 0 b(X x s )ds + σ t 0 X x s dW s .
(3.1)

On the exact process

We have the following Lemma 3.1. Let us assume (H1) and (H3). We set ν = 2b(0) σ 2 -1 > 1. For any p such that 1 < p < ν, for any t ∈ [0, T ] and any x > 0,

E (X x t ) -1 ≤ C(T )x -1 and E (X x t ) -p ≤ C(T )t -p or E (X x t ) -p ≤ C(T, p)x -p .
Moreover for all µ ≤ ν 2 σ 2 8 ,

E exp µ t 0 (X x s ) -1 ds ≤ C(T ) 1 + x -ν/2 , (3.2)
where the positive constant C(T ) is a non-decreasing function of T and does not depend on x.

Proof. As b(x) ≥ b(0) -Kx, The Comparison Theorem gives that, a.s. X x t ≥ Y x t , for all t ≥ 0, where (Y x t , t ≤ T ) is the CIR process solving

Y x t = x + t 0 (b(0) -KY x s ) ds + σ t 0 Y x s dW s . (3.3)
In particular, E exp(µ

t 0 (X x s ) -1 ds) ≤ E exp(µ t 0 (Y x s ) -1 ds). As b(0) >
∂u ∂t (t, x) ≤C(1 + x), (3.4)
u L ∞ ([0,T ]×[0,+∞)) + k=1 ∂ k u ∂x k L ∞ ([0,T ]×[0,+∞)) ≤C (3.5)
and u(t, x) satisfies

   ∂u ∂t (t, x) + b(x) ∂u ∂x (t, x) + σ 2 2 x ∂ 2 u ∂x 2 (t, x) = 0, (t, x) ∈ [0, T ) × [0, +∞), u(T, x) = f (x), x ∈ [0, +∞). (3.6)
In all what follows, we will denote ∞ the norm on L ∞ spaces. Before to prove the Proposition 3.2, we introduce some notations and give preliminary results: for any λ ≥ 0 and any x > 0, we denote by (X x t (λ), 0 ≤ t ≤ T ), the [0, +∞)-valued process solving

X x t (λ) = x + λσ 2 t + t 0 b(X x s (λ))ds + σ t 0 X x s (λ)dW s . (3.7)
Equation (3.7) has a non-exploding unique strong solution. Moreover, for all t ≥ 0, X x t (λ) ≥ X x t . The coefficients are locally Lipschitz on (0, +∞), with locally Lipschitz first order derivatives. Then X x t (λ) is continuously differentiable (see e.g. Theorem V.39 in [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]), and if we denote J x t (λ) = dX x t dx (λ), the process (J x t (λ), 0 ≤ t ≤ T ) satisfies the linear equation

J x t (λ) = 1 + t 0 J x s (λ)b (X x s (λ))ds + t 0 J x s (λ) σdW s 2 X x s (λ) . ( 3.8)
By Lemma 3.1, the process (

t 0 dWs √ X x s (λ)
, 0 ≤ t ≤ T ) is a locally square integrable martingale.

Then, for all λ ≥ 0, J x t (λ) is given by (see e.g. Theorem V.51 in [START_REF] Protter | Stochastic Integration and Differential Equations[END_REF]),

J x t (λ) = exp t 0 b (X x s (λ))ds + σ 2 t 0 dW s X x s (λ) - σ 2 8 t 0 ds X x s (λ)
.

(3.9)

Lemma 3.3. Assume (H3). The process (M x t (λ), 0 ≤ t ≤ T ) defined by

M x t (λ) = exp σ 2 t 0 dW s X x s (λ) - σ 2 8 t 0 ds X x s (λ)
is a P-martingale. Moreover, sup t∈[0,T ] E (J x t (λ)) ≤ C where the positive constant C does not depend on x . Proof. By Lemma 3.1, (M x t (λ), 0 ≤ t ≤ T ) satisfies the Novikov criterion. Under (H2), b is a bounded function and

E [J x t (λ)] = E exp t 0 b (X x s (λ))ds M x t (λ) ≤ exp( b ∞ T ). Let (Z (λ,λ+ 1 2 ) t , 0 ≤ t ≤ T ) defined by Z (λ,λ+ 1 2 ) t = exp -σ 2 t 0 1 √ X x s (λ) dX x s (λ) σ √ X x s (λ) - b(X x s (λ))+(λ+ 1 2 )σ 2 σ √ X x s (λ) ds σ 2 8 t 0 ds X x s (λ) .
(3.10) By the Girsanov Theorem, under the probability Q λ+ 1 2 such that dQ λ+

1 2 dP Ft = 1 Z (λ,λ+ 1 2 ) t , the process (B λ+ 1 2 t = t 0 dX x s (λ) σ √ X x s (λ) - b(X x s (λ))+(λ+ 1 2 )σ 2 σ √ X x s (λ) ds, t ∈ [0, T ]) is a Brownian motion on (Ω, F T , Q λ+ 1 2
). Indeed, we have that

X x t (λ) = x + (λ + 1 2 )σ 2 t + t 0 b(X x s (λ))ds + σ t 0 X x s (λ)dB λ+ 1 2 s . Hence, L Q λ+ 1 2 (X x . (λ)) = L P (X x . (λ+ 1
2 )) and from the Lemma 3.3,

Z (λ,λ+ 1 2 ) t = exp(-σ 2 t 0 dB λ+ 1 2 s √ Xs(λ) - σ 2 8 t 0 ds Xs(λ) ) is a Q λ+ 1 2 -martingale.
The following proposition allows us to compute the derivatives of u(t, x). Proposition 3.4. Assume (H1), (H2) and (H3). Let g(x), h(x) and k(x) be some bounded C 1 functions, with bounded first derivatives. For any λ ≥ 0, let v(t, x) be the R-valued function defined, on [0, T ] × R * + , by

v(t, x) = E g(X x t (λ)) exp( t 0 k(X x s (λ))ds) + t 0 E h(X x s (λ)) exp( s 0 k(X x θ (λ))dθ) ds. Then v(t, x) is of class C 1 with respect to x and ∂v ∂x (t, x) = E exp( t 0 k(X x s (λ))ds) g (X x t (λ))J x t (λ) + g(X x t (λ)) t 0 k (X x s (λ))J x s (λ)ds + t 0 E exp( s 0 k(X x θ (λ))dθ) h (X x s (λ))J x s (λ) + h(X x s (λ)) s 0 k (X x θ (λ))J x θ (λ)dθ ds.
The proof is postponed in the Appendix B.

Proof of Proposition 3.2. First, we note that

u(t, x) = Ef (X x T -t ) is a continuous function in x and bounded by f ∞ . Let us show that u is in C 1,4 ([0, T ] × [0, +∞)). f being in C 4 (R), by the Itô's formula, u(t, x) = f (x) + T -t 0 E (b(X x s )f (X x s )) ds + σ 2 2 T -t 0 E (X x s f (X x s )) ds + σE T -t 0 X x s f (X x s )dW s .
f is bounded and (X x t ) have moments of any order. Then we obtain that

∂u ∂t (t, x) = -E b(X x T -t )f (X x T -t ) + σ 2 2 X x T -t f (X x T -t ) .
Hence, ∂u ∂t is a continuous function on [0, T ] × [0, +∞) and (3.4) follows by Lemma 2.1.

By Proposition 3.4, for x > 0, u(t, x) = Ef (X x T -t ) is differentiable and ∂u ∂x (t, x) = E f (X x T -t (0))J x T -t (0) .
Hence, by using the Lemma 3.

3, ∂u ∂x (t, x) ≤ f ∞ E J x T -t (0) ≤ C f ∞ . We introduce the probability Q 1 2 such that dQ 1 2 dP Ft = 1 Z (0, 1 2 ) t . Denoting by E 1 2 the expectation under the probability Q 1 2 , we have ∂u ∂x (t, x) = E 1 2 f (X x T -t (0))Z (0, 1 2 ) T -t J x T -t (0) .
From (3.9), as

W t = B 1 2 t + t 0 σ 2 √ X x s (0)
ds, we notice that

J x t (0) = exp t 0 b (X x s (0)ds + σ 2 t 0 dB 1 2 s X x s (0) + σ 2 8 t 0 ds X x s (0)
and that

Z (0, 1 2 ) T -t J x T -t (0) = exp T -t 0 b (X x s (0))ds , from the definition of Z (0, 1 2 ) t in (3.10). Hence, ∂u ∂x (t, x) = E 1 2 f (X x T -t (0)) exp T -t 0 b (X x s (0))ds . As L Q 1 2 (X x • (0)) = L P (X x • ( 1 2 )),
for x > 0, we finally obtain the following expression for ∂u ∂x (t, x):

∂u ∂x (t, x) = E f (X x T -t ( 1 2
)) exp

T -t 0 b (X x s ( 1 2 ))ds . (3.11) Now the right-hand side of (3.11) is a continuous function on [0, T ] × [0, +∞) so that u ∈ C 1,1 ([0, T ] × [0, +∞)). Moreover for x > 0, by Proposition 3.4, ∂u ∂x (t, x) is continuously differentiable and ∂ 2 u ∂x 2 (t, x) = E f (X x T -t ( 1 2 ))J x T -t ( 1 2 ) exp
T -t 0 b (X x s ( 1 2 ))ds +E f (X x T -t ( 1 2
)) exp

T -t 0 b (X x s ( 1 2
))ds

T -t 0 b (X x s ( 1 2 ))J x s ( 1 2
)ds .

(3.12)

As previously, we can conclude that ∂ 2 u ∂x 2 (t, x) is bounded uniformly in x. In order to obtain an expression for ∂ 2 u ∂x 2 (t, x) continuous in [0, T ] × [0, +∞) and also to compute the third derivative, we need to transform the expression in (3.12) in order to avoid again the appearance of the derivative of J x t ( 1 2 ) that we do not control. Thanks to the Markov property and the time homogeneity of the process

X x t ( 1 2 )), for any s ∈ [0, T -t], E f (X x T -t ( 1 2
)) exp

T -t s b (X x u ( 1 2 )) F s = E f (X y T -t-s ( 1 2
)) exp

T -t-s 0 b (X y u ( 1 2
))

y=X x s ( 1 2 ))
By using (3.11), we get

E[f (X x T -t ( 1 2
)) exp(

T -t s b (X x u ( 1 2 )))/F s ] = ∂u ∂x (t + s, X x s ( 1 2
))). We introduce this last equality in the second term of the right-hand side of (3.12):

E f (X x T -t ( 1 2
)) exp

T -t 0 b (X x u ( 1 2
))du

T -t 0 b (X x s ( 1 2 ))J x s ( 1 2 )ds = E T -t 0 E f (X x T -t ( 1 2
)) exp

T -t s b (X x u ( 1 2
))du

F s × exp s 0 b (X x u ( 1 2 ))du b (X x s ( 1 2 ))J x s ( 1 2 )ds = T -t 0 E ∂u ∂x (t + s, X x s ( 1 2 ))) exp s 0 b (X x u ( 1 2 ))du b (X x s ( 1 2 ))J x s ( 1 2
) ds.

Coming back to (3.12), this leads to the following expression for ∂ 2 u ∂x 2 (t, x):

∂ 2 u ∂x 2 (t, x) = E f (X x T -t ( 1 2 ))J x T -t ( 1 2 ) exp
T -t 0 b (X x s ( 1 2
))ds

+ T -t 0 E ∂u ∂x (t + s, X x s ( 1 2 ))) exp s 0 b (X x u ( 1 2 ))du b (X x s ( 1 2 ))J x s ( 1 2
) ds.

We introduce the probability

Q 1 such that dQ 1 dP Ft = 1 Z ( 1 2 ,1) t . Then, ∂ 2 u ∂x 2 (t, x) = E 1 Z ( 1 2 ,1) T -t f (X x T -t ( 1 2 ))J x T -t ( 1 2 ) exp T -t 0 b (X x s ( 1 2
))ds

+ T -t 0 E 1 Z ( 1 2 ,1) s ∂u ∂x (t + s, X x s ( 1 2 ))) exp s 0 b (X x u ( 1 2 ))du b (X x s ( 1 2 ))J x s ( 1 2 ) ds.
Again for all θ ∈ [0, T ], we have that Z

( 1 2 ,1) θ J x θ ( 1 2 ) = exp θ 0 b (X x u ( 1 2
))du and

∂ 2 u ∂x 2 (t, x) =E 1 f (X x T -t ( 1 2 )) exp 2 T -t 0 b (X x s ( 1 2
))ds

+ T -t 0 E 1 ∂u ∂x (t + s, X x s ( 1 2 ))) exp 2 s 0 b (X x u ( 1 2 ))du b (X x s ( 1 2
)) ds.

As L Q 1 (X x . ( 1 2 )) = L P (X x . ( 1
)), we finally obtain the following expression for ∂ 2 u ∂x 2 (t, x):

∂ 2 u ∂x 2 (t, x) = E f (X x T -t (1)) exp 2 T -t 0 b (X x s (1))ds + T -t 0 E ∂u ∂x (t + s, X x s (1))) exp 2 s 0 b (X x u (1))du b (X x s (1)) ds (3.13) from which, we deduce that u ∈ C 1,2 ([0, T ] × [0, +∞)). As J x s (1) = dX x s (1) dx
exists and is given by (3.9), for x > 0, ∂ 2 u ∂x 2 (t, x) is continuously differentiable (see again Proposition 3.4) and In view of (3.4) and (3.5), one can adapt easily the proof of the Theorem 6.1 in [START_REF] Friedman | Stochastic Differential Equations and Applications[END_REF] and show that u(t, x) solves the Cauchy problem (3.6).

∂ 3 u ∂x 3 (t, x) = E exp 2 T -t 0 b (X x s (1))ds × f (3) (X x T -t (1))J x T -t (1) + 2f (X x T -t (1)) T -t 0 b (X x s (1))J x s (1)ds + T -t 0 E exp 2 s 0 b (X x u (1)du × J x s (1) ∂ 2 u ∂x 2 (t + s, X x s (1)))b (X x s (1)) + ∂u ∂x (t + s, X x s (1)))b (3) (X x s (1)) +2 ∂u ∂x (t + s, X x s (1)))b (X x s (1)) s 0 b (X x u (1))J x u (

On the approximation process

According to (2.1) and (2.4), the discrete time process (X) associated to (X) is

   X 0 = x 0 , X t k+1 = X t k + b(X t k )∆t + σ X t k (W t k+1 -W t k ) , (3.15)
k = 0, . . . , N -1, and the time continuous version (X t , 0 ≤ t ≤ T ) satisfies

X t = x 0 + t 0 sgn(Z s )b(X η(s) )ds + σ t 0 sgn(Z s ) X η(s) dW s + 1 2 L 0 t (X), (3.16)
where sgn(x) = 1 -2 l l (x≤0) , and for any t ∈ [0, T ],

Z t = X η(t) + (t -η(t))b(X η(t) ) + σ X η(t) (W t -W η(t) ).
(3.17)

In this section, we are interested in the behavior of the processes (X) and (Z) visiting the point 0. The main result is the following Proposition 3.5. Let α = 1 2 . Assume (H1). For ∆t sufficiently small (∆t ≤ 1/(2K)), there exists a constant C > 0, depending on b(0), K, σ, x 0 and T but not in ∆t, such that

E L 0 t (X) -L 0 η(t) (X) F η(t) ≤ C∆t exp - X η(t) 16σ 2 ∆t and EL 0 T (X) ≤ C ∆t x 0 b(0) σ 2 . (3.18)
The upper bounds above, for the local time (L 0 • (X)), are based on the following technical lemmas: Lemma 3.6. Assume (H1). Assume also that ∆t is sufficiently small (∆t ≤ 1/(2K) ∧ x 0 ). Then for any γ ≥ 1, there exists a positive constant C, depending on all the parameters b(0), K, σ, x 0 , T and also on γ, such that

sup k=0,...,N E exp - X t k γσ 2 ∆t ≤ C ∆t x 0 2b(0) σ 2 (1-1 2γ )
.

Lemma 3.7. Assume (H1). For ∆t sufficiently small (∆t ≤ 1/(2K)), for any t ∈ [0, T ],

P Z t ≤ 0 X η(t) ≤ 1 2 exp - X η(t) 2(1 -K∆t) -2 σ 2 ∆t .
As 2(1 -K∆t) -2 > 1 when ∆t ≤ 1/(2K), the combination of Lemmas 3.6 and 3.7 leads to

P Z t ≤ 0 ≤ C ∆t x 0 b(0) σ 2 . (3.19)
We give successively the proofs of Lemmas 3.6, 3.7 and Proposition 3.5.

Proof of Lemma 3.6. First, we show that there exits a positive sequence (µ j , 0 ≤ j ≤ N ) such that, for any k ∈ {1, . . . , N },

E exp - X t k γσ 2 ∆t ≤ exp   -b(0) k-1 j=0 µ j ∆t   exp (-x 0 µ k ) .
We set µ 0 = 1 γσ 2 ∆t . By (3.15), as -b(x) ≤ -b(0) + Kx, for all x ∈ R, we have that

E exp -µ 0 X t k ≤ E exp -µ 0 X t k-1 + (b(0) -KX t k-1 )∆t + σ X t k-1 ∆W t k . ∆W t k and X t k-1 being independent, E exp(-µ 0 σ X t k-1 ∆W t k ) = E exp( σ 2 2 µ 2 0 ∆tX t k-1 ). Thus E exp -µ 0 X t k ≤ exp (-µ 0 b(0)∆t) E exp -µ 0 X t k-1 1 -K∆t - σ 2 2 µ 0 ∆t = exp (-µ 0 b(0)∆t) E exp -µ 1 X t k-1 ,
where we set µ 1 = µ 0 (1 -K∆t -σ 2 2 µ 0 ∆t). Consider now the sequence (µ j ) i∈N defined by

µ 0 = 1 γσ 2 ∆t , µ j = µ j-1 1 -K∆t -σ 2 2 µ j-1 ∆t , j ≥ 1.
(3.20)

An easy computation shows that if γ ≥ 1 and ∆t ≤ 1 2K , then (µ j ) is a positive and decreasing sequence. For any j ∈ {0, . . . , k -1}, by the same computation we have

E exp -µ j X t k-j ≤ exp (-b(0)µ j ∆t) E exp -µ j+1 X t k-j-1
and it follows by induction that

E exp -µ 0 X t k ≤ exp   -b(0) k-1 j=0 µ j ∆t   exp (-x 0 µ k ) . (3.21)
Now, we study the sequence (µ j , 0 ≤ j ≤ N ). For any α > 0, we consider the non-decreasing function

f α (x) := x 1+αx , x ∈ R. We note that (f α • f β )(x) = f α+β (x).
Then, for any j ≥ 1, the sequence (µ j ) being decreasing, µ j ≤ µ j-1 -σ 2 2 ∆tµ j µ j-1 , and

µ j ≤ f σ 2 2 ∆t (µ j-1 ) ≤ f σ 2 2 ∆t f σ 2 2 ∆t (µ j-2 ) ≤ . . . ≤ f σ 2 2 j∆t (µ 0 ). (3.22)
The next step consists in proving, by induction, the following lower bound for the µ j :

µ j ≥ µ 1 1 1 + σ 2 2 ∆t(j -1)µ 0 -K ∆t(j -1)µ 0 1 + σ 2 2 ∆t(j -1)µ 0 , ∀j ≥ 1. ( 3
µ j+1 = µ j 1 - σ 2 2 ∆tµ j -K∆tµ j ≥µ j 1 - σ 2 2 ∆tf σ 2 2 j∆t (µ 0 ) -K∆tf σ 2 2 j∆t (µ 0 ) ≥µ j 1 + σ 2 2 ∆t(j -1)µ 0 1 + σ 2 2 ∆tjµ 0 -K ∆tµ 0 1 + σ 2
2 ∆tjµ 0 and we conclude by using (3.23) for µ j . Now, we replace µ 0 by its value 1 γσ 2 ∆t in (3.23) and obtain that µ j ≥ 2γ-1 ∆tγσ 2 (2γ-1+j) -2K σ 2 , for any j ≥ 0. Hence,

k-1 j=0 ∆tµ j ≥ 1 γσ 2 k-1 j=0 2γ -1 2γ -1 + j - 2Kt k σ 2 ≥ 1 γσ 2 k 0 2γ -1 2γ -1 + u du - 2KT σ 2 ≥ 2γ -1 γσ 2 ln 2γ -1 + k 2γ -1 - 2KT σ 2 .
Coming back to (3.21), we obtain that

E exp - X t k γσ 2 ∆t ≤ exp b(0) 2KT σ 2 exp x 0 2K σ 2 × 2γ -1 2γ -1 + k 2b(0) σ 2 (1-1 2γ ) exp - x 0 ∆tγσ 2 (2γ -1) (2γ -1 + k) .
Finally, we use the inequality x α exp(-x) ≤ α α exp(-α), for all α > 0 and x > 0. It comes that

E exp - X t k γσ 2 ∆t ≤ exp b(0) 2KT σ 2 exp x 0 2K σ 2 × 2b(0) ∆tγ x 0 (1 - 1 2γ ) 2b(0) σ 2 (1-1 2γ ) exp - 2b(0) σ 2 (1 - 1 2γ
)

≤ C ∆t x 0 2b(0) σ 2 (1-1 2γ )
where we set

C = (b(0)(2γ -1)) 2b(0) σ 2 (1-1 2γ ) exp 2 σ 2 b(0)(KT -1 + 1 2γ ) + x 0 K .
Proof of Lemma 3.7. Under (H1), b(x) ≥ b(0) -Kx, for x ≥ 0. Then, by the definition of (Z) in (3.17),

P Z t ≤ 0 ≤ P   W t -W η(t) ≤ -X η(t) (1 -K(t -η(t))) -b(0)(t -η(t)) σ X η(t) , X η(t) > 0   .
By using the Gaussian inequality P(G ≤ β) ≤ 1/2 exp(-β 2 /2), for a standard Normal r.v. G and β < 0, we get

P Z t ≤ 0 ≤ 1 2 E exp - (X η(t) (1 -K(t -η(t))) + b(0)(t -η(t))) 2 2σ 2 (t -η(t))X η(t) l l {X η(t) >0}
from which we finally obtain that

P Z t ≤ 0 ≤ 1 2 E[exp(- X η(t) 2(1 -K∆t) -2 σ 2 ∆t )].
Proof of Proposition 3.5. By the occupation time formula, for any t ∈ [t k , t k+1 ), 0 ≤ k ≤ N , for any bounded Borel-measurable function φ, P a.s

R φ(z) L z t (X) -L z t k (X) dz = R φ(z) L z t (Z) -L z t k (Z) dz = t t k φ(Z s )d Z, Z s = σ 2 t t k φ(Z s )X t k ds.
Hence, for any x > 0, an easy computation shows that

R φ(z)E L z t (X) -L z t k (X) X t k = x dz = σ 2 t t k xE φ(Z s ) X t k = x ds = σ R φ(z) t t k √ x 2π(s -t k ) exp - (z -x -b(x)(s -t k )) 2 2σ 2 x(s -t k ) ds dz.
Then, for any

z ∈ R, E L z t (X) -L z t k (X) X t k = x = σ t t k √ x 2π(s -t k ) exp - (z -x -b(x)(s -t k )) 2 2σ 2 x (s -t k ) ds.
In particular for z = 0 and t = t k+1 ,

E L 0 t k+1 (X) -L 0 t k (X) X t k = x = σ ∆t 0 √ x √ 2πs exp - (x + b(x)s) 2 2σ 2 x s ds.
From (H1), b(x) ≥ -Kx, with K ≥ 0. Then,

E L 0 t k+1 (X) -L 0 t k (X) F t k ≤ σ ∆t 0 X t k √ 2πs exp - X t k (1 -Ks) 2 2σ 2 s ds.
For ∆t sufficiently small, 1 -K∆t ≥ 1/2 and

E L 0 t k+1 (X) -L 0 t k (X) F t k ≤ σ ∆t 0 X t k √ 2πs exp - X t k 8σ 2 ∆t ds.
Now we use the upper-bound a exp(-

a 2 2 ) ≤ 1, ∀a ∈ R, to get E L 0 t k+1 (X) -L 0 t k (X) ≤ σ 2 ∆t 0 2 √ ∆t √ πs E exp - X t k 16σ 2 ∆t ds ≤ 4σ 2 ∆t √ π sup k=0,...,N E exp - X t k 16σ 2 ∆t .
We sum over k and apply the Lemma 3.6 to end the proof.

Proof of Theorem 2.3

We are now in position to prove the Theorem 2.3. To study the weak error Ef (X T )-Ef (X T ), we use the Feynman-Kac representation of the solution u(t, x) to the Cauchy problem (3.6), studied in the Proposition 3.2: for all (t, x) ∈ [0, T ] × (0, +∞), Ef (X x T -t ) = u(t, x). Thus the weak error becomes

Ef (X T ) -Ef (X T ) = E u(0, x 0 ) -u(T, X T )
with (X) satisfying (3.16). Applying the Itô's formula a first time, we obtain that

E u(0, x 0 ) -u(T, X T ) = - T 0 E ∂u ∂t (s, X s ) + sgn(Z s )b(X η(s) ) ∂u ∂x (s, X s ) + σ 2 2 X η(s) ∂ 2 u ∂x 2 (s, X s ) ds -E T 0 sgn(Z s )σ X η(s) ∂u ∂x (s, X s )dW s -E T 0 1 2 ∂u ∂x (s, X s )dL 0 (X) s .
From Proposition 3.2 and Lemma 2.1, we easily check that the stochastic integral (

• 0 sgn(Z s ) X η(s) ∂u ∂x (s, X s )dW s )
is a martingale. Furthermore, we use the Cauchy problem (3.6) to get

E u(0, x 0 ) -u(T, X T ) = - T 0 E b(X η(s) ) -b(X s ) ∂u ∂x (s, X s ) + σ 2 2 X η(s) -X s ∂ 2 u ∂x 2 (s, X s ) ds -E T 0 1 2 ∂u ∂x (s, X s )dL 0 (X) s + T 0 2E l l {Zs≤0} b(X η(s) ) ∂u ∂x (s, X s ) ds.
From Proposition 3.5,

E T 0 ∂u ∂x (s, X s )dL 0 (X) s ≤ ∂u ∂x ∞ E L 0 T (X) ≤ C ∆t x 0 b(0) σ 2
.

On the other hand, by Lemma 3.7 for any s

∈ [0, T ], 2E l l {Zs≤0} b(X η(s) ) ∂u ∂x (s, X s ) ≤ ∂u ∂x ∞ E (b(0) + KX η(s) ) exp - X η(s) 8σ 2 ∆t .
As for any x ≥ 0, x exp(-x 16σ 2 ∆t ) ≤ 16σ 2 ∆t, we conclude, by Lemma 3.6, that

T 0 2E l l {Zs≤0} b(X η(s) ) ∂u ∂x (s, X s ) ds ≤ C ∆t x 0 b(0) σ 2 .
Hence,

E u(0, x 0 ) -u(T, X T ) ≤ T 0 E b(X η(s) ) -b(X s ) ∂u ∂x (s, X s ) + σ 2 2 X η(s) -X s ∂ 2 u ∂x 2 (s, X s ) ds + C ∆t x 0 b(0) σ 2
.

By applying the Itô's formula a second time ( ∂u ∂x is a C 3 function with bounded derivatives),

E b(X s ) -b(X η(s) ∂u ∂x (s, X s ) = E s η(s) sgn(Z θ )b(X η(s) ) b(X θ ) -b(X η(s) ∂ 2 u ∂x 2 (s, X θ ) + b (X θ ) ∂u ∂x (s, X θ ) dθ + E s η(s) σ 2 2 X η(s) b(X θ ) -b(X η(s) ∂ 3 u ∂x 3 (s, X θ ) + 2b (X θ ) ∂ 2 u ∂x 2 (s, X θ ) + b (X θ ) ∂u ∂x (s, X θ ) dθ + E s η(s) 1 2 b(0) -b(X η(s) ∂ 2 u ∂x 2 (s, 0) + b (0) ∂u ∂x (s, 0) dL 0 θ (X) so that E b(X s ) -b(X η(s) ∂u ∂x (s, X s ) ≤ C∆t 1 + sup 0≤θ≤T E|X θ | 2 + E (1 + |X η(s) |) L 0 s (X) -L 0 η(s) (X)
and we conclude by Lemma 2.1 and Proposition 3.5 that

E b(X s ) -b(X η(s) ∂u ∂x (s, X s ) ≤ C   ∆t + ∆t x 0 b(0) σ 2   .
By similar arguments, we show that

E X s -X η(s) ∂ 2 u ∂x 2 (s, X s ) ≤ C   ∆t + ∆t x 0 b(0) σ 2  
which ends the proof of Theorem 2.3.

4 The case of processes with 1/2 < α < 1

Preliminary results

In this section, (X t ) denotes the solution of (2.9) starting at x 0 at time 0 and (X x t ), starting at x ≥ 0 at time 0, is the unique strong solution to

X x t = x + t 0 b(X x s )ds + σ t 0 (X x s ) α dW s . (4.1)

On the exact solution

We give some upper-bounds on inverse moments and exponential moments of (X t ).

Lemma 4.1. Assume (H1). Let x > 0. For any 1/2 < α < 1, for any p > 0, there exists a positive constant C, depending on the parameters of the model (4.1) and on p such that

sup t∈[0,T ] E (X x t ) -p ≤ C(1 + x -p ).
Proof. Let τ n be the stopping time defined by τ n = inf{0 < s ≤ T ; X x s ≤ 1/n}. By the Itô's formula,

E (X x t∧τn ) -p =x -p -pE t∧τn 0 b(X x s )ds (X x s ) p+1 + p(p + 1) σ 2 2 E t∧τn 0 ds (X x s ) p+2(1-α) ≤x -p + pK t 0 E 1 (X x s∧τn ) p ds + E t∧τn 0 p(p + 1) σ 2 2 1 (X x s ) p+2(1-α) -p b(0) (X x
s ) p+1 ds . It is possible to find a positive constant C such that, for any x > 0,

p(p + 1) σ 2 2 1 x p+2(1-α) -p b(0) x p+1 ≤ C.

An easy computation shows that

C = p (2α -1) σ 2 2 (p + 2(1 -α)) σ 2 2b(0) p+2(1-α) 2α-1
is the smallest one satisfying the upper-bound above. Hence,

E (X x t∧τn ) -p ≤ x -p + CT + pK t 0 sup θ∈[0,s] E (X x θ∧τn ) -p ds
and by the Gronwall Lemma

sup t∈[0,T ] E (X x t∧τn ) -p ≤ x -p + CT exp(pKT ).
We end the proof, by taking the limit n → +∞.

Lemma 4.2. Assume (H1).

(i) For any a ≥ 0, for all 0 ≤ t ≤ T , a.s. (X x t ) 2(1-α) ≥ r t (a), where (r t (a), 0 ≤ t ≤ T ) is the solution of the CIR Equation:

r t (a) = x 2(1-α) + t 0 (a -λ(a)r s (a))ds + 2σ(1 -α) t 0 r s (a)dW s with λ(a) = 2(1 -α)K + (2α -1) 2α-1 a + σ 2 (1 -α)(2α -1) b(0) 2(1-α) 1 2α-1 . ( 4.2)
(ii) For all µ ≥ 0, there exists a constant C(T, µ) > 0 with a non decreasing dependency on T and µ, depending also on K, b(0), σ, α and x such that

E exp µ T 0 ds (X x s ) 2(1-α) ≤ C(T, µ). (4.3)
(iii) The process (M x t , 0 ≤ t ≤ T ) defined by

M x t = exp ασ t 0 dW s (X x s ) 1-α -α 2 σ 2 2 t 0 ds (X x s ) 2(1-α) (4.4)
is a martingale. Moreover for all p ≥ 1, there exists a positive constant C(T, p) depending also on b(0), σ and α, such that

E sup t∈[0, T ] (M x t ) p ≤ C(T, p) 1 + 1 x αp . (4.5)
Proof. Let Z t = (X x t ) 2 (1-α) . By the Itô's formula,

Z t = x 2(1-α) + t 0 β(Z s )ds + 2(1 -α)σ t 0 Z s dW s ,
where, for all x > 0, the drift coefficient β(x) is defined by

β(x) = 2(1 -α)b(x 1 2(1-α) )x -2α-1 2(1-α) -σ 2 (1 -α)(2α -1).
From (H1), b(x) ≥ b(0) -Kx and for all x > 0, β(x) ≥ β(x), where we set

β(x) = 2(1 -α)b(0)x -2α-1 2(1-α) -2(1 -α)Kx -σ 2 (1 -α)(2α -1).
For all a ≥ 0 and λ(a) given by (4.2), we consider f (x) = β(x) -a + λ(a)

x. An easy computation shows that f (x) has one minimum at the point

x = b(0)(2α-1) λ(a)-2(1-α)K 2(1-α) . Moreover, f (x ) = b(0) 2(1-α) (2α -1) 2α-1 (λ(a) -2(1 -α)K) 2α-1 -a + σ 2 (1 -α)(2α -1)
and when λ(a) is given by (4.2), f (x ) = 0. We conclude that β(x) ≥ a-λ(a)x and (i) holds by the Comparison Theorem for the solutions of one-dimensional SDE. As a consequence,

E exp µ T 0 ds (X x s ) 2(1-α) ≤ E exp µ T 0 ds r s (a)
.

We want to apply the Lemma A.2, on the exponential moments of the CIR process. To this end, we must choose the constant a such that a ≥ 4(1-α) 2 σ 2 and µ ≤ ν 2 (a)(1-α) 2 4σ 2

8

, for ν(a) as in Lemma A.2. An easy computation shows that a = 4(1-α)

2 σ 2 ∨ 2(1 -α) 2 σ 2 + (1 -α)σ2
√ 2µ is convenient and (ii) follows by applying the Lemma A.2 to the process (r t (a), 0 ≤ t ≤ T ). Thanks to (ii), the Novikov criteria applied to M x t is clearly satisfied. Moreover, by the integration by parts formula.

M x t = X x t x α exp t 0 -α b(X x s ) X x s + α(1 -α) σ 2 2 1 (X x s ) 2(1-α) ds ≤ X x t x α exp(KT ) exp t 0 -α b(0) X x s + α(1 -α) σ 2 2
1 (X x s ) 2(1-α) ds . To end the proof, notice that it is possible to find a positive constant λ such that, for any

x > 0, -b(0)α x + σ 2 α(1-α) 2 1 x 2(1-α) ≤ λ. An easy computation shows that λ = α 2 (2α -1) (1 -α) 3-2α σ 2 b(0) 2(1-α) 1 2α-1 . is convenient. Thus, M x t ≤ X x t x α
exp ((K + λ)T ) and we conclude by using the Lemma 2.1.

On the associated

u L ∞ ([0,T ]×[0,+∞)) + ∂u ∂x L ∞ ([0,T ]×[0,+∞)) ≤ C
and for all x > 0, sup

t∈[0,T ] ∂u ∂t (t, x) ≤ C(1 + x 2α ),
and sup

t∈[0,T ] 4 k=2 ∂ k u ∂x k (t, x) ≤ C 1 + 1 x q(α) ,
where the constant q(α) > 0 depends only on α. Moreover, u(t, x) satisfies

   ∂u ∂t (t, x) + b(x) ∂u ∂x (t, x) + σ 2 2 x 2α ∂ 2 u ∂x 2 (t, x) = 0, (t, x) ∈ [0, T ] × (0, +∞), u(T, x) = f (x), x ∈ [0, +∞). (4.6)
The following Proposition 4.4 allows us to compute the derivatives of u(t, x). Equation (4.1) has locally Lipschitz coefficients on (0, +∞), with locally Lipschitz first order derivatives. Then X x t is continuously differentiable and if we denote J x t = dX x t dx , the process (J x t , 0 ≤ t ≤ T ) satisfies the linear equation

J x t = 1 + t 0 J x s b (X x s )ds + t 0 ασJ x s dW s (X x s ) 1-α . (4.7)
Proposition 4.4. Assume (H1) and (H2). Let g(x), h(x) and k(x) be some C 1 functions on (0, +∞) such that, there exist p 1 > 0 and p 2 > 0,

∀x > 0, |g(x)| + |g (x)| + |h(x)| + |h (x)| + |k (x)| ≤ C 1 + x p1 + 1 x p 2 , |k(x)| ≤ C 1 + 1 x 2(1-α) .
Let v be the R-valued function defined on [0, T ] × (0, +∞) by

v(t, x) = E g(X x t ) exp( t 0 k(X x s )ds) + t 0 E h(X x s ) exp( s 0 k(X x θ )dθ) ds.
Then v(t, x) is of class C 1 with respect to x and

∂v ∂x (t, x) =E exp( t 0 k(X x s )ds) g (X x t )J x t + g(X x t ) t 0 k (X x s )J x s ds + t 0 E exp( s 0 k(X x θ )dθ) h (X x s )J x s + h(X x s ) s 0 k (X x θ )J x θ dθ ds.
The proof is postponed in the Appendix B.

Proof of Proposition 4.3. Many arguments are similar to those of the proof of Proposition 3.2. Here, we restrict our attention on the main difficulty which consists in obtaining the upper bounds for the spatial derivatives of u(t, x) up to the order 4. By Lemma 4.1, for

Bossy & Diop

x > 0, (

t 0 dWs (X x s ) 1-α , 0 ≤ t ≤ T
) is a locally square integrable martingale. Then J x t is given by

J x t = exp t 0 b (X x s )ds + ασ t 0 dW s (X x s ) 1-α - σ 2 α 2 2 t 0 ds (X x s ) 2(1-α)
Or equivalently

J x t = exp t 0 b (X x s )ds M t ,
∂u ∂x (t, x) = E f (X x T -t )J x T -t . Then, | ∂u ∂x (t, x)| ≤ f ∞ exp(KT ).
The integration by parts formula gives

J x t = (X x t ) α x α exp t 0 b (X x s ) - αb(X x s ) X x s + σ 2 α(1 -α) 2(X x s ) 2(1-α) ds .
We apply again the Proposition 4.4 to compute ∂ 2 u ∂x 2 (t, x): for any x > 0,

dJ x t dx = - αJ x t x + α(J x t ) 2 X x t + J x t t 0 b (X x s ) - αb (X x s ) X x s + αb(X x s ) (X x s ) 2 - σ 2 α(1 -α) 2 (X x s ) 3-2α J x s ds
and

∂ 2 u ∂x 2 (t, x) = E f (X x T -t )(J x T -t ) 2 - α x ∂u ∂x (t, x) + αE (J x T -t ) 2 X x T -t f (X x T -t ) +E f (X x T -t )J x T -t T -t 0 b (X x s ) - αb (X x s ) X x s + αb(X x s ) (X x s ) 2 - σ 2 α(1 -α) 2 (X x s ) 3-2α
J x s ds .

(4.9)

By using the Cauchy-Schwarz Inequality with Lemma 4.1 and estimate (4.8), the second term on the right-hand side is bounded by

f ∞ E sup t∈[0,T ] (J x t ) 2 T -t 0 b (X x s ) - αb (X x s ) X x s + αb(X x s ) (X x s ) 2 - σ 2 α(1 -α) 2 (X x s ) 3-2α ds ≤ C(T ) 1 + 1 x 2(1+α) .
INRIA By using similar arguments, it comes that

∂ 2 u ∂x 2 (t, x) ≤ C(T ) 1 + 1 x 2+2α .
We apply again the Proposition 4.4 to compute ∂ 3 u ∂x 3 (t, x) from (4.9) and next ∂ 4 u ∂x 4 (t, x), the main difficulty being the number of terms to write. In view of the expression of dJ x s dx , each term can be bounded by C(T )(1 + x -2(n-1)-nα ), where n is the derivation order, by using the Cauchy-Schwarz Inequality and the upper-bounds E sup

t∈[0,T ] (J x t ) p ≤ C(T )(1 + x -αp ) and sup t∈[0,T ] E(X x t ) -p ≤ C(1 + x -p ).

On the approximation process

When 1/2 < α < 1, according to (2.1) and (2.4), the discrete time process (X) associated to (X) is

X 0 = x 0 , X t k+1 = X t k + b(X t k )∆t + σX α t k (W t k+1 -W t k ) , k = 0, . . . , N -1,
Its time continuous version (X t , 0 ≤ t ≤ T ) satisfies

X t = x 0 + t 0 sgn(Z s )b(X η(s) )ds + σ t 0 sgn(Z s )X α η(s) dW s + 1 2 L 0 t (X), (4.10)
where for any t ∈ [0, T ], we set

Z t = X η(t) + (t -η(t))b(X η(t) ) + σX α η(t) (W t -W η(t) ), (4.11)
so that, for all t ∈ [0, T ], X t = |Z t |.

In the sequel, we will use the following notation:

O exp (∆t) = C(T ) exp - C ∆t α-1 2 ,
where the positive constants C and C(T ) are independent of ∆t but can depend on α, σ and b(0). C(T ) is non-decreasing in T . The quantity O exp (∆t) decreases exponentially fast with ∆t.

In this section, we are interested in the behavior of the processes (X) or (Z) near 0. We work under the hypothesis (H3'): x 0 > b(0) √ 2 ∆t. We introduce the stopping time τ defined by

τ = inf s ≥ 0; X s < b(0) 2 ∆t . (4.12)
Under (H3'), we are able to control probabilities like P(τ ≤ T ). This is an important difference with the case α = 1/2.

Bossy & Diop

Lemma 4.5. Assume (H1), (H2) and (H3'). Then

P (τ ≤ T ) ≤ O exp (∆t). (4.13)
Proof. The first step of the proof consists in obtaining the following estimate:

∀k ∈ {0, . . . , N }, P X t k ≤ b(0) √ 2 ∆t ≤ O exp (∆t). (4.14)
Indeed, as b(x) ≥ b(0) -Kx for x ≥ 0, for k ≥ 1,

P X t k ≤ b(0) √ 2 ∆t ≤ P W t k -W t k-1 ≤ -X t k-1 (1 -K∆t) -b(0)(1 -1 √ 2 )∆t σX α t k-1 , X t k-1 > 0 .
As ∆t is sufficiently small, by using the Gaussian inequality P(G ≤ β) ≤ 1/2 exp(-β 2 /2), for a standard Normal r.v. G and β < 0, we get

P X t k ≤ b(0) √ 2 ∆t ≤ E   exp   - X t k-1 (1 -K∆t) + b(0)(1 -1 √ 2 )∆t 2 2σ 2 X 2α t k-1 ∆t    l l {Xt k-1 >0}    ≤ E   exp   - X 2(1-α) t k-1 8σ 2 ∆t   exp - b(0)(1 -1 √ 2 ) 2σ 2 X 2α-1 t k-1 l l {Xt k-1 >0}   .
By separating the events X t k-1 ≥ √ ∆t and X t k-1 < √ ∆t in the expectation above, we obtain

P X t k ≤ b(0) √ 2 ∆t ≤ exp - 1 8σ 2 ∆t α + exp - b(0)(1 -1 √ 2 ) 2σ 2 (∆t) α-1 2 = O exp (∆t).
Now we prove (4.13). Notice that

P (τ ≤ T ) ≤ N -1 k=0 P inf t k <s≤t k+1 Z s ≤ b(0) 2 ∆t , X t k > b(0) 2 ∆t .
For each k ∈ {0, 1, . . . , N -1}, by using (4.14) and b(x) ≤ b(0) -Kx, we have

P inf t k <s≤t k+1 Z s ≤ b(0) 2 ∆t, X t k > b(0) 2 ∆t = P inf t k <s≤t k+1 Z s ≤ b(0) 2 ∆t, X t k > b(0) 2 ∆t, X t k ≤ b(0) √ 2 ∆t + P inf t k <s≤t k+1 Z s ≤ b(0) 2 ∆t, X t k > b(0) √ 2 ∆t ≤ P X t k ≤ b(0) √ 2 ∆t + P inf t k <s≤t k+1 Z s ≤ b(0) 2 ∆t, X t k > b(0) √ 2 ∆t ≤ O exp (∆t) + E l l Xt k > b(0) √ 2 ∆t P inf 0<s≤∆t x 1-α σ + b(0) -Kx σx α s + B s ≤ b(0)∆t 2σx α x=Xt k
, where (B t ) denotes a Brownian motion independent of (W t ). The proof is ended if we show that

P inf 0<s≤∆t x 1-α σ + (b(0) -Kx) σx α s + B s ≤ b(0)∆t 2σx α ≤ O exp (∆t), for x ≥ b(0) √ 2 ∆t.
We use the following formula (see [START_REF] Borodin | Handbook of Brownian Motion : Facts and Formulae. Probability and its applications[END_REF]): if (B µ t , 0 ≤ t ≤ T ) denotes a Brownian motion with drift µ, starting at y 0 , then for all y ≤ y 0 ,

P inf 0<s<t B µ s ≤ y = 1 2 erfc y 0 -y √ 2t + µ √ t √ 2 + 1 2 exp(2µ(y -y 0 ))erfc y 0 -y √ 2t - µ √ t √ 2 ,
where erfc(z) =

√ 2 √ π ∞ √ 2z exp -y 2 2
dy, for all z ∈ R. We set µ = (b(0)-Kx)

σx α , y 0 = x 1-α σ and we choose y = b(0)∆t 2σx α satisfying y ≤ y 0 , if x > b(0) √ 2 ∆t. Then P inf 0<s≤∆t x 1-α σ + (b(0) -Kx) σx α s + W s ≤ b(0)∆t 2σx α = 1 2 erfc x -b(0)∆t 2 σx α √ 2∆t + (b(0) -Kx) √ ∆t σx α √ 2 + 1 2 exp - 2(b(0) -Kx) σ 2 x 2α x - b(0)∆t 2 erfc x -b(0)∆t 2 σx α √ 2∆t - (b(0) -Kx) √ ∆t σx α √ 2 := A(x) + B(x).
For any z ≥ 0, erfc(z) ≤ exp(-z 2 ). Then, if ∆t is sufficiently small, we have

A(x) ≤ exp - (x(1 -K∆t) + b(0) 2 ∆t) 2 2σ 2 x 2α ∆t ≤ exp - x 2(1-α) 8σ 2 ∆t Bossy & Diop and, for x ≥ b(0) √ 2 ∆t, A(x) ≤ exp -2 α b(0) 2(1-α) 16σ 2 ∆t 2α-1 = O exp (∆t). Now we consider B(x). If x ≥ 3 2 b(0)∆t (1+K∆t)
, then as for A(x), we have

B(x) ≤ exp - 2(b(0) -Kx) σ 2 x 2α x - b(0)∆t 2 exp - (x -b(0)∆t 2 -(b(0) -Kx)∆t) 2 σ 2 x 2α 2∆t = exp - (x(1 -K∆t) + b(0)∆t 2 ) 2 σ 2 x 2α 2∆t ≤ exp - x 2(1-α) 8σ 2 ∆t and B(x) ≤ exp -2 α b(0) 2(1-α) 16σ 2 ∆t 2α-1 = O exp (∆t), for x ≥ 3 2 b(0)∆t (1+K∆t) . If b(0) √ 2 ∆t ≤ x < 3 2 b(0)∆t (1+K∆t) , then 2(b(0)-Kx) σ 2 x 2α (x -b(0)∆t 2 ) ≥ b(0) 2 ∆t( 1 √ 2 -1 2 ) σ 2 x 2α and B(x) ≤ exp - 2(b(0) -Kx) σ 2 x 2α x - b(0)∆t 2 ≤ exp - b(0) 2 ∆t( 1 √ 2 -1 2 ) σ 2 x 2α . For x ≥ b(0) √ 2 ∆t, we get B(x) ≤ exp - 2 2α b(0) 2(1-α) ( 1 √ 2 -1 2 )(1+K∆t) 2α 3 2α σ 2 (∆t) 2α-1 = O exp (∆t).
Lemma 4.6. Assume (H1), (H2) and (H3'). Let τ be the stopping time defined in (4.12). For all p ≥ 0, there exists a positive constant C depending on b(0), σ, α, T and p but not on ∆t, such that

∀t ∈ [0, T ], E 1
Z p t∧τ ≤ C 1 + 1 x p 0 . (4.15)
Proof. First, we prove that

∀t ∈ [0, T ], P Z t ≤ X η(t) 2 ≤ O exp (∆t). (4.16)
Indeed, while proceeding as in the proof of Lemma 4.5, we have

P Z t ≤ X η(t) 2 ≤ E exp - X η(t) (1 -2K(t -η(t))) + 2b(0)(t -η(t)) 2 8σ 2 (t -η(t))X 2α η(t)
.

By using (a + b) 2 ≥ a 2 + 2ab, with a = X η(t) (1 -2K(t -η(t))) and b = 2b(0)(t -η(t)), P Z t ≤ X η(t) 2 ≤ E   exp   - X 2(1-α) η(t) (1 -2K∆t) 2 8σ 2 ∆t   exp - b(0)(1 -2K∆t) 2σ 2 X 2α-1 η(t)   .
For ∆t sufficiently small,

P Z t ≤ X η(t) 2 ≤ E   exp   - X 2(1-α) η(t) 32σ 2 ∆t   exp - b(0) 4σ 2 X 2α-1 η(t)   .
By separating the events X η(t) ≥ √ ∆t and X η(t) < √ ∆t in the expectation above, we obtain

P Z t ≤ X η(t) 2 ≤ exp - 1 32σ 2 (∆t) α + exp - b(0) 4σ 2 (∆t) α-1 2 = O exp (∆t).
Now we prove (4.15). Notice that Z t∧τ = X t∧τ , by the Itô's formula 1

Z p t∧τ = 1 x p 0 -p t∧τ 0 b(X η(s) ) Z p+1 s ds -pσ t∧τ 0 X α η(s) Z p+1 s dW s + p(p + 1) σ 2 2 t∧τ 0 X 2α η(s) Z p+2 s ds.
Taking the expectation and using again b(x) ≥ b(0) -Kx, we have

E 1 Z p t∧τ ≤ 1 x p 0 -pE t∧τ 0 b(0) Z p+1 s ds + pKE t∧τ 0 X η(s) Z p+1 s ds + p(p + 1) σ 2 2 E t∧τ 0 X 2α η(s) Z p+2 s ds .
By the definition of τ in (4.12),

E t∧τ 0 X η(s) Z p+1 s ds = E t∧τ 0 l l (Zs≤ X η(s) 2 ) X η(s) Z p+1 s ds + E t∧τ 0 l l (Zs> X η(s) 2 ) X η(s) Z p+1 s ds ≤ 2 b(0)∆t p+1 T sup t∈[0, T ] P Z t ≤ X η(t) 2 1/2 sup t∈[0, T ] E X 2 η(t) 1/2 + 2 t 0 E 1 Z p s∧τ ds.
We conclude, by the Lemma 2.1 and the upper-bound (4.16) that

E t∧τ 0 X η(s) Z p+1 s ds ≤ C + 2 t 0 E 1 Z p s∧τ ds.
Similarly,

E t∧τ 0 X 2α η(s) Z p+2 s ds = E t∧τ 0 l l (Zs≤ X η(s) 2 ) X 2α η(s) Z p+2 s ds + E t∧τ 0 l l (Zs> X η(s) 2 ) X 2α η(s) Z p+2 s ds ≤ E t∧τ 0 l l (Zs≤ X η(s) 2 ) X 2α η(s) Z p+2 s ds + 2 2α E t∧τ 0 ds Z p+2(1-α) s
.

Bossy & Diop

By using again the Lemma 2.1 and the upper-bound (4.16), we have

E t∧τ 0 l l (Zs≤ X η(s) 2 ) X 2α η(s) Z p+2 s ds ≤ T 2 b(0)∆t p+2 sup t∈[0, T ]   P Z t ≤ X η(t) 2 E X 4α η(t)   ≤ T 2 b(0)∆t p+2 O exp (∆t) ≤ C.
Finally,

E 1 Z p t∧τ ≤ 1 x p 0 + E t∧τ 0 - pb(0) Z p+1 s + 2 2α-1 p(p + 1)σ 2 Z p+2(1-α) s ds + C t 0 E 1 Z p s∧τ ds + C.
We can easily check that there exists a positive constant C such that, for all z > 0, -pb(0)

z p+1 + p(p+1)2 2α-1 σ 2 z p+2(1-α) ≤ C. Hence E 1 Z p t∧τ ≤ 1 x p 0 + C t 0 E 1 Z p s∧τ ds + C
and we conclude by applying the Gronwall Lemma.

Proof of Theorem 2.5

As in the proof of Theorem 2.3, we use the Feynman-Kac representation of the solution of the Cauchy problem (4.6), studied in the Proposition 4.3: for all (t, x) ∈ [0, T ] × (0, +∞), Ef (X x T -t ) = u(t, x). Thus, the weak error becomes

Ef (X T ) -Ef (X T ) = E u(0, x 0 ) -u(T, X T ) .

Let τ be the stopping time defined in (4.12). By Lemma 4.5,

E u(T ∧ τ, Z T ∧τ ) -u(T, X T ) ≤ 2 u L ∞ ([0,T ]×[0,+∞]) P (τ ≤ T ) ≤ O exp (∆t).
We bound the error by

E u(T, X T ) -u(0, x 0 ) ≤ |E u(T ∧ τ, Z T ∧τ ) -u(0, x 0 ) | + O exp (∆t)
and we are now interested in E u(T ∧ τ, Z T ∧τ ) -(u(0, x 0 ) . Let L and L z the second order differential operators defined for any C 2 function g(x) by

Lg(x) = b(x) ∂g ∂x (x) + σ 2 2 x 2α ∂ 2 g ∂x 2 (x) and L z g(x) = b(z) ∂g ∂x (x) + σ 2 2 z 2α ∂ 2 g ∂x 2 (x).
From Proposition 4.3, u is in C 1,4 ([0, T ]×(0, +∞)) satisfying ∂u ∂s (t, x)+Lu(t, x) = 0. X t has bounded moments and the stopped process (X t∧τ )=(Z t∧τ ) has negative moments. Hence, applying the Itô's formula,

E u(T ∧ τ, X T ∧τ ) -u(0, x 0 ) = E T ∧τ 0 L X η(s) u -Lu (s, X s )ds, Notice that ∂ θ (L z u -Lu) + b(z)∂ x (L z u -Lu) + σ 2 2 z 2α ∂ 2 x 2 (L z u -Lu) = L 2 z u -2L z Lu + L 2 u
and by applying again the Itô's formula between η(s) and s to L X η(s) u -Lu (s, X s ),

E u(T ∧ τ, X T ∧τ ) -u(0, x 0 ) = T 0 s η(s) E l l (θ≤τ ) L 2 X η(s) u -2L X η(s) Lu + L 2 u (θ, X θ ) dθds. L 2 z u -2L z Lu + L 2 u (θ, x
) combines the derivatives of u up to the order four with b and its derivatives up to the order two and some power functions like the z 4α or x 2α-2 . When we value this expression at the point (z, x) = (X η(s∧τ ) , X θ∧τ ), with the upper bounds on the derivatives of u given in the Proposition 4.3 and the positive and negative moments of X given in the Lemmas 2.1 and 4.6, we get

E l l (θ≤τ ) L 2 X η(s) u -2L X η(s) Lu + L 2 u (θ, X θ ) ≤ C 1 + 1 x q(α) 0
which implies the result of Theorem 2.5.

A On the Cox-Ingersoll-Ross model

In [START_REF] Cox | A theory of the term structure of the interest rates[END_REF], Cox, Ingersoll and Ross proposed to model the dynamics of the short term interest rate as the solution of the following stochastic differential equation

dr x t = (a -br x t )dt + σ √ r x t dW t , r x 0 = x ≥ 0, (A.1)
where (W t , 0 ≤ t ≤ T ) is a one-dimensional Brownian motion on a probability space (Ω, F, P), a and σ are positive constants and b ∈ R. For any t ∈ [0, T ], let F t = σ(s ≤ t, W s ).

Lemma A.1. For any x > 0 and any p > 0,

E 1 (r x t ) p = 1 Γ(p) 2b σ 2 (1-e -bt ) p × 1 0 (2 -θ)θ p-1 (1 -θ) 2a σ 2 -p-1 exp -2bxθ σ 2 (e bt -1) dθ, (A.2)
where

Γ(p) = +∞ 0 u p-1 exp(-u)du, p > 0, denotes the Gamma function. Moreover, if a > σ 2 E 1 r x t ≤ e bt x (A.3)
and, for any p such that

1 < p < 2a σ 2 -1, E 1 (r x t ) p ≤ 1 Γ(p) 2e |b|t σ 2 t p or E 1 (r x t ) p ≤ C(p, T ) 1 x p , (A.4)
where C(p, T ) is a positive constant depending on p and T .

Proof. By the definition of the Gamma function, for all x > 0 and p > 0,

x -p = Γ(p) -1 +∞ 0 u p-1 exp(-ux)du, so that E 1 (r x t ) p = 1 Γ(p) +∞ 0 u p-1 E exp(-ur x t )du.
The Laplace transform of r x t is given by

E exp(-ur x t ) = 1 (2uL(t) + 1) 2a/σ 2 exp - uL(t)ζ(t, x) 2uL(t) + 1 ,
where L(t) = σ 2 4b (1 -exp(-bt)) and ζ(t, x) = 4xb σ 2 (exp(bt)-1) = xe -bt /L(t), (see e.g. [START_REF] Lamberton | Introduction to Stochastic Calculus Applied to Finance[END_REF]). Hence,

E 1 (r x t ) p = 1 Γ(p) +∞ 0 u p-1 (2uL(t) + 1) 2a/σ 2 exp - uL(t)ζ(t, x) 2uL(t) + 1 du.
By changing the variable θ = 2 uL(t) 2uL(t)+1 in the integral above, we obtain

E 1 (r x t ) p = 1 2 p Γ(p)L(t) p 1 0 (2 -θ)θ p-1 (1 -θ) 2a σ 2 -p-1 exp - xe -bt θ 2L(t)
dθ, from which we deduce (A.2). Now if a > σ 2 , we have for p = 1 where ν = 2a σ 2 -1 ≥ 0.

E 1 r x t ≤ 1 2L(t) 1 0 exp - xe -bt θ 2L(t) dθ ≤ e bt x and for 1 < p < 2a σ 2 -1, E 1 (r x t ) p ≤ 1 2 p Γ(p)L(t) p = 2 p |b| p σ 2p Γ(p)(1 -e -|b|t
Proof. For any t ∈ [0, T ], we set

H t = 2 σ √ r x t , so that E exp( ν 2 σ 2 8 t 0 ds r x s ) = E exp( ν 2 2 t 0 ds H 2 s
).

The process (H t , t ∈ [0, T ]) solves

dH t = 2a σ 2 - 1 2 dt H t - b 2 H t dt + dW t , H 0 = 2 σ √ x.
For any t ∈ [0, T ], we set

B t = H t -H 0 - t 0 2a σ 2 -1 2 ds
Hs . Let (Z t , t ∈ [0, T ]) defined by

Z t = exp - t 0 b 2 H s dB s - b 2 8 t 0 H 2 s ds .
By the Girsanov Theorem, under the probability Q such that dQ dP Ft

= 1 Zt , (B t , t ∈ [0, T ]) is a Brownian motion. Indeed (H t , t ∈ [0, T ]) solves dH t = 2a σ 2 - 1 2 dt H t + dB t , t ≤ T, H 0 = 2 σ √ x
and under Q, we note that (H t ) is a Bessel process with index ν = 2a σ 2 -1. Moreover, by the integration by parts formula,

t 0 2H s dB s = H 2 t -H 2 0 -4a σ 2 t and Z t = exp - b 4 H 2 t - b 2 8 t 0 H 2 s ds + b σ 2 x + t ba σ 2 ≤ exp b σ 2 x + T ba σ 2 . Now, denoting by E Q the expectation relative to Q, E exp ν 2 2 t 0 ds H 2 s = E Q exp ν 2 2 t 0 ds H 2 s Z t ≤ exp b σ 2 x + T ba σ 2 E Q exp ν 2 2 t 0 ds H 2 s . Let E (ν) 2 σ

√

x denotes the expectation relative to P (ν)

2 σ √
x , the law on C(R + , R + ) of the Bessel process with index ν, starting at 2 σ √

x. The next step uses the following change of probability measure, for ν ≥ 0 (see Proposition 2.4 in [START_REF] Geman | Bessel processes, asian options, and perpetuities[END_REF]).

P (ν) 2 σ √ x σ(Rs,s≤t) = σR t 2 √ x ν exp - ν 2 2 t 0 ds R 2 s P (0) 2 σ √ x σ(Rs,s≤t)
, where (R t , t ≥ 0) denotes the canonical process on C(R + , R + ). Then, we obtain that

E exp ν 2 2 t 0 ds H 2 s ≤ exp b σ 2 x + T ba σ 2 E (ν) 2 σ √ x exp ν 2 2 t 0 ds R 2 s ≤ exp b σ 2 x + T ba σ 2 E (0) 2 σ √ x σR t 2 √ x ν . It remains to compute E (0) 2 σ √ x σR t 2 √ x ν . Let (W 1 t , W 2 t , t ≥ 0) be a two dimensional Brow- nian motion. Then E (0) 2 σ √ x σR t 2 √ x ν = σ 2 √ x ν E (W 1 t ) 2 + (W 2 t + 2 √ x σ ) 2 ν 2
and an easy computation shows that E (0)

2 σ √ x σR t 2 √ x ν ≤ C(T ) 1 + x -ν 2 .
B Proofs of Propositions 3.4 and 4.4

Proof of Proposition 3.4. To simplify the presentation, we consider only the case when k(x) and h(x) are nil. For any > 0 and x > 0, we define for all t ∈ [0, T ], the process

J x, t = 1 (X x+ t -X x t ), satisfying J x, t = 1 + t 0 φ s J x, s ds + t 0 ψ s J x, s dW s , with φ s = 1 0 b (X x s + θ J x, t ) dθ and ψ s = 1 0 σdθ 2 √ X x s + θJ x, t
. Under (H3), the trajectories (X x t , 0 ≤ t ≤ T ) are strictly positive a.s. (see Remark 2.2). By Lemma A.1, t 0 ψ s dW s is a martingale. Then J x, t is explicitly given by

J x, t = exp t 0 φ s ds + t 0 ψ s dW s - 1 2 t 0 (ψ s ) 2 ds . We remark that t 0 σ 2 √ X x s dW s = 1 2 log X x t x - t 0 1 2 b(X x s ) X x s ds and J x, t ≤ C X x t x exp - t 0 1 2 b(X x s ) X x s ds - 1 2 t 0 (ψ s ) 2 ds + t 0 ψ s - σ 2 X x s dW s .
We upper-bound the moments E(J x, t ) α , α > 0. As b(x) ≥ b(0) -Kx, for any p,

(J x, t ) α ≤C X x t x α 2 exp   - t 0 α 2 b(0) X x s ds - α 2 t 0 (ψ s ) 2 ds + t 0 α 2 p 2 ψ s - σ 2 X x s 2 ds   × exp   t 0 α ψ s - σ 2 X x s dW s - t 0 α 2 p 2 ψ s - σ 2 X x s 2 ds  
and by the Hölder Inequality for p > 1, we have

E(J x, t ) α ≤C E X x t x αp 2(p-1) exp αp 2(p -1) - t 0 b(0) X x s ds + αp t 0 σ 2 4X x s ds p-1 p
.

Then, for any 0 < α < 4, for any p > 1 such that αp ≤ 4,

E(J x, t ) α ≤ C E X x t x αp 2(p-1) p-1 p . (B.1)
The same computation shows that for the same couple (p, α) and for any 0

≤ β ≤ p-1 p , E (J x, t ) α (X x t ) α 2 +β ≤ C x α 2 E (X x t ) -βp p-1 p-1 p , (B.2)
which is bounded according to Lemma 3.1. Hence, by (B.1) for (α, p) = (2, 2), there exists a positive constant C such that

E(X x+ t -X x t ) 2 ≤ C 2 , ∀ t ∈ [0, T ],
and X x+ t tends X x t in probability. We consider now the process (J x t , t ∈ [0, T ]) solution of (3.8). Applying the integration by parts formula in (3.9), we obtain that J

x t = X x t x exp( t 0 (b (X x s ) - b(X x s ) 2X x s + σ 2 8 1 X x s
)ds), from which by using (H3), we have

J x t ≤ X x t x exp - t 0 (b(0) - σ 2 4 ) ds 2X x θ exp(KT ) ≤ C X x t x . (B.3) Moreover, J x t -J x, t = t 0 b (X x s )(J x s -J x, s )ds + t 0 σ 2 X x s (J x s -J x, s )dW s + t 0 (b (X x s ) -φ s )J x, s ds + t 0 ( σ 2 X x s -ψ s )J x,
s dW s .

Bossy & Diop

We study the convergence of E(J x t -J x, t ) 2 as tends to 0. We set E x t := |J x t -J x, t |. By the Itô's formula,

E(E x t ) 2 =E
Then, by using the Lemma 3.1 with the Markov Inequality,

P(τ λ < T ) = P(γ(T ) ≥ λ) ≤ exp(- λ 2 )E exp T 0 σ 2 ds 8X x s ≤ C exp(- λ 2
).

Choosing λ = -log( r ) for a given r > 0, we have that

P(τ λ < T ) ≤ C r 2 and
E(E x T ) 2 ≤ E(E x τ λ ) 2 + C r 4 .
With (B.4), we can easily check that for any bounded stopping time τ ≤ T ,

E(E x τ ) 2 ≤ T 0 exp(C(T -s)) E τ 0 σ 2 ds 4X x s (E x s ) 2 + C √ and for τ λ , E(E x τ λ ) 2 ≤ C 1 E τ λ 0 (E x s ) 2 dγ(s) + C 0 √ ,
for some positive constants C 0 and C 1 , depending on T . After the change of time u = γ(s), we can apply the Gronwall Lemma

E(E x τ λ ) 2 ≤ C 1 E λ 0 (E x τu ) 2 du + C 0 √ ≤ T C 0 √ exp(C 1 λ).
With the choice r = (4C 1 ) -1 and λ = -log( r ), we get E(E x τ λ ) 2 ≤ T C 0 1 4 . As T is arbitrary in the preceding reasoning, we conclude that E|J x t -J x, t | tends to 0 with for all t ∈ [0, T ]. Consider now

g(X x+ t ) -g(X x t ) -g (X x t )J x t = J x, t 1 0 g (X x t + αJ x, t )dα -J x t g (X x t ) = (J x, t -J x t ) 1 0 g (X x t + αJ x, t )dα + J x t 1 0 (g (X x t + αJ x, t ) -g (X x t )) dα := A + B .
EA ≤ g ∞ E|J x t -J x, t |, which tends to zero with . B is a uniformly integrable sequence. g is a continuous function. By the Lebesgue Theorem, as X x+ t tends X x t in probability, B tends to 0 with . As a consequence, E( g(X x+ t ) -g(X x t ) ) tends to E[g (X x t )J x t ] when tends to 0.

Proof of Proposition 4.4. The proof is very similar to the proof of Proposition 3.4. Again, we consider only the case when h(x) and k(x) are nil. Let J x, t = 1 (X C End of the proof of Proposition 3.2

To compute ∂ 4 u ∂x 4 (t, x), we need first to avoid the appearance of J x t (1) in the expression of ∂ 3 u ∂x 3 (t, x). We transform the expression of ∂ 3 u ∂x 3 (t, x) in (3.14), in order to obtain ∂

Bossy & Diop

To eliminate J x t , we introduce the probability Q

T -t 0 b (X x s ( 3 2
))ds

× 3f (3) (X x T -t ( 3 2 )
)

T -t 0 b (X x s ( 3 2 ))J x s ( 3 2 )ds + f (3) (X x T -t ( 3 2 ))J x T -t 3 2 )
+

T -t 0 E exp 3 s 0 b (X x u ( 3
)

)b (3) (X x s ( 3 2
))

+ ∂u ∂x (t + s, X x s ( 3 2
))b (4)

(X x s ( 3 2 )) ds,
from which we can conclude on (3.5).

(

  H0) α ∈ [1/2, 1). (H1) The drift function b is such that b(0) > 0 and satisfies the Lipschitz condition |b(x) -b(y)| ≤ K|x -y|, ∀ (x, y) ∈ R 2 .

( H2 )

 H2 The drift function b(x) is a C 4 function, with bounded derivatives up to the order 4. Bossy & Diop 2.1.1 Convergence rate when α = 1/2

  1)du ds. (3.14) By Lemma 3.3, the derivatives of f and b being bounded up to the order 3, we get immediately that | ∂ 3 u ∂x 3 (t, x)| ≤ C uniformly in x. The computation of the fourth derivative uses similar arguments. We detail it in the Appendix C.

  .23) (3.23) is clearly true for j = 1. Suppose (3.23) holds for j. By (3.20) and (3.22)

Kolmogorov PDE Proposition 4 . 3 .

 43 Let 1/2 < α ≤ 1. Let f be a R-valued C 4 bounded function, with bounded spatial derivatives up to the order 4. We consider the R-valued function defined on [0, T ] × [0, +∞) by u(t, x) = Ef (X x T -t ). Then under (H1) and (H2), u is in C 1,4 ([0, T ] × (0, +∞)) and there exists a positive constant C depending on f , b and T such that

≤ C 1 + x -ν 2 ,

 12 ) p which gives (A.4), by noting that (1 -e -|b|t ) ≥ |b|te -|b|t . Lemma A.2. If a ≥ σ 2 /2 and b ≥ 0, there exists a constant C depending on a, b, σ and T , such that sup t∈[0,T ]

3 u ∂x 3

 3 (t, x) as a sum of terms of the formE exp x u (1))du J x s (1)Λ(X x s (1)) dsfor some functions β(x), Γ(x), Λ(x). In this first step, to simplify the writing, we write X x s instead of X x s (1). Two terms are not of this form in (3.14): x θ )dθ b (X x u )du ds .Finally, replacing I and II in (3.14), we get∂ 3 u ∂x 3 (t, x) =E exp 2

  ) 2 ds) and, by using Lemma 4.2(ii), one easily concludes that E(J x, t ) α ≤ C and consequently that X x+ t converges to X x t in L 2 (Ω). Then, by applying the Lebesgue Theorem, E|B | tends to 0. Moreover, E|A | ≤g ∞ E|J x, t -J x t | 2 .We can proceed as in the proof of the Proposition 3.4, to show that E|J x, t -J x t | 2 tends to 0, but now the moments E(J x, t ) α , α > 0 are bounded and the Lemma 4.1 ensures that the E|X x t | -p , p > 0 are all bounded.

	with φ s =	1 0 b (X x t + θ J x, s )dθ and ψ t =	1 0	ασdθ t +θ J x, (X x s ) 1-α . For any C 1 function g(x) with
	bounded derivative, we have
	g(X x+ t	) -g(X x t ) -g (X x t )J x t = J x, t			1	g (X x t + θJ x, t )dθ -J x t g (X x t )
												0
				1									1
	= (J x, t	-J x t )	g (X x t + θJ x, t )dθ + J x t	(g (X x t + θJ x, t ) -g (X x t )) dθ
				0									0
	:= A + B .
	E(J x, t ) t 0	ψ s dW s -	α 2	0	t	(ψ s
													x+ t	-X x t ), given also
	by
			J x, t	= exp	0	t	φ s ds +	0	t	ψ s dW s -	1 2	0	t	(ψ s ) 2 ds ,
	RR no 5396

α ≤ exp(α b ∞ t)E exp(α

  3/2 such that dQ 3/2 • instead of X x (1) • . Finally, asL Q 3/2 (X x (1)) = L P (X x ( 32)), we obtain the following expression for ∂ 3 u ∂x 3 (t, x): ) exists and is given by (3.9). By the Proposition 3.4, ∂ 3 u ∂x 3 (t, x) is continuously differentiable and ∂ 4 u ∂x 4 (t, x) =E exp 3

	J x s ( 3 2															dP	Ft	= 1 t Z (1, 3 2	)	. Then
	∂ 3 u ∂x 3 (t, x) =E 3/2 exp 2		0	T -t	b (X x s )ds f (3) (X x T -t )Z T -t J x (1, 3 2 ) T -t
	+	T -t	E 3/2 exp 2	s	b (X x u )du Z s (1, 3 2 )	J x s
		0														0
									× 3	∂ 2 u ∂x 2 (t + s, X x s )b (X x s ) +	∂u ∂x	(t + s, X x s )b (3) (X x s )	ds.
	Again, we note that Z t (1, 3 2 )	J x t = exp	t 0 b (X x u )du and
	∂ 3 u ∂x 3 (t, x) =E 3/2 exp 3	0	T -t	b (X x s )ds f (3) (X x T -t )
			T -t												s
	+				E 3/2 exp 3	b (X x u )du
		0													0
									× 3	∂ 2 u ∂x 2 (t + s, X x s )b (X x s ) +	∂u ∂x	(t + s, X x s )b (3) (X x s )	ds
	where we write X x
	∂ 3 u ∂x 3 (t, x) =E exp 3	0	T -t	b (X x s (	3 2	))ds f (3) (X x T -t (	3 2	))
	+	0	T -t	E exp 3			0	s	b (X x u (	3 2	)du	3	∂ 2 u ∂x 2 (t + s, X x s (	3 2	))b (X x s (	3 2	))
																+	∂u ∂x	(t + s, X x s (	3 2	))b (3) (X x s (	3 2	))	ds.

RR no 5396

Bossy & Diop

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