Julien Baptiste
email: julien.baptiste@ceremade.dauphine.fr

Julien Grepat
email: julien.grepat@gmail.com.

Emmanuel Lepinette
email: emmanuel.lepinette@ceremade.dauphine.fr

Approximation of non-Lipschitz SDEs by Picard iterations

Keywords: Pricing, European Options MSC: 60G44, G11-G13

In this paper, we propose an approximation method based on Picard iterations deduced from the Doléans-Dade exponential formula. Our method allows to approximate trajectories of Markov processes in a large class, e.g. solutions to non-Lipchitz SDEs. An application to the pricing of Asian-style contingent claims in the CEV model is presented and compared to other methods of the literature.

Introduction

In frictionless models without arbitrage opportunities, it is possible to define the price of a contingent claim as the initial value of the unique portfolio process whose terminal value coincides with the payoff, see [START_REF] Dana | Financial markets in continuous time[END_REF]Section 3.2.2].

The price of a replicable contingent claim may be deduced as an expectation as soon as the price process is a local martingale and the replicating portfolio is a martingale under some equivalent probability measure. If the price dynamics is simple enough, e.g. in the Black and Scholes model, it is possible to obtain an explicit expression for the Call or Put option prices.

When sophisticated models are considered for a sake of realism [START_REF] Brigo | A mixed-up smile[END_REF][START_REF] Dupire | Pricing and hedging with smiles[END_REF][START_REF] Hull | The pricing of options on assets with stochastic volatilities[END_REF], one can not expect to compute easily option prices which do not admit pricing formulas in closed form. For this purpose, Monte-Carlo methods have been widely developed to simulate trajectories of diffusion processes and deduce approximations of European option prices [START_REF] Bouleau | Numerical methods for stochastic processes[END_REF][START_REF] Kloeden | Numerical solution of stochastic differential equations[END_REF].

An alternative way is to study the density function of the terminal price value as in the Black and Scholes model [START_REF] Black | The pricing of options and corporate liabilities[END_REF]. This is also possible for the CEV model [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF] where analytic expressions of European Call and Put option prices are deduced. Nevertheless, it is not possible to evaluate the price of an Asian option since a knowledge of the whole trajectory is required.

Moreover, the Monte Carlo discretization method seems to be useless as the diffusion coefficient is not Lipschitz so that the Euler scheme should explode.

In [START_REF] Funahashi | A chaos expansion approach under hybrid volatility models[END_REF][START_REF] Kijima | A chaos expansion approach for the pricing of contingent claims[END_REF], the asset price process is alternatively approximated through a Picard iteration scheme followed by approximations based on Wiener-Ito chaos expansions. Only the first three terms in the expansions are considered as significative and the associated density functions are approximated.

The accuracy of the suggested numerical method illustrated by numerical implementations is highlighted even if it is not confirmed by mathematical arguments. A direct discretization scheme based on Doléans-Dade formula is also used (without mathematical proof of the convergence) in [START_REF] Mehrdoust | Efficient Monte Carlo option pricing under CEV model[END_REF] to simulate trajectories in the CEV model. Inspired by these works, our goal is to study the method based on Picard iterations to approximate Asian option prices when the risky asset dynamics is a stochastic differential equation with non Lipschitz-coefficients.

The outline of our work is as follows. By a truncation, we first approximate the stochastic process of interest by a diffusion process with Lipschitz coefficients. Second, we introduce a Picard iteration technique that approximates the process as a Doléans-Dade exponential. Finally, a Monte Carlo discretization scheme is implemented. Comparatively to the classical Euler scheme, we obtain a convergence order n instead of √ n where n is the number of discrete dates. The method may be applied to the pricing of a large class of European and Asian options. We illustrate it by computing prices of average Put options in the CEV model where the diffusion coefficient is not Lipschitz.

Comparing with the exact formulas and other techniques, our method appears to provide accurate results. Moreover, as soon as we consider an average Put option with more than three dates, our method is less time-consuming than the method combining Monte Carlo simulations and analytic expressions. Nevertheless, in some cases, a simple truncated Euler scheme appears to be rather efficient whereas the direct discretization scheme based on the Doléans-Dade formula without truncation does not converge. For some other cases, e.g. the CEV model with β > 1, we clearly observe that the simple Euler scheme explodes but, in that case the backward Euler-Maruyama scheme is the most efficient.

Model

Notations

We denote by R the set of all real numbers.

The smallest σ-algebra containing a family of subsets A is denoted by σ(A).

The set of all bounded real-valued functions on [0, T ] is denoted by B([0, T ]).

Let (Ω, F, P ) be a stochastic basis with a complete filtration F = (F t ) t∈[0,T ] satisfying the usual hypothesis and T > 0 is fixed. We consider a standard Brownian motion W adapted to F. For any predictable process H, we consider the stochastic integral of H with respect to W denoted by

H • W t := t 0 H u dW u , t ∈ [0, T ],
when existence holds. The Doleans-Dade exponential is denoted by

E t (H) = exp H • W t - 1 2 t 0 H 2 u du , t ∈ [0, T ].
In the following, S 0 is the deterministic initial value at time 0 of a process S modelling an asset price in a financial market model. In the following, we consider uniform partitions (τ n ) n≥0 of the interval [0, T ]. The grid τ n is composed of n + 1 dates where n is called the order of the discretization scheme. It is defined by

τ n = {t 0 = 0, t 1 = T /n, t 2 = 2T /n, • • • t n = T }.
We use the abuse of notation t n i = t i = (T /n)i, i = 0, • • • , n, for a sake of simplicity. If X is a stochastic process defined on [0, T ], we denote by X (n) the piecewise constant process defined by

X (n) t = X t i , t ∈ [t i , t i+1 ), X (n)
T = X T .

The stochastic differential equation

Let (Ω, F, F = (F t ) t∈[0,T ] , P) be a continuous-time stochastic basis satisfying the usual assumptions and supporting a standard Brownian motion W , i.e.

F t := σ(W s : s ≤ t)∨N where N designates the P-negligible sets. In the case of a financial market model, there is a risk-free bond given by the interest rate r 0 and a risky asset price stochastic process S.

By [8, Theorem 2.2 p 104], recall that we have the following.

dS(t) = S(t)σ(t, S(t))dW t + S(t)r(t, S(t))dt, S(0) = S 0 > 0, (2.1)
admits a unique strong solution S. Moreover, it is positive.

In the sequel, we shall consider locally Lipschitz coefficients r and σ and we assume that r is bounded. We suppose that S = 0 on [τ ∧ T, T ] where τ := inf{t : S t = 0}, meaning that 0 is an absorbing point. The s.d.e. (2.1) admits a unique non negative solution under the conditions of the proposition above but we only suppose one of the following Conditions 2.2 or 2.3.

Condition 2.2.

There exists constants C κ > for every κ > 0, such that

|σ(t, x ∧ κ) -σ(u, x ∧ κ)| ≤ C κ |t -u|, (t, u, x) ∈ (R + ) 2 × (0, ∞), |r(t, x ∧ κ) -r(u, x ∧ κ)| ≤ C κ |t -u|, (t, u, x) ∈ (R + ) 2 × (0, ∞), |σ(t, x ∧ κ) -σ(t, y ∧ κ)| ≤ C κ |x -y|, (t, x, y) ∈ R + × (0, ∞) 2 , |r(t, x ∧ κ) -r(t, y ∧ κ)| ≤ C κ |x -y|, (t, x, y) ∈ R + × (0, ∞) 2 , |r(t, x ∧ κ)| + |σ(t, x ∧ κ)| ≤ C κ , (t, x, y) ∈ R + × (0, ∞) 2 .
Condition 2.3. There exists constants C κ > for every κ > 0, such that

|σ(t, x ∨ κ) -σ(u, x ∨ κ)| ≤ C κ |t -u|, (t, u, x) ∈ (R + ) 2 × (0, ∞), |r(t, x ∨ κ) -r(u, x ∨ κ)| ≤ C κ |t -u|, (t, u, Y ) ∈ (R + ) 2 × (0, ∞), |σ(t, e X ∨ κ -1 ) -σ(t, e Y ∨ κ -1 )| ≤ C κ |X -Y |, (t, X, Y ) ∈ R × R 2 , |r(t, e X ∨ κ -1 ) -r(t, e Y ∨ κ -1 )| ≤ C κ |X -Y |, (t, X, Y ) ∈ R × R 2 , |r(t, x ∨ κ)| + |σ(t, y ∨ κ)| ≤ C κ , (t, x, y) ∈ R + × (0, ∞) 2 .
Moreover, when P (τ < T ) > 0, we suppose that the mappings x → σ(t, x) and x → r(t, x) are differentiable on (0, ∞) for all t ∈ [0, T ] and x∇ x σ(t, x),

x∇ x r(t, x) are bounded on [κ -1 , ∞) by C κ .
Remark 2.4. Under Condition 2.2, S admits moments at any order, see the proof in Section 8. Notice that this assumption is satisfied by the CEV model dS t = σS β t dW t and, more generally, when σ is of the form σ(t, x) = ν(t)x β(t)-1 , x > 0, where ν and β ≥ 1 are bounded functions and ν is Lipschitz.

Condition 2.3 is satisfied by the CEV model, e.g. when σ is of the form σ(t, x) = ν(t)x β(t)-1 where ν is Lipschitz and bounded and 1 -α ≤ β ≤ 1, α ∈ R + . This also includes the CIR model and more generally mean-reverting SDEs as used in finance, see numerical simulations below.

In finance, under a risk-neutral probability measure for S, e.g. when the market is complete, we are interested in approximating the price of a contingent claim whose payoff G(S)1 τ >T depends on the whole trajectory S = (S t ) t∈[0,T ] .

Note that the delivered payoff G(S)1 τ >T is 0 when S T = 0, i.e. when the firm having issued the risky asset failed.

We suppose that the functional G : B([0, T ]) → R + satisfies the Lipschitz condition:

|G(α) -G(β)| ≤ C||α -β||, α, β ∈ B([0, T ]), (2.2)
where C > 0 and . is the norm of uniform convergence on [0, T ], i.e.

f := sup t∈[0,T ] |f (t)|, f ∈ B([0, T ]).
If we assume that P is the unique risk-neutral probability measure in a complete market, the smallest price v of the contingent claim G(S)1 τ >T is evaluated under P by v = Ee -r 0 T G(S)1 τ >T where E is the expectation under P and r 0 is the constant risk-free interest rate.

In the following, a stochastic process r : u → r u models an interest rate of the market. The price of the associated bond with initial value 1 is

F t (r) := exp t 0 r u du , t ∈ [0, T ]. (2.3)
We consider a class of payoff functions G adapted to the numerical scheme we propose. For a sake of simplicity, we make the abuse of notation r u = r(u, S u ) as well as σ u = σ(u, S u ) so that the notations F (r) and E(σ) make sense.

Definition 2.5. We say that the payoff function G is adapted to the discretization scheme defined by the partition τ n if there exists α(n) ≥ 1, p, q ∈ N such that, for all Lipschitz functions σ and r satisfying

T 0 EF 2p u (r)E 2p u (σ)du < ∞,
we have, for some function f ,

|EG(S r,σ ) -EG(S (n) r,σ )| ≤ f (G, σ, r, p, q) α(n) T 0 [1 + ES 2p r,σ (u)du 1/q
, where S r,σ = F (r)E(σ) and f depends on σ, r, p through the quantities

max x |σ(t, x)x| p /(1 + |x| 2p ) < ∞ and max x |r(t, x)x| p /(1 + |x| 2p ) < ∞.
Recall that F (r) is given by (2.3) and, in the definition above, the process S r,σ = F (r)E(σ) is interpreted as a price. Observe that the European call payoff function is adapted to the discretization whatever α(n) is. Indeed, the payoff function of the European call option with strike K is G(S) where

G(α) = (α(T )-K) + , α ∈ B([0, T ]
) and, since G only depends on the terminal value of α, we have

G(S r,σ ) = G(S (n) r,σ ).
The payoff of the average strike Asian call option with strike K is G(S) where

G(α) = 1 T T 0 α(t)dt -K + , α is B([0, T ])-measurable.
Lemma 2.6. The average strike Asian call option function G with strike K is adapted to the discretization with α(n) = n.

All the proofs are postponed to Section 8.

Approximation of the stochastic process

The approximation scheme we propose is obtained in three steps. By a truncation, we first approximate the price process by a diffusion process S κ , κ > 0, with bounded and Lipschitz volatility. By Picard iterations, a recursive scheme is introduced to approximate S κ by S κ,m , m ≥ 1, Doléans-Dade solutions to stochastic differential equations defined by iterated and uniformly bounded volatilities. At last, we discretize S κ,m in time.

Volatility truncation

Let κ ∈ R + be such that κ ≥ sup (t,x)∈R 2 r(t, x) and κ > S 0 ∨ 1 when Condition 2.2 holds and κ -1 < min(max r, S 0 ∨ 1) when Condition 2.3 is satisfied. Consider Y κ the unique solution to the s.d.e.

Y κ (0) = 0, (3.4) dY κ (t) = σ κ (t, S 0 e Yκ(t) )dW t + γ κ (t, S 0 e Yκ(t) )dt, where σ κ (u, x) := σ(u, x ∧ κ) and r κ (u, x) := r(u, x ∧ κ) when Condition 2.2 holds. If Condition 2.3 holds, we define σ κ (u, x) := σ(u, x ∨ κ -1 ) and r κ (u, x) := r(u, x ∨ κ -1 ). At last, we define γ κ (t, x) = r κ (t, x) -1 2 σ 2 κ (t, x). Observe that S κ := S 0 e Yκ satisfies, for all t ∈ [0, T ],

S κ (0) = S 0 , S κ (t) := S 0 exp t 0 σ κ (u, S κ (u))dW u + t 0 γ κ (u, S κ (u))du .
Lemma 3.1. Suppose that Conditions 2.2 and 2.3 hold respectively. Then, S κ converges pointwise to S as κ → ∞ on the interval [0, T ] ( on the interval [0, τ ) respectively).

In the following, we use the following assumption: Condition 3.2. There exists p > 0 and a constant C p such that

E sup t∈[0,T ] S 2p t + sup κ E sup t∈[0,T ] S 2p κ (t) ≤ C p , ∀κ > 0. (3.5)
By the Lemmas 3.4 and 3.5 proven in appendix, we deduce that Condition 3.2 is satisfied for the large class of models satisfying the following Condition 3.3 above even if the coefficients of (2.1) are unbounded.

Condition 3.3. There exists L > 0 and α ∈ (0, 1] such that the following conditions are satisfied under Conditions 2.2 and 2.3 respectively:

(I): 0 < σ 2 (t, x) ≤ L(1 + | ln(ln(x))|1 x>1 ), (t, x) ∈ R × R, |r(t, x)| ≤ L, (t, x) ∈ R × R, (II): |σ(t, x)| + |r(t, x)| ≤ L |x| α + L, x > 0. Lemma 3.4.
Assume that Condition 3.3 is satisfied. Then, for all p ≥ 0, there exists a constant C p independent of κ such that E sup u≤T (S κ (u)) p ≤ C p . Lemma 3.5. Assume that Condition 3.3 is satisfied. Then, for all p ≥ 0, there exists a constant C such that E sup u≤T (S u ) p ≤ C p . Lemma 3.6. Suppose that Condition 3.2 is satisfied for some p > 0. Moreover, assume that Condition 3.3 (II) holds if σ κ = σ(•, • ∨ κ -1 ). Then, for all l ≥ 1, l ∈ N, there exists a constant C l,p such that

σ κ = σ(•, • ∧ κ) ⇒ E sup t∈[0,T ] |S κ (t) -S t | p ≤ C l,p κ l , σ κ = σ(•, • ∨ κ -1 ) ⇒ E sup t∈[0,τ ∧T ] |S κ (t) -S t | p ≤ C p α(τ, κ),
where α(τ, κ) → 0, as κ → ∞.

This result shows that S may be approximated by S κ in the L p sense, p > 0, as κ → ∞.

Picard iterations

In this section, we construct iterated Doleans-Dade processes S κ,m , m ∈ N, such that S κ,m converge to S κ when m → ∞. To do so, we introduce the following scheme:

S κ,0 := S 0 , S κ,m+1 (t) := F t (r κ,m+1 )E t (σ κ,m+1 ), t ∈ [0, T ],
where

σ κ,m+1 (t) := σ κ t, S κ,m (t) , m ≥ 0, t ∈ [0, T ], r κ,m+1 (t) := r κ t, S κ,m (t) , m ≥ 0, t ∈ [0, T ].
Notice that we have

S κ,m+1 (t) = S 0 exp t 0 σ κ u, S κ,m (u) dW u + t 0 γ κ u, S κ,m (u) du .
Let us introduce

Y κ,m (t) := log S κ,m (t) -log S 0 , t ∈ [0, T ].
This process satisfies the following s.d.e.

dY κ,m+1 (t) = σ κ t, S κ,m (t) dW t + γ κ t, S κ,m (t) dt.

We shall prove the convergence of S κ,m to S κ in L 2 as m → ∞.

sup u≤T Y κ,m+1 (u) -Y κ,m (u) 2 ≤ C(1 + S 0 )(C κ ) m+1 √ T m+1 (m + 1)! , (3.6)
sup u≤T Y κ,m+1 (u) -Y κ,m (u) 4 ≤ C(1 + S 0 )(C κ ) m+1 (T 1/4 ) m+1 (m + 1)!) 1/4 , (3.7)
where C is a constant which does not depend on κ and C κ is a constant which depends on κ.

Corollary 3.8. There exists constants C, C κ independent of m such that

E sup u≤t Y κ,∞ (u) -Y κ,m (u) 2 ≤ C(1 + S 2 0 ) ∞ j=m+1 (C κ ) j j! , E sup u≤t Y κ,∞ (u) -Y κ,m (u) 4 ≤ C(1 + S 4 0 ) ∞ j=m+1 (C κ ) j j! .
Lemma 3.9. For every κ, there exists constants C, C κ such that

E sup t∈[0,T ] S κ,m+1 (t) -S κ,m (t) 2 ≤ C κ E sup t∈[0,T ] (Y κ,m (t) -Y κ,m-1 (t)) 4 ≤ C κ (C κ ) m √ m! .
By the lemma above, we deduce that S κ,m converges to S κ,∞ in L 2 with respect to the uniform convergence.

Lemma 3.10. For every κ, there exists a constant C κ such that

E sup t∈[0,T ] S κ,∞ (t) -S κ,m (t) 2 ≤ C κ ∞ j=m (C κ ) j √ j! .
Note that the limit processes satisfy

dY κ,∞ (t) = σ κ t, S κ,∞ (t) dW t + γ κ (t, S κ,∞ (t))dt, (3.8) dS κ,∞ (t) = S κ,∞ (t)σ κ t, S κ,∞ (t) dW t + S κ,∞ (t)r κ t, S κ,∞ (t) dt. (3.9)
It follows that S κ,∞ = S κ and Y κ,∞ = Y κ since uniqueness holds.

Volatility discretization based on Picard iterations.

In the following, we suppose without loss of generality that T /n ≤ 1 if n is large enough. We recursively define the processes S κ,m,n , σ κ,m+1,n , m ≥ 0, as follows:

S κ,0,n := S 0 , σ κ,1,n (t) := σ κ (t i , S 0 ), r κ,1,n (t) := r κ (t i , S 0 ), t i ≤ t < t i+1 , S κ,m,n (t) := F t (r κ,m,n )E t (σ κ,m,n ), m ≥ 1, t ∈ [0, T ], σ κ,m+1,n (t) = σ κ t (n) , S (n) κ,m,n (t) , m ≥ 1, , t ∈ [0, T ], r κ,m+1,n (t) = r κ t (n) , S (n) κ,m,n (t) , m ≥ 1, t ∈ [0, T ].
where we use the notation t (n) = t i if t i ≤ t < t i+1 and we recall that

E t (σ κ,m,n ) = exp t 0 σ κ u (n) , S (n)
κ,m-1,n (u) dW u - 1 2 t 0 σ 2 κ u (n) , S (n)
κ,m-1,n (u) du , F t (r κ,m,n ) = S 0 exp t 0 r κ u (n) , S (n) κ,m-1,n (u) du , m ≥ 1, t ∈ [0, T ].
We also define Y κ,m,n = ln(S κ,m,n /S 0 ).

Lemma 3.11. For all κ > 0, there exists constants α κ , C κ > 0 such that for all m, n ≥ 1:

E sup t∈[0,T ] S κ,m (t) -S κ,m,n (t) 2 ≤ α κ C m κ n .

Main result

In the following, we follow the convention that τ κ,m,n = +∞ if P (τ ≤ T ) = 0.

Otherwise, τ κ,m,n is defined as

τ κ,m,n := inf{t : S (n) κ,m,n (t) ≤ κ -1 }.
Theorem 3.12. Assume that Conditions 2.2 or 2.3 hold and suppose that G is Lipschitz and consistent of order n with respect to discretization. Suppose that τ = +∞ a.s. under Condition 2.2. Then, for κ > 0 and m, n ∈ N, there are some positive constants α κ , C κ and sequences ε κ > 0, ε κ,m > 0, and ε κ,m,n such that

|EG(S)1 τ >T -EG(S (n) κ,m,n )1 τ κ,m,n >T | ≤ ε κ + ε κ,m + α κ (C κ ) m n + ε κ,m,n ,
where

ε κ → 0 as κ → ∞, ε κ,m → 0 as m → ∞, ε κ,m,n → 0 as n → ∞. Moreover, ε κ,m,n = 0 if P (τ ≤ T ) = 0.
Remark 3.13. In practice, we first fix κ large enough and we fix m large enough so that κ,m is sufficiently small. At last, we choose n large enough.

The algorithm's complexity allowing to compute this scheme is then mn.

Notice that in the case where G(S) only depends on S T and G(0) = 0, then

EG(S) = EG(S)1 τ >T which is approximated by EG(S (n) κ,m,n )1 τ κ,m,n >T .

Numerical example: the CEV model

The Constant Elasticity of Variance (CEV) model is defined by the dynamics

dS t = σS β t dW t , t ∈ [0, T ], (4.10)
where the parameter β is called the elasticity. We suppose that σ = σ LN S 1-β 0 where σ LN is the effective lognormal volatility. This implies that the volatility depends on β as in [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF], see also [START_REF] Delbaen | A note on option pricing for constant elasticity of variance model[END_REF][START_REF] Schroder | Computing the constant elasticity of variance option pricing formula[END_REF] for more details on this model.

Consider the financial market model defined by a Bond B ≡ 1 and the risky asset price S. We recall that S is a martingale if

β ≤ 1. If β > 1, S is a strict local martingale, i.e.
is not a martingale. In both cases, there is no arbitrage opportunity when using predictable strategies generating portfolio processes which are bounded from below, see [4, Section 3.2]. Indeed, any self-financing portfolio process which is bounded from below is a supermartingale.

Let h T be a European or Asian contingent claim. Suppose that h T is bounded, which is the case for a Put option. Then, by the predictable representation theorem, the martingale

V t = E(h T |F t ), t ∈ [0, T ], is a self-financing portfolio
process such that V T = h T . We deduce that the unique price for the payoff h T is well defined as the expectation V 0 = Eh T . In the following, we illustrate our method by estimating V 0 = Eh T when β > 1 and β < 1 respectively.

The price for the Call option may be easily deduced from the Put-Call parity and the analytic expression of the expectations ES t , t ∈ [0, T ], see [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF].

The CEV model with elasticity

β > 1 Let us introduce δ = (1-2β)(1-β) -1 .
We suppose that δ > 2 or equivalently β > 1 so that S is a strictly positive process. The price of the Call option with strike K > 0 is then given in [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF] by

p Call (S 0 , K, σ, δ, T ) = S 0 Γ( δ 2 -1, S 0 2T ) -χ 2 ( S 0 T , δ -2, K T ) -Kχ 2 ( K T , δ, S 0 T ),
where

K := K 2(1-α) σ 2 (1-α) 2 , Γ is the normalised incomplete gamma function Γ(z, x) = 1 Γ(z) x 0 t z-1 e -t dt,
p P ut (S 0 , K, σ, δ, T ) = p Call (S 0 , K, σ, δ, T ) -S 0 Γ(1 - δ 2 , S 0 2T ) + K.
In [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF], the price of the Put option is computed analytically for several values of β > 1 when T = 1, S 0 = 100, σ LN = 20% and K = 90, 100, 110. These analytical values are compared to the ones obtained by simulating the terminal value

X T := S 2(1-β) T σ 2 (1-β) 2 .
In the following, we compare these results to the values we obtain when using our method. Actually, we can not expect to obtain the convergence of S κ towards S in the L p sense, p ≥ 1, since Condition 3.2 does not a priori hold for p ≥ 1, see also Condition 3.3 (I). Indeed, ES κ (t) = S 0 for all t since S κ is a square integrable martingale. On the other hand, ES t < S 0 for all t > 0 as shown in [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF] since S is a strict local martingale. Therefore, ES κ (t) does not converge to ES t as κ → ∞. Nevertheless, when the payoff function g ≥ 0 is bounded, the dominated convergence theorem yields that g(S κ ) → g(S) in the L p sense as κ → ∞, p ≥ 1, which is enough for our purpose. Precisely, since S = S κ on τ κ ≥ T , we get that E g(S κ ) -g(S)

p ≤ (max g) p P sup

t∈[0,T ] S t ≥ κ → 0, as κ → ∞.
This implies that Theorem 3.12 still hold.

Let us compare the Picard method to the analytic one and to the backward Euler-Maruyama scheme (BEM), see [START_REF] Neuenkirch | First order strong approximations of scalar SDEs defined in a domain[END_REF]. In the following, we choose m = 3 and σ κ (u, x) := σ(u, x ∧ κ) where

κ = S 0 1 + 5 1 + β 2 .
This choice has been empirically chosen to improve the performance of our results for the Put option prices and then implemented for the Asian options.

Recall that the volatility of the model also depends on β. If β = 1.5, we observed that the approximation error is not smaller if we increase to much κ. The reason is very simple: choosing κ very large is equivalent to suppose that κ = +∞ as the maximum of S over all simulated trajectories is finite since we fix a finite number of dates n + 1 and a finite number of simulations N . Theoretically, if κ is very large, we need to increase the number of dates n as the Lipschitz constant is larger, which is more time consuming.

We choose N = 15000 simulations and n = 1000 so that the discretization step is T /n. We consider the case K = 100. The presented results are averages we obtain by repeating more than 1000 times the same simulations so that the relative standard deviation of the results is smaller than 5% of the estimated averages.

Naturally, the computation time needed for the analytic method (< 0.01) is negligible contrary to the needed time for other methods which require to simulate all the trajectories.

In the following, we compare our method to the Monte Carlo one of [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF] and the BEM method as well (see [15, Section 2, Section 3.4]), requiring that the radius of the 95% confidence interval is less than 0.025 when applying the law It is also natural to compare our method to a simple Doléans-Dade scheme deduced from the dynamics (4.10) where the volatility function σ

(x) = σx β-1 is replaced by σ κ (x) = σ(x ∧ κ) β-1 .
Inspired by the Doleans-Dade formula, we consider the Doléans-Dade scheme

S n t i+1 = S n t i exp σ κ (S n t i )∆W t i+1 - σ 2 κ (S n t i ) 2 ∆t i+1 , S n 0 = S 0 , i = 0, • • • , n-1.
In the following table, we compare the approximate prices to the analytic ones for several values of β by implementing the scheme until the error is smaller than the one observed in Table 1 by increasing N with the bound N ≤ 5.10 6 . We choose n = 1000 so that the time step is T /n. The strike is K = 100 and κ = 400. The error is still computed as the ratio between the difference of the two prices and the smallest one. We observe that the results are similar for smaller κ = 120 as the errors are around 10% and computation times are about 520 seconds. If κ = 800, we get similar results. The performance is still good for β = 5 but also β = 4

while it is bad for β = 5.5 contrary to the case κ = 400. The best results we can get seem to be when κ = +∞, i.e. without any truncation, but remains bad. With a larger n = 10000 and N = 15000, errors go from 2.5% to 10.24% but computational times are larger than 14 seconds. With only n = 1000, computation times are smaller, precisely about 1.5 seconds, but the errors are similar i.e. from 2% to 20%. So, seemingly, it is more efficient to increase N rather than n if we aim to decrease the error. In any case, comparing with the results of Table 1, it is clear that our methods outperforms the simple truncated Euler scheme.

As to a direct discretization from (4.10), i.e. S n t i+1 = S n t i + Σ(S n t i )∆W t i+1 , the Euler scheme with n = 1000 and N = 15000 explodes without truncation for β ≥ 3 as we may expect. For β ∈ {1.5, 2, 2.5}, the scheme appears to be efficient as the error is small (≤ 1.4%) and the computation time is also smaller ( 1.6 seconds). With the truncation Σ(x) = σ(x∧κ) β where κ = 300, the errors are about 1.5% for the first values of β > 1 while the errors go to 4.6% for β = 5 to 769% for β = 7. Therefore, a truncation from the simple Euler scheme seems to be conceivable only for values of β smaller than 5.

At last, we shall see that our method outperforms the classical one based on Monte Carlo simulations as soon as we consider an Asian Put option of the

form (K -(S T 1 -• • • S T J )/J) + where J ≥ 3 and T 1 < T 2 < • • • < T J = T .
Let us first consider the case J = 2. In that case, by conditioning, we rewrite

E(2K -S T 1 -S T 2 ) + = EP (S T 1 )
where, by the Markov property, P (s) is the price of the Put option of strike 2K -s in the CEV model with initial value s and horizon date T 2 -T 1 . As shown in [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF], it is possible to directly simulate the distribution of S T 1 . We then analytically deduce the price P (S T 1 ) for each simulated value of S T 1 . By the law of large numbers, we then deduce an approximation of EP (S T 1 ). With the same parameters as before, we obtain the following results when T 1 = T /2 and T 2 = T = 1. Table 7. Asian Put option price with three dates using Picard iterations and the Monte Carlo methods combined with the analytic expression respectively such that the 95% confidence interval radius is smaller than 0.05.

Conclusion: our method appears to outperform the technique based on

the Monte Carlo methods combined with the analytic expression for Asian options with more than three dates as well as the truncated Euler scheme.

Notice that computational times may be shortened in the Picard scheme by replacing the time-consuming computation of the exponential function by the first terms of its Taylor expansion. Nevertheless, the BEM method appears to be the most efficient even if, with three dates, the error in Table 7 is smaller for the Picard method. In the case where β = 2, the simple truncated Euler scheme seems to be efficient provided that β ≤ 5 as convergence errors and computation times are small. In the following section, we implement the CEV model for β < 1. In that case, the BEM method does not work well.

The CEV model with elasticity β ∈ (0, 1)

Recall that Condition 2.3 holds when β ∈ (0, 1). Nevertheless, the required property that there exists an absorbing point only holds when β ≥ 0.5 [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF].

In this section, the parameters are S 0 = 100, σ LN = 0.5, T = 4, K = 100, N = 15000, n = 20 and m = 2. The parameter κ is empirically fixed to κ = S 0 (0.78 -0.4 * β 2 ) for the Picard scheme and κ = 300 for the truncated Euler scheme.

When implementing the BEM method, the implicit scheme requires to solve an equation of the form g(X) = 0, e.g. using the Newton method. To do so, we fix an interval [a, b] we expect the solution X * to belong to. Unfortunately, the algorithm stops as soon as a is too large or b too small as [a, b] does not contain X * . We observe this phenomenon even with a = 10 -8 and b = 10 8 so that we can not obtain any result.

We consider the Put option.

The methodology we follow consists in increasing the number of simulations N until the estimated price belongs to the interval of radius 1% and centred at the analytic expression of the price. Otherwise, we stop N to 2.10 5 . Repeating 1000 times the procedure, we deduce an average of the estimated price for each method and the needed time in milliseconds as well. Table 10. Asian Put option prices with two dates using Picard iterations (P), truncated Euler scheme (E) and the Monte Carlo methods combined with the analytic expression (C). The errors are computed between the Picard method (P) and the Euler scheme (E) respectively with respect to the combined method (C).

Clearly, the combined method derived from [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF] remains the most performant for two dates. Notice that the Picard method outperforms the truncated one.

At last, with the combined method, it takes several hours to estimate the price of an Asian option with three dates T i/3, i = 1, 2, 3 and T = 4.

It only takes 78 seconds for the Picard method and the estimated price is 27.8456.

Conclusion: in the case where β ∈ [0, 1), the Picard method seems to be the best method comparatively to Euler truncated scheme, the BEM method or the combined method based on Monte Carlo simulations of the terminal price for more than three dates as the computation time is smaller and the errors are satisfactory. Otherwise, for one or two dates, the combined method derived from [START_REF] Lindsay | Simulation of the CEV process and the local martingale property[END_REF] appears to be efficient.

The Aït-Sahalia model

The Aït-Sahalia model (see [START_REF] Neuenkirch | First order strong approximations of scalar SDEs defined in a domain[END_REF]Section 3.6]) satisfies the s.d.e.

dS t = σS ρ t dW t + α -1 S -1 t -α 0 + α 1 S t -α t S r t dt,
where ρ, r > 1,

α i ≥ 0, i = -1, • • • , 2 and σ > 0. The initial value is S 0 = 0.2
and the other parameters are T = 1, K = 0 for the Call option so that we shall estimate E(S T ).

We choose α -1 = 0.1 and

α i = 0, if i = 0, • • • , 2.
We fix σ = 0.2 and the BEM method that requires more time. Clearly, the Picard method is not efficient on this model in the sense that the needed time to obtain a small error is significantly larger.

r = ρ = 2.

Example of mean reverting SDEs

Let us consider the dynamics dS t = (1 -S t )dt + γS β t dW t , where W is a standard Brownian motion and β ∈ [0.5, 1). Here we choose β = 1. Using the Lamperti transformation, we simulate the process X t = S -β+1 t when implementing the BEM method. We noticed that the BEM scheme explodes because of the inversion problem that needs to be solved and the simple Euler scheme explodes in particular when β is closed to 0.5. Otherwise, an improved convergent Milstein scheme is proposed by [START_REF] Desmond | Convergence, non-negativity and stability of a new Milstein scheme with applications to finance[END_REF]. As mentioned in [START_REF] Desmond | Convergence, non-negativity and stability of a new Milstein scheme with applications to finance[END_REF], the discretization step needs to be smaller than γ -2 for the approximation process to be positive. This is indeed what we observed so that the needed time to compute appears to be significantly large for γ large enough. In that case, the Picard method appears to be more efficient. Here we compare the two methods with γ = 1 so that it is possible to get accurate estimations with the improved Milstein scheme in a reasonable computation time. The goal here is to present the accuracy of the Picard method. With the discrete

dates t n k = (T /n)k, k = 0, • • • , n
, the improved Milstein is given by:

S n t n k+1 = S n t n k + (1 -S n t n k+1 )∆t n k+1 + [S n t n k ] β ∆W t n k + 1 2 (β -1)[S n t n k ] 2β-1 (∆W t k ) 2 -∆t k .
We propose to compare the results obtained by implementing the Picard method without truncature, i.e. κ = +∞, to the Milstein scheme as a reference. To do so, we first implement the Milstein scheme. We then compute the averaged needed time over 1000 experiments for the Picard method to provide an approximation at a relative distance from the Milstein estimation less than 1%. As shown in the following

Conclusion

The conclusion of this work is that it is not possible to say in general what the 'best' simulation technique is. Through the examples we present, the same technique may appear as very efficient or, on the contrary, inappropriate.

The Picard technique seems to be appropriate for models where the diffusion coefficient is sublinear, e.g. for asian option pricing, while the BEM method is efficient when the diffusion coefficient is super linearly growing.

Proofs

Throughout the paper, we shall use the following convention: from a line to the next one, constants K, κ or C may designate different constants which are independent of any other current variables except maybe the fixed parameters of the model as the maturity date T or Lipschitz constants. In the contrary case, we shall use notations like C m to stress the dependence on some parameter m.

Proof of Lemma 2.6. Let us denote S = S r,σ = F (r)E(σ). Using the integration by part formula, we obtain that

S t t = t 0 S u du + t 0 udS u , t ∈ [0, T ].
On the other hand, a direct computation shows that

T 0 S (n) u du = - T 0 udS (n) u + T S (n)
T .

Since S

T = S T , we deduce that

T 0 S u du - T 0 S (n) u du = T 0 ud(S (n) u -S u ).
Observe that

T 0 udS (n) u = n i=1 t n i ∆S t n i , = n i=1 t n i t n i t n i-1 σ(u, S u )S u dW u + n i=1 t n i t n i t n i-1 r(u, S u )S u du, = T 0 I (n) (u)σ(u, S u )S u dW u + T 0 I (n) (u)r(u, S u )S u du,
where

I (n) (u) = t i if u ∈ [t i-1 , t i ). It follows that T 0 S u du- T 0 S (n) u du = T 0 (I (n) (u)-u)σ(u, S u )S u dW u + T 0 (I (n) (u)-u)r(u, S u )S u du
As |u (n) (u)-u| ≤ T /n and r and σ are supposed to be Lipschitz, we deduce by the Cauchy-Schwarz inequality and the Ito isometry the following sequence of inequalities:

E|G(S)) -G(S (n) )| ≤ C T E T 0 (I (n) (u) -u)σ(u, S u )S u dW u +C T E T 0 (I (n) (u) -u)r(u, S u )S u du , E|G(S) -G(S (n) )| ≤ C T C(σ) T 0 (I (n) (u) -u) 2 E(1 + S 4 u )du +C T (r) T 0 |I (n) (u) -u|E(1 + S 2 u )du, ≤ C G C(σ) n T 0 (1 + ES 4 u )du + C T (r) n T 0 E(1 + S 2 u )du, ≤ C G C(σ) n T 0 (1 + ES 4 u )du + C T (r) n T 0 E(1 + S 2 u ) 2 du.
The conclusion follows. 2 Lemma 8.1. Suppose that Condition 2.3 holds. Let κ > 0. Then, there exists a constant C κ depending on κ such that for all x, y > 0 and t ∈ R + ,

|σ κ (t, x) -σ κ (t, y)| ≤ C κ |x -y|.
Proof. Under Conditions 2.3, we obtain the following chain of inequalities

|σ κ (t, x) -σ κ (t, y)| = |σ κ (t, x ∨ κ -1 ) -σ κ (t, y ∨ κ -1 )|, ≤ C κ | ln(x ∨ κ -1 ) -ln(y ∨ κ -1 )|, ≤ C κ | ln(x) ∨ ln(κ -1 ) -ln(y) ∨ ln(κ -1 )|, ≤ 2C κ | ln( √ x) ∨ ln(κ -1/2 ) -ln( √ y) ∨ ln(κ -1/2 )|, ≤ C κ | √ x - √ y| ≤ C κ |x -y|.
The last inequalities are due to the fact that the mapping z → ln(z) ∨ ln(κ -1/2 ) is Lipschitz and the square root function is Holder continuous of order 1/2. 2 Lemma 8.2. Suppose that Condition 2.2 holds. Let κ > 0. Then, there exists a constant C κ depending on κ such that for all X, Y ∈ R 2 , x, y > 0

and t ∈ R + , |σ κ (t, e X ) -σ κ (t, e Y )| ≤ C κ |X -Y |, |σ κ (t, x) -σ κ (t, y)| ≤ C κ |x -y|.
Proof.

Using the assumption and the fact that the mapping X → e X ∧κ is Lipschitz, we get that

|σ κ (t, e X ) -σ κ (t, e Y )| = |σ κ (t, e X ∧ κ) -σ κ (t, e Y ∧ κ)|, ≤ C κ |e X ∧ κ -e Y ∧ κ| ≤ C κ |X -Y |.
On the other hand, we have

|σ κ (t, x) -σ κ (t, y)| = |σ κ (t, x ∧ κ) -σ κ (t, y ∧ κ)|, ≤ C κ |x ∧ κ -y ∧ κ| ≤ C κ |x ∧ κ -y ∧ κ|, ≤ C κ |x -y|.
The last inequalities hold since |x ∧ κ -y ∧ κ| is bounded by √ 2κ and the

mapping x → x ∧ κ is Lipschitz. 2
Proof of Lemma 3.1. Suppose that Condition 2.2 holds. Consider the sequence of stopping times

τ κ := inf{t : S t ≥ κ}. (8.11)
Since the stopped processes S τ κ and (S κ ) τ κ satisfy the same s.d.e., we de-

duce that S t = S κ (t) on t ∈ [0, τ κ ]. As τ κ → ∞,
dM u = M u σ κ (u, S κ (u))dW u .
As r is supposed to be bounded, we deduce that S κ (u) ≤ CS 0 M u . Let (τ n ) be a sequence of stopping times such that M τ n is a true martingale. Then, ES κ (u∧τ n ) ≤ CS 0 and finally ES κ (u) ≤ CS 0 by virtue of by Fatou's lemma. Recall that σ κ is bounded. Using the Doleans-Dade representation of S κ (u), we deduce that S κ (u) admits finite moments at any order. It follows that the Ito integral in the inequality above is a square integrable martingale.

Therefore,

E(S κ (u)) q = (S 0 ) q + u 0 qE(S κ (r)) q r κ (r, S κ (r))dr, + 1 2 q(q -1) u 0 E(S κ (r)) q σ 2 κ (r, S κ (r))dr.
We then use the inequality r ≤ 1 + r 2 and by Assumption 3.3 (II), we deduce that g j+1 (u) := E(S κ (u)) 2(j+1)α satisfies

g j+1 (u) ≤ max j≤N (S 0 ) 2(j+1)α + C(N 2 α 2 + N α) u 0 g j (r)dr +C N α + N 2 α 2 u 0 g j+1 (r)dr
Suppose that g j ≤ C j where C j does not depend on κ. Then, by Gronwall's lemma, we deduce that g j+1 ≤ C j+1 where C j+1 does not depend on κ. The conclusion follows with j = N since the number of steps N (p) is finite and only depends on p. 2 Corollary 8.6. Assume that Condition 3.3 is satisfied. Then, for all p ≥ 0, there exists a constant C p independent of κ such that E sup u≤T (S κ (u)) p ≤ C p .

Proof. Let us now suppose that Condition (II) holds. In this case,

Since • L p 1 ≤ • L p 2 if 1 ≤ p 1 ≤ p 2 ,
0 ≤ S p κ (u)(σ p (u, S κ (u) ∨ κ -1 ) + r p (u, S κ (u) ∨ κ -1 )) ≤ C(S p κ (u) + S (1-α)p κ (u)).
Since 1 -α ≥ 0, we use Lemma 8.5 and conclude as previously. 2

Since S κ converges pointwise on [0, τ ] to S, S = S κ on [0, τ κ ] where τ κ → τ , and S t = 0 on t ≥ τ , we apply Fatou's Lemma and deduce the following. S κ (t) -S t p 1 τ κ <T .

In the case where τ k is defined by (8.11), we deduce that

E sup t∈[0,T ] S κ (t) -S t p ≤ E sup t∈[0,T ] S κ (t) -S t p 1 sup t∈[0,T ] St≥κ ≤ E sup t∈[0,T ] S κ (t) -S t 2p E sup t∈[0,T ] S t 2l κ 2l .
The conclusion follows. In the case where σ κ = σ(•, • ∨ κ -1 ), let τ k be defined by (8.12). Then,

E sup t∈[0,τ ∧T ] S κ (t) -S t p = E sup t∈[0,T ] S κ (t) -S t p 1 τ κ <t≤τ ∧T ≤ E sup t∈[0,T ] S κ (t) -S t p 1 τ κ <t≤τ 1 τ ≤T +E sup t∈[0,T ] S κ (t) -S t p 1 inf t∈[0,T ] St≤κ -1 1 τ >T .
Notice that the second term of the last inequality above is bounded as follows:

E sup t∈[0,T ] S κ (t) -S t p 1 inf t∈[0,T ] St≤κ -1 1 τ >T ≤ E sup t∈[0,T ] S κ (t) -S t 2p γ(τ, κ),
where γ(τ, κ)

:= P (inf t∈[0,T ] S t ≤ κ -1 ; τ > T ) → 0 as κ → ∞. If t ∈ [τ κ , τ ],
where τ < T , then

S κ (t) = κ -1 + t τ κ S κ (u)σ κ (u, S κ (u))dW u + t τ κ S κ (u)r κ (u, S κ (u))du, S t = κ -1 + t τ κ S(u)σ(u, S(u))dW u + t τ κ
S(u)r(u, S(u))du.

Under Condition (II) of Condition 3.3, we have

|S κ (u)σ κ (u, S κ (u))| + |S κ (u)r κ (u, S κ (u))| ≤ C(|S κ (u)| + |S κ (u)| 1-α ), |S κ (u)σ κ (u, S κ (u))| + |S κ (u)r κ (u, S κ (u))| ≤ C(|S κ (u)| + |S κ (u)| 1-α ).
Using the Burkholder-Davis-Gundy inequalities and the Cauchy-Schwarz inequality, we deduce that the term δ(τ, κ) = E sup t∈[0,T ] S κ (t)-S t p 1 τ κ <t<τ 1 τ ≤T is bounded as follows:

δ(τ, κ) ≤ C sup κ E sup t∈[0,T ] |S κ (t)| 2p + E sup t∈[0,T ] |S(t)| 2p E|τ κ ∧ T -τ ∧ T | p .
By the Lebesgue theorem, since τ κ → τ , we deduce that E|τ κ ∧T -τ ∧T | p → 0 as κ → ∞. The conclusion follows. 2 Lemma 8.9. Let (g m ) m be a sequence of positive functions defined on an interval [0, T ], T > 0, such that for some C > 0, we have:

g m+1 (t) ≤ C t 0 g m (u)du, 0 ≤ g 0 ≤ C. Then sup t∈[0,T ] g m (t) ≤ C m t m m! sup t∈[0,T ] g 0 (t).
Proof. The proof relies on the following recursive reasoning. Set

C g := sup t∈[0,T ] g 0 (t) ≤ C. Suppose that sup t∈[0,T ] g n-1 (t) ≤ C n-1 t n-1 (n -1)! C g .
Then, we have

g n (t) ≤ C t 0 C n-1 u n-1 (n -1)! C g du.
The claim follows. 2

Proof of Lemma 3.7. If α ≥ 2, recall that for all x, y ≥ 0:

x α/2 -y α/2 ≤ α 2 (max(|x|; |y|)) α/2-
)| ≤ C κ |X -Y |, hence σ κ,m+1 (u) -σ κ,m+1 (u) + r κ,m+1 (u) -r κ,m+1 (u) ≤ C κ Y κ,m (u) -Y κ,m-1 (u) .
We deduce that

E sup u≤t Y κ,m+1 (u) -Y κ,m (u) 2 ≤ C κ t 0 E sup u≤s Y κ,m (u) -Y κ,m-1 (u) 2 ds,
where C κ is a constant depending on κ. Using Lemma 8.9, we get that

E sup u≤t Y κ,m+1 (u) -Y κ,m (u) 2 ≤ (C κ ) m+1 E sup u≤t Y κ,1 (u) -Y κ,0 (u) 2 t m m! , ≤ C(C κ ) m+1 t m+1 (m + 1)! (1 + S 2 0 ), ( 8
E sup u≤t Y κ,∞ (u) -Y κ,m (u) 2 ≤ ∞ j=m 2 j E sup u≤t Y κ,j+1 (u) -Y κ,j (u) 2 .
We deduce the first inequality from (8.13) and, similarly, the second one. The conclusion follows. 2

Proof of Lemma 3.9. Applying the Burkholder-Davis-Gundy and Jensen inequalities, we get that where we recall that σ κ,m (s) := σ κ (s, S κ,m-1 (s)). The first term in the r.h.s.

E sup u∈[0,t] S κ,m+1 (u) -S κ,m (u) 2 ≤ CE t 0 (S κ,m+1 (
of the inequality is bounded from above as follows:

E t 0 (S κ,m+1 (s)σ κ,m+1 (s) -S κ,m (s)σ κ,m (s)) 2 ds ≤ 2E T 0 S 2 κ,m+1 (s) [σ κ,m+1 (s) -σ κ,m (s)] 2 ds + 2E T 0 (σ m,κ (s)(S κ,m+1 (s) -S κ,m (s))) 2 ds.
S κ,m+1 (u) -S κ,m (u) 2 ≤ C κ E sup u≤T Y κ,m (u) -Y κ,m-1 (u) 4 + C κ t 0 E sup u≤s (S κ,m+1 (u) -S κ,m (u)) 2 ds.
Using Gronwall's Lemma, we deduce that

E sup t∈[0,T ] (S κ,m+1 (t) -S κ,m (t)) 2 ≤ C κ E sup t≤T |Y κ,m (t) -Y κ,m-1 (t)| 4 .

The conclusion follows. 2

Proof of Lemma 3.10. By virtue of Lemma 8.10, we get the inequality

E sup t∈[0,T ] S κ,∞ (t) -S κ,m (t) 2 ≤ ∞ j=m 2 j E sup u∈[0,t] S κ,j+1 (u) -S κ,j (u) 2 ≤ C κ ∞ j=m (C κ ) j √ j! . 2
Proof of Lemma 3.11. Since σ κ and r κ are bounded by a constant depending on κ, we deduce by the Burkholder-Davis-Gundy inequalities and the Gronwall's lemma a constant c κ > 0 such that for all m, n,

E sup t∈[0,T ] S κ,m (t) + S κ,m,n (t) 4 ≤ c κ . (8.15)
For a sake of simplicity, we suppose that r = 0 as the proof is similar for the general case. Therefore,

S κ,1 (t) = S 0 + t 0 S κ,1 (u)σ κ (u, S 0 )dW u , S κ,1,n (t) = S 0 + t 0 S κ,1,n (u)σ κ (u (n) , S 0 )dW u .
Let us introduce g m (t) := E sup u∈[0,t] S κ,m (u) -S κ,m,n (u) 2 , m ≥ 1. By the Doob's inequality,

g 1 (t) ≤ 4 t 0 S κ,1,n (u)σ κ (u (n) , S 0 ) -S κ,1 (u)σ κ (u, S 0 ) 2 du, g 1 (t) ≤ 8 t 0 σ 2 κ (u (n) , S 0 ) (S κ,1,n (u) -S κ,1 (u)) 2 du + 8 t 0 S 2 κ,1 (u) σ κ (u (n) , S 0 ) -σ κ (u, S 0 ) 2 du. Since σ κ is bounded by a constant C κ and |σ κ (u (n) , S 0 ) -σ κ (u, S 0 )| ≤ T /n,
we deduce by (8.15) that

g 1 (t) ≤ C κ n 2 + C κ t 0 g(u)du, t ∈ [0, T ].
By the Gronwall's lemma, we deduce that g 1 (T ) ≤ C κ /n 2 for some constant C κ which does not depend on n. More generally, if m ≥ 2, we have

S κ,m (t) = S 0 + t 0 S κ,m (u)σ κ (u, S κ,m-1 (u))dW u , S κ,m,n (t) = S 0 + t 0 S κ,m,n (u)σ κ (u (n) , S (n)
κ,m-1,n (u))dW u .

Therefore, by the Doob's inequality,

g m (t) ≤ 4 t 0 S κ,m,n (u)σ κ (u (n) , S (n)
κ,m-1,n (u)) -S κ,m (u)σ κ (u, S κ,m-1 (u)) 2 du, g m (t) ≤ 8 t 0 σ 2 κ (u (n) , S (n)
κ,m-1,n (u)) (S κ,m,n (u) -S κ,m (u)) 2 du + 8 t 0 S 2 κ,m (u) σ κ (u (n) , S (n) κ,m-1,n (u)) -σ κ (u, S κ,m-1,n (u)) 2 du + 8 t 0 S 2 κ,m (u) (σ κ (u, S κ,m-1,n (u)) -σ κ (u, S κ,m-1 (u))) 2 du.
Recall that σ κ is bounded by a constant C κ and

|σ κ (u (n) , S κ,m-1,n (u)) -σ κ (u, S κ,m-1,n (u))| ≤ C κ /n.
Therefore, we may obtain upper bounds for the two first terms of the r.h.s. in the inequality above as we do for m = 1. Precisely, we also need to estimate E(S

(n) κ,m-1,n (u) -S κ,m-1,n (u)) 2 .
To do so, it suffices to recall that

E sup t∈[[0,T ]] S 4 κ;m (t) ≤ C κ for some constant C κ which does not depend on m hence, if u ∈ [t n i , t n i+1 ), i = 0, • • •
, n, we deduce by the Ito isometry and the Cauchy-Schwarz inequality that E(S

(n) κ,m-1,n (u) -S κ,m-1,n (u)) 2 ≤ C κ n -1
where C κ only depends on κ.

At last, an upper bound for the third term is obtained using the Cauchy-Schwarz inequality, (8.15) and the following inequality (see Lemmas 8. Proof. Suppose that E|1 τ S >T -1 τ S n >T | → l > 0 for a subsequence. Since | inf t∈[0,T ] S t -inf t∈[0,T ] S n t | ≤ sup t∈[0,T ] S t -S n t , we may also suppose that inf t∈[0,T ] S n t converges to inf t∈[0,T ] S t a.s. for a common subsequence. In the case where τ > T , i.e. inf t∈[0,T ] S t ≥ a, as P (inf t∈[0,T ] S t = a) = 0, we may assume that inf t∈[0,T ] S t > a hence inf t∈[0,T ] S n t > a for n large enough and, finally τ S n > T . If τ ≤ T , inf t∈[0,T ] S n t < a and then, similarly, we obtain that As the mapping h(x) = x ∨ κ -1 is not differentiable at point κ -1 , we first approximate h by

h n (x) = κ -1 1 (-∞,κ -1 -n -1 ] (x) + x1 [κ -1 +n -1 ,∞) (x) + κ -1 + n 4 (x -κ -1 + n -1 ) 2 1 [κ -1 -n -1 ,K+n -1 ] (x).
We have the following properties:

0 ≤ h n (x) -h(x) ≤ 1 4n , |(h n ) (x) -h (x)| ≤ 1 [κ -1 -n -1 ,κ -1 +n -1 ] (x).
We then consider the solution Y n κ of the s.d.e. (3.4) where we replace σ κ by σ n κ (t, x) := σ(t, h n (x)).

We may show that sup t∈[0,T ] |Y n κ (t) -Y κ (t)| → 0 in L 2 as n → ∞. Therefore, since Y n κ admits a derivative in the Malliavin sense, we also deduce that Y κ ∈ D 1,2 . Moreover, for all s ≤ t, making n → ∞, we get that D s Y κ (t) = σ κ (s, S 0 e Yκ(s) ) + t s D s Y κ (u)h (S 0 e Yκ(u) )S 0 e Yκ(u) ∇ x σ κ (t, S 0 e Yκ(u) )dW u , + t s D s Y κ (u)h (S 0 e Yκ(u) )S 0 e Yκ(u) ∇ x γ κ (t, S 0 e Yκ(u) )du, where γ κ = r κ -σ 2 κ /2. By the assumptions, e.g. X∇ X σ(t, X) is bounded if X ≥ κ -1 and h (X) = 0 if X ≤ κ -1 , and the Gronwall's lemma, we deduce

Since G is supposed to be Lipschitz, the first term (8.16) is bounded from above by ε κ as stated in Lemma 3.6. The second term (8.17

Proposition 2 . 1 .

 21 Let σ : [0, T ]×R → R and r : [0, T ]×R → R be Lipschitz functions. The stochastic differential equation (s.d.e.)

Lemma 3 . 7 .

 37 The sequence Y κ,m converges in L 4 to Y κ,∞ and

and χ 2

 2 (x, k, λ) is the cumulative density function of the non-central chisquared distribution with degrees of freedom k and non-centrality parameter λ. The price of the Put option with strike K is deduced as

2 Lemma 8 . 3 .

 283 we then conclude. Under Condition 2.3, we follow the same arguments with τ κ := inf{t : S t ≤ κ -1 }. (8.12) Note that τ κ → τ as κ → ∞. Assume that Condition 3.3 (I) is satisfied. Then, there exists a constant C independent of κ such that sup u≤T ES κ (u) ≤ C. Proof. By definition, we have S κ (u)/F u (r κ (•, S κ (•))) ≤ M u where M 0 = 1 and M is a local martingale which is solution to the s.d.e.

2 Lemma 8 . 4 .

 284 Assume that Assumption 3.3 (I) is satisfied. Then, for all p such that p ≤ 0 or p ≥ 1/2, there exists a constant C p independent of κ such that sup u≤T E(S κ (u)) p ≤ C p .

  and (S κ (u)) p ≤ 1 + S κ (u) when p ∈ [0, 1], we may assume without loss of generality that p ∈ 2N. Let us first suppose that Condition (I) holds. We recall thatS κ (t) = S 0 + t 0 σ(u, S κ (u) ∧ κ)S κ (u)dW u + t 0 r(u, S κ (u) ∧ κ)S κ (u)du.Since 0 ≤ σ 2 (t, x) ≤ C(1 + x) and r is bounded, we easily deduce that E sup u≤T (S κ (u)) p ≤ C p by the Burkholder-Davis-Gundy and Jensen inequalities as well as Lemma 8.4.

0 (

 0 s)σ κ,m+1 (s) -S κ,m (s)σ κ,m (s)) 2 ds, +2E t S κ,m+1 (s)r κ,m+1 (s) -S κ,m (s)r κ,m (s)) 2 ds,

  |σ κ (u, S κ,m-1,n (u)) -σ κ (u, S κ,m-1 (u)| ≤ C κ |S κ,m-1,n (u) -S κ,m-1 (u)|.

2 Lemma 8 . 11 . 2 →

 28112 Therefore,g m (t) ≤ C κ n + C κ g m-1 (t) + C κ T 0 g m (u)du.By the Gronwall's lemma, we then deduce a constant C κ which only depends on κ such thatg m (T ) ≤ C κ n + C κ g m-1 (T ).By induction, we deduce that, for all m ≥ 2,g m (T ) ≤ n -1 m j=1 (C κ ) j ≤ α κ (C κ ) m n -1 ,for some constant α κ . Let S be a positive stochastic process such that P (inf t∈[0,T ] S t = a) = 0 for all a ∈ (0, S 0 ). Let (S n ) n∈N be a sequence of stochastic processes such that E sup t∈[0,T ] S t -S n t 0 as n → ∞. Let us define τ S := inf{t : S t < a} and τ S n := inf{t : S n t < a}. Then, E|1 τ S >T -1 τ S n >T | → 0 as n → ∞.

κ - 1 )

 1 under Condition 2.3 with P (τ ≤ T ) > 0. The random variables Z κ = inf t∈[0,T ] Y κ (t), κ > 0, and Z κ,m = inf t∈[0,T ] Y κ,m (t), m ≥ 1, admit a density with respect to the Lebesgue measure. Proof. The proof is based on [19][Proposition 2.1.11]. To do so, we need verify the conditions of [19][Proposition 2.1.10].

  |D s Y κ (t)| p < ∞, whatever p ≥ 2. By the Ito formula, we obtain an expression of [D s Y κ (t)] 2 for t ≥ s as a stochastic integral. By the stochastic Fubini theorem, we then also deduce a representation of t → t 0 [D s Y κ (t)] 2 ds as a stochastic integral. Using the Gronwall's lemma, we finally deduce that E sup t≤T t 0 [D s Y κ (t)] 2 ds < ∞, i.e. the conditions of [19][Proposition 2.1.10] hold. Moreover, as (D s Y κ (t)) t≥s is a Doléans-Dade exponential, it is strictly positive and we finally deduce that [19][Proposition 2.1.11] applies, i.e. Z κ = inf t∈[0,T ] Y κ (t) admits a density with respect to the Lebesgue measure. The second statement is shown similarly by induction for m = 0, 1, • • • . 2Proof of Theorem 3.12. Notice that we have|EG(S)1 τ >T -EG(S (n) κ,1,n )1 τ κ,1,n >T | ≤ E|G(S) -G(S κ )|1 τ >T (8.16) +E|G(S κ ) -G(S κ,m )| (8.17) +E|G(S κ,m ) -G(S κ,m,n )| (8.18) +E|G(S κ,m,n ) -G(S (n) κ,m,n ) (8.19) +E|G(S)||1 τ >T -1 τ κ,1,n >T |, .

  ) is bounded from above by the error ε κ,m given by Lemma 3.10. The term (8.18) is bounded from above according to Lemma 3.11. The consistency property with respect to discretization satisfied by G allows us to estimate the fourth term(8.19).As to the last term, notice that G is of linear growth since it is Lipschitz. By Corollary 8.7, we deduce from the Cauchy-Schwarz inequality a constant C such thatE|G(S)||1 τ >T -1 τ κ,1,n >T | ≤ C E|1 τ >T -1 τ κ,1,n >T |. It remains to show that E|1 τ >T -1 τ κ,m,n >T | → 0 a.s. as κ, m, n → ∞. Let us introduce τ κ := inf{t : S κ (t) < κ -1 }. Since S and S κ coincides on [0, τ κ ] and τ κ = τ κ > T if κ is large enough and τ > T , we deduce that |1 τ >T -1 τ κ >T | → 0 if τ > T as κ → ∞. If τ ≤ T , we have τ κ ≤ τ so that |1 τ >T -1 τ κ >T | = 0. Therefore, E|1 τ >T -1 τ κ >T | → 0 as κ → +∞. It remains to show that E|1 τ κ >T -1 τ κ,m,n >T | → 0 a.s. as κ, m, n → ∞.Let us introduce τ κ,m := inf{t : S κ,m (t) < κ -1 }. By Proposition 3.10, Lemma 8.13 and Lemma 8.11, for fixed κ, E|1 τ κ >T -1 τ κ,m >T | → 0 as m → ∞. Therefore, it remains to show that E|1 τ κ,m >T -1 τ κ,m,n >T | → 0 a.s. as n → ∞. By virtue of Lemma 3.11, Lemma 8.13 and Lemma 8.12, E|1 τ κ,m >T -1 τ κ,m,n >T | → 0 a.s. as n → ∞. 2

Table 1 .

 1 Put option prices using Picard iterations and the analytic expression respectively for different values of β > 1.

	β	Picard	Time in s.	BEM	Time in s.	Analytic	Err. (P)	Err. (BEM)
	1.5	7.8602	6.38		7.8839	0.78			7.9696	1.4%	1.08%
	2	7.8995	6.19		7.8930	0.76			7.9789	1.01%	1.08%
	2.5	7.9948	6.09		7.9087	0.79			7.9960	0.02%	1.09%
	3	8.1332	6.08		8.9329	0.76			8.0211	1.4%	1.10%
	3.5	8.2779	6.06		7.9686	0.80			8.0560	2.76%	1.09%
	4	8.4006	6.06		8.0200	0.81			8.1033	3.67%	1.03%
	4.5	8.4886	6.13		8.0841	0.80			8.1639	3.98%	0.98%
	5	8.5348	6.08		8.1583	0.78			8.2345	3.65%	0.93%
	5.5	8.5344	5.936		8.2365	0.78			8.3084	2.72%	0.87%
	6	8.5015	5.906		8.3086	0.80			8.3786	1.47%	0.84%
	6.5	8.4499	6.096		8.3715	0.78			8.4396	0.13%	0.81%
	7	8.3821	6.401		8.4228	0.82			8.4882	1.26%	0.77%
	β	Picard	Comput. Time	BEM	Time	MC [32]	Time	Error
	1.5	7.961482891	268.748	7.9572	31.73	7.958596698	20.756	0.03%
	2	8.001945176	276.356	7.9692	28.28	7.968447144	12.571	0.42%
	2.5	8.096442435	286.754	7.9841	26.05	7.981469166	13.68	1.44%
	3	8.231139717	298.65	8.0099	23.60	8.002688168	13.098	2.86%
	3.5	8.382673524	317.32	8.0483	22.09	8.050143346	10.78	4.14%
	4	8.507583601	338.079	8.0948	20.57	8.091346416	7.908	5.15%
	4.5	8.594539064	345.449	8.1543	19.06	8.164748506	5.855	5.17%
	5	8.631951141	351.436	8.2231	17.13	8.234170007	3.698	4.84%

Table 2 .

 2 Put

option prices using Picard iterations, BEM and Monte Carlo methods respectively for different values of β > 1 so that the radius of the 95% confidence interval is less than 0.025. of large number. The simple Monte Carlo simulation is less time consuming.

Nevertheless, we shall see that the two other methods outperform the Monte Carlo simulations as soon as we consider an Asian Put option with several dates.

Table 3 .

 3 Put option prices with truncated Doléans-Dade scheme for different values of β.

	β	Price (truncated Doléans-Dade scheme)	Comput. time in seconds	N	Error
	1.5	7.1412	535	5.10 6	11.6%
	2	7.1978	555	5.10 6	10.85%
	2.5	7.2544	553	5.10 6	10.22%
	3	7.3134	549	5.10 6	9.68%
	3.5	7.3779	556	5.10 6	9.19%
	4	7.4524	564	5.10 6	8.73%
	4.5	7.5999	541	5.10 6	7.42%
	5	8.0123	0.014	136	3.65%
	5.5	8.4954	0.013	136	2.72%
	6	10.7683	523	5.10 6	28.5%
	6.5	11.934	518	5.10 6	41.4%
	7	12.9042	518	5.10 6	52%

Table 4 .

 4 Asian

	β	P	Time in s.	BEM (B)	Time	MC	Time	Error P-C	Error B-C
	1.5	6.293809919	171.191	6.2947	20.49	6.307557882	10.375	0.22%	0.20%
	2	6.314878756	173.881	6.3012	18.93	6.311235573	3.961	0.06%	0.16%
	2.5	6.373160059	179.582	6.3126	17.68	6.320525289	3.429	0.84%	0.12%
	3	6.460785842	186.615	6.3216	16.33	6.331600424	3.015	2.04%	0.16%
	3.5	6.559108888	198.54	6.3398	15.48	6.347881289	2.524	3.33%	0.13%
	4	6.655375191	205.6	6.3670	14.41	6.366022368	2.317	4.55%	-0.02%
	4.5	6.734410841	216.076	6.3994	13.56	6.390785176	2.13	5.38%	-0.13%
	5	6.781869529	219.877	6.4358	12.62	6.42642745	2.009	5.54%	-0.15%

Put option prices with two dates using Picard iterations (P), the Monte Carlo methods combined with the analytic expression (MC) and a truncated Euler scheme (E) respectively. The errors are computed between the Picard method (P) and the Euler scheme (E) respectively with respect to the combined method (C).

Table 5 .

 5 Asian Put option prices with two dates using Picard iterations and the Monte Carlo methods combined with the analytic expression respectively such that the 95% confidence interval radius is smaller than 0.025.As we may observe, it is still advantageous to combine the Monte Carlo method with the analytic one in the case of an Asian option with only two dates. Moreover, we still observe that the simple truncated Euler scheme is efficient for β ≤ 5 while it produces a large convergence error if β > 5 contrary to the Picard iterations. the Asian Put option of strike 3K -s in the CEV model with initial value s, intermediate date T 2 -T 1 and horizon date T 3 -T 1 . By the Monte Carlo method, we simulate S T 1 so that we deduce P (S T 1 ) using the pricer for two dates we have introduced before. In that case, the complexity explodes as we need to run two successive Monte Carlo schemes. With the same parameters but only with β = 1.5 or β = 7 and T 1 = T /3, T 2 = 2T /3 and T 3 = T = 1,

	we get the following results:
	β	P	Time	BEM	Time	MC	Time	Error P-C	Euler (E)	Error E-C	Error B-C
	1.5	5.3788	61.73	5.6768	7.87	5.2698	8333	2.07%	5.8172	10.39%	7.72%
	7	5.6278	61.83	5.9535	8.36	5.5188	2306	1.98%	10.2290	85.35%	7.88 %

Let us now suppose that J = 3. By the Markov property, we rewrite after conditioning E(3K -S T 1 -S T 2 -S T 3 ) + = EP (S T 1 ) where P (s) is the price of

Table 6 .

 6 Asian Put option price with three dates when β = 1.5 or β = 7 using Picard iterations, the Monte Carlo methods combined with the analytic expression and the truncated Euler scheme respectively. The errors are computed between the Picard method (P) and the Euler scheme (E) respectively with respect to the combined method (C).

	β	P	Time	BEM	Time	CI	MC	Time	Error P-C	Error B-C
	1.5	5.345	1322	5.7254	17.724	0.05	5.26	372089	1.62%	8.85%

Table 8 .

 8 Put option prices using truncated Euler Scheme and the analytic expression respectively for different values of β < 1.

	β	Truncated Euler scheme	Time (ms)	Analytic	Time (ms)	Error	# of simulations
	0	37.5155	18.254	39.0452	< 0.01	3.92%	8618
	0.1	38.8543	4.662	39.0089	< 0.01	0.40%	1891
	0.2	38.9200	1.806	38.9307	< 0.01	0.03%	745
	0.3	38.8460	1.106	38.8210	< 0.01	0.06%	460
	0.4	38.7139	1.164	38.6962	< 0.01	0.05%	487
	0.5	38.5853	1.099	38.5753	< 0.01	0.03%	456
	0.6	38.4624	2.287	38.4724	< 0.01	0.03%	930
	0.7	38.2547	4.537	38.3928	< 0.01	0.36%	1866
	0.8	37.4803	12.511	38.3368	< 0.01	2.23%	5198
	0.9	36.9813	17.533	38.3041	< 0.01	3.45%	7479
	Let us now present the approximated prices obtained by the Picard method
	with n = 20.

Table 9 .

 9 Put option prices using Picard iterations and the analytic expression respectively for different values of β < 1.As we may observe, the Picard method outperforms the truncated euler method, i.e. provides smaller error with less computation time. For two dates Asian options we choose n = 40 in order to work with the same mesh.

	β	(P)	Time (ms)	MC (C)	Time (ms)	Error P-C	Euler (E)	Time (ms)	Error E-C
	0,5	28,8773	266,07	32,5530	0,103	12,73%	28,0664	171,11	15,99%
	0,6	31,3302	11,391	31,3744	0,124	0,14%	30,9665	18,045	1,32%
	0,7	30,7832	5,097	30,7640	0,137	0,06%	30,7794	2,305	0,05%
	0,8	30,6667	2,84	30,6624	0,149	0,01%	30,6813	3,402	0,06%
	0,9	30,6594	4,208	30,6532	0,194	0,02%	30,6334	3,826	0,06%

Table 11 .

 11 Estimation of ES T with T = 6 and S 0 = 1 for several values of β.

	β	Milstein Picard Error Time (s)
	0.5	0.9968	0.9980 0.12%	1.103
	0.55	0.9969	0.9974 0.05%	0.631
	0.6	0.9969	0.9977 0.07%	0.633
	0.65	0.9969	0.9976 0.07%	0.643
	0.7	0.9968	0.9977 0.09%	0.656
	0.75	0.9967	0.9979 0.12%	0.760
	0.8	0.9965	0.9972 0.07%	0.835
	0.85	0.9961	0.9974 0.13%	0.886
	0.9	0.9955	0.9965 0.10%	0.979
	0.95	0.9946	0.9960 0.14%	1.129
	1	0.9932	0.9945 0.12%	0.853

table, the estimations are very accurate.

  Corollary 8.7. Assume that Condition 3.3 is satisfied. Then, for all p ≥ 0,there exists a constant C such that E sup u≤T (S u ) p ≤ C p . Under Condition 3.3 (I), since 0 ≤ σ 2 (t, x) ≤ C(1 + x), we then deduce that E sup u≤T Y 2u < ∞ as the coefficient r is bounded. In particular, the process• 0 σ(t, S t )dW t is a true martingale.Proof of Lemma 3.6. If σ κ = σ(•, • ∧ κ), we consider the stopping time τ k

	Remark 8.8. Recall that
	dY t = σ(t, S t )dW t + r(t, S t )dt -	1 2	σ 2 (t, S t )dt.
	defined by (8.11). We obtain that
	E sup	S κ (t) -S t	p = E sup	S κ (t) -S t	p 1 τ κ <t
	t∈[0,T ]		t∈[0,T ]
			≤ E sup
			t∈[0,T ]

  1 |x -y|.|σ κ (t, e X ) -σ(t, e Y )| + |r κ (t, e X ) -r(t, e Y

	We deduce by Conditions 2.2 and 2.3 that
		t	2		t
		σ 2 κ,m+1 (s) -σ 2 κ,m (s)ds	≤ CC κ		(σ κ,m+1 (s) -σ κ,m (s)) 2 ds,
		0		0
	by virtue of Jensen's inequality. Together with the Burkholder-Davis-Gundy
	inequalities, we obtain that
	E sup	Y κ,m+1 (u) -Y κ,m (u)	2 ≤ CC κ E	t	sup	σ κ,m+1 (u) -σ κ,m (u)	2 ds
	u≤t			0		u≤s
					t
			+ CC κ E		sup	r κ,m+1 (u) -r κ,m (u) ds.
				0		u≤s
	By Conditions 2.2 or 2.3, with Lemma 8.2,

  Lemma 8.10. Let a 1 , • • • a k be real numbers. we have the following inequality For any real numbers a, b, the inequality 2ab ≤ a 2 + b 2 implies that(a + b) 2 ≤ 2a 2 + 2b 2 . Therefore, (a + b) 4 ≤ (2a 2 + 2b 2 ) 2 ≤ 8a 4 + 8b4 . We conclude by induction. 2 Proof of Corollary 3.8. Using Lemma 8.10, we obtain the inequality

	k	2	k	k	4	k
	a l	≤	2 l a 2 l ,	a l	≤	8 l a 4 l .
	l=1		l=1	l=1		l=1
	Proof.

.13)

where C does not depend on m and κ. We deduce Inequality

(3.6)

. Similarly, we obtain Inequality (3.7) and the claim follows. 2

  The second one is similarly bounded. Recall the inequalities |σ κ | + |r κ | ≤ C κ , κ > 0, and recall that, by assumption or by Lemma 8.2,|σ κ (t, e X ) -σ κ (t, e Y )| + |r κ (t, e X ) -r κ (t, e Y )| ≤ C κ |X -Y |.Using the stochastic integral representation of S κ,m+1 and S κ,m respectively,

	(8.14) Moreover, using Gronwall's lemma, since σ we obtain by means of Cauchy-Schwarz's inequality, Jensen's inequality and the Ito isometry that E sup u∈[0,t]

κ and r κ are bounded by a constant depending on κ, we deduce the existence of C p,κ > 0 such that for all p sup m E sup t∈[0,T ] S 2p κ,m (t) ≤ C p,κ .

Proof. Since r is bounded, the following inequality holds for all fixed q ≥ 0:

(S κ (u)) p ≤ C q S p 0 N u e 1 2 (q-p) u 0 σ 2 κ (r,Sκ(r))dr ,

where

Using the inequality 0 ≤ ab ≤ (a 2 + b 2 ), we get that (S κ (u)) p ≤ CS p 0 Ñu + CS p 0 e (q-p) u 0 σ 2 (r,Sκ(r)∧κ)dr , where Ñ = N 2 is a local martingale if we choose q = 2p 2 . Moreover, as the function x → e (q-p)ux is convex, the Jensen inequality and the hypothesis yields that

where k is a constant. Using the property (log(x ∨ 1)) k(q-p)T ≤ C p x, ∀x ≥ 0, and Lemma 8.3, we deduce that E(S κ (u)) p ≤ C p . 2 Lemma 8.5. Assume that Condition 3.3 (II) is satisfied. Then, for all p such that p ≥ 0, there exists a constant C p independent of κ such that

it is enough to show the lemma when p = 2N α. We show by induction that the statement holds for every p = 2jα, 0 ≤ j ≤ N . This is trivial if j = 0.

Applying the Ito formula when q = 2(j + 1)α, j ≥ 0, we get that (S κ (u)) q = (S 0 ) q + u 0 q(S κ (r)) q σ κ (r, S κ (r))dW r + r 0 q(S κ (r)) q r κ (r, S κ (r))dr,

and the conclusion follows. Note that the same property holds if we replace the infinimum by the supremum of S over I.

On the set {inf t∈[0,T ] S t < a}, as S is continuous, we may construct by a measurable selection argument a measurable interval

ε > 0 a.s., such that sup t∈I S t < a. Otherwise, we arbitrarily set

Suppose that for a subsequence E|1 τ S >T -1


James Lovelock's Gaia hypothesis: "A New Look at Life on Earth" ... for the Life and the Earth sciences

1 James Lovelock's Gaia hypothesis: "A New Look at Life on Earth" ... for the Life and the Earth sciences.

Sébastien Dutreuil

James Lovelock (b. 1919) was described by the curators of an exhibition at London's Science Museum in 2014 as a "scientist, inventor and maverick." 1 He was clearly an eclectic inventor from the very beginning of his career as a research engineer in the 1940's. He was an accomplished scientist before formulating the Gaia hypothesis in the 1970's, with pioneering work in analytical chemistry, biochemistry and cryobiology. He was perhaps a maverick, when he quit academia in 1964, at age 45 to settle as an "independent scientist." But the Gaia hypothesis, his major accomplishment, is that of a dreamer. As he described it, Gaia was for those who like to walk or simply stand and stare, to wonder about the Earth and the life it bears, and to speculate about the consequences of our own presence here. 2 From the very beginning, the Gaia hypothesis was thought as something big, challenging the paradigmatic views prevailing in the earth and life sciences, redefining the boundaries and questions of these disciplines, providing a new conception of life, nature and the Earth.

But for many biologists, the Gaia hypothesis was just the dream of a "romantic and new-ager" 3 wandering, astray from science. Here is what the microbiologist John Postgate says about it: Gaia -the great mother Earth! The planetary organism! Am I the only biologist to suffer a nasty twitch, a feeling of unreality, when the media invite me yet again to take her seriously? 4 Was Gaia just a misty dream? Was it only an evocative metaphor comparing Earth with an organism, as it has often been resumed to? By postulating a new entity emerging from the interconnection of life and geological processes, the Gaia hypothesis not only had a revolutionary influence on the constitution of a new scientific field of the Earth sciences, but on the way we collectively think about nature.

Dreaming about life in the solar system ... and back to Earth

Lovelock received, on October, 19 th , 1961, a letter of invitation from NASA to work at the Jet Propulsion Laboratory (JPL), Pasadena, as a consultant engineer on instruments related to space exploration -a chromatograph for the 1964 Mariner B mission to Mars. As a science-fiction reader, Lovelock was thrilled and accepted.

As he commonly tells the story, the Gaia hypothesis traces back to a very practical problem he encountered while working at JPL: how would one detect life on a further planet, such as Mars or Venus? In 1965, Lovelock turned his back from the biochemical approaches to life, prevailing in exobiology, to hinge on physical, thermodynamical approaches. 5 He pointed out that Earth's atmosphere is in great thermodynamic disequilibrium: e.g. methane and oxygen coexist in proportion orders of magnitude out of what thermodynamic equilibrium would predict. And this, he underlined, is the consequence of living beings' influence on their planetary environment (constantly producing methane and oxygen). Important thermodynamic disequilibrium would thus be a sign for the presence of life. This proposal challenged and redirected research in the nascent field of exobiology. Back to Earth, the crucial recognition that life massively influences its planetary environment soon lead Lovelock to the development of the Gaia hypothesis. In 1968, at a NASA meeting on origins of life, Lovelock met Lynn Margulis (1938 -2011), at that time a young microbiologist. They collaborated to develop the Gaia hypothesis in a series of co-written papers from 1973 through 1978, before the publication of Lovelock's book. 6 The Gaia hypothesis was meant to account for the long term stability of Earth's environment, which had kept the Earth habitable by life for billions of years in spite of external perturbations, such as the increase of solar luminosity. The Gaia hypothesis accounted for this stability by positing that: the ensemble of living organisms which constitute the biosphere might act as a single entity to regulate chemical composition, surface pH, and possibly also climate. 7 It is these supposed regulatory properties which then lead Lovelock to compare Gaia, the entity composed of the living beings and the geological environment with which they interact, with a living, homeostatic entity. 8 After 1978, Lovelock and Margulis published separately. They came from two very different intellectual backgrounds 9 -Margulis clung explicitly to a 19 th century romantic and naturalist tradition, foreign to Lovelock's chemical and cybernetic background -and had different views about Gaia. After 1979, Gaia became Lovelock's major and central scientific concern. For Margulis, Gaia was an instance of a more general notion, that of symbiosis, encompassing cellular and microbial associations as much as relationships between living beings at a global scale. For this and other reasons, Lovelock is often presented as the author of the Gaia hypothesis. Yet Margulis had a decisive role in the 1970's: she drew Lovelock's attention to microbes' major ecological role, she brought Gaia into evolutionary biology, and she had an important role in the diffusion of Gaia in American counterculture. 10

The standard account: evolutionary biology ridiculed Gaia as

pseudo-science

The standard account of the reception of the Gaia hypothesis, the most popular in the life sciences, claims that it was rejected by scientists after famous critiques made by evolutionary biologists Ford Doolittle (see Chapter 7) and Richard Dawkins in the early 1980's. 11 The supposed idea that living organisms may act in order to regulate a larger whole seemed to reintroduce agency in the natural world at the wrong level of the biological hierarchy, and reminded evolutionary biologists of the heated debate of the 1960's and 1970's over the explanation of biological altruism, where benefit to the whole emerged from the altruistic actions of the individuals. 12 These early critiques paved the way for the denunciation of Gaia as pseudo-science, barely good for neo-pagan worshipers of mother Earth.

The idea that the Earth or the cosmos is (like) a living creature finds roots in stoic philosophy and was dismissed with the rise of modern and mechanistic science, and partly revived in the Naturphilosophie of 19 th century German romanticism. For contemporary scientists, and evolutionary biologists in particular, Gaia also reminded them of the idea, linked with natural theology, that there is a "balance of nature".

The Gaia hypothesis was altogether considered to be an extreme form of holism and of naively benevolent views of nature: a metaphor at best, pseudo-scientific mysticism at worse. 13 What is particularly remarkable when looking at Gaia's reception in evolutionary biology, is its homogeneity in the entire field: Gaia has been dismissed by Richard Dawkins but also, and sometimes on the ground of strikingly similar arguments, by people who usually disagree about every single other matter with Dawkins, such as Stephen Jay Gould and Richard Lewontin. 14 The fame of these biologists' critiques ultimately contributed to the diffusion of this standard account about Gaia.

In spite of its popularity, nothing of this standard account of Lovelock's hypothesis sounds right. It gives the impression that Lovelock was some kind of guru of a neo-pagan community, or a retired romantic in the countryside writing green poetry about a re-enchanted nature, or an evolutionary biologists with odd ideas. As we shall see, the reality of Lovelock's approach could not be more different. So before the 1960's, Lovelock was already an accomplished scientist and a gifted engineer. In 1964 he quit academia to "bury [himself] in the country village of Bowerchalke," 15 in the southwest of England. Reflecting an important romantic theme, he constantly presents himself with an ethos of a solitary and creative thinker, doing his best when working alone, out of every institutional constraint and the bureaucracy of contemporary science, and finding his inspiration in the walks he made in the countryside.

Lovelock

But make no mistake about what this "independent scientist" status meant concretely. The first thing he purchased was a Hewlett-Packard 9800 to solve differential equations for his ECD. In Bowerchalke, he did not settle into a library with rare alchemical books, but a home made laboratory in the garage with chromatographs and electronic circuits.

He quickly obtained a formal attachment to the University of Reading, in the department of Cybernetics, since one of his papers had been refused because he only had a private address, and no institutional one. To pay for his living expenses and his own research expenses, like other "scientific entrepreneurs" of the 1960's and 1970's, he worked as a consultant engineer in big industries: Shell and Hewlett Packard mostly -not the typical places where you expect to find beardy wizards making naked incantation to mother nature [START_REF] Shapin | The scientific life: a moral history of a late modern vocation[END_REF] .

If Lovelock had a disciplinary home, it was chemistry. He was trained early on as a practical chemist while a laboratory assistant, a job he obtained after high school where, he says, he learned "to regard accuracy in measurements as almost sacred." 17 The 1960's marked, for Lovelock, the transition from the study of the chemistry of living bodies (biochemistry), to the chemistry of Earth's surface (geochemistry), a field foreign to him before the late 1960's.

Aside from chemistry, the other intellectual matrix which played a prominent role in shaping Lovelock's researches was cybernetics.

When forced to represent what Gaia is, to draw what his vision of Gaia is, Lovelock does not call for the graphic artists Cameron hired for

Avatar to picture a network of interrelated animated entities, he designs an electronic circuit (figure 1). Electronic circuits were central components in most of the small-scale devices he invented as an engineer. And first order cybernetics, the science of thermostats, systems and feedbacks, occupies a central place in Gaian publications, when it comes to propose a mechanism which would maintain Earth's stability. emissions with a chromatographs. These DMS measures were for him linked with the Gaia hypothesis. In 1973, in a famous paper published in Nature, he suggested that the important emission of DMS by algae were essential for the closing of the global sulphur cycle. [START_REF] James Lovelock | Atmospheric dimethyl sulphide and the natural sulphur cycle[END_REF] The idea that Gaia traces back to Lovelock's thinking about life detection has been prominently put to the fore by Lovelock. Yet after the mid 1960's and up to the early 1980's, Lovelock central scientific activities were not focused on this issue but on global pollution, a problem rendered central to Lovelock's thinking and preoccupations through his work as a consultant for the greatest chemical and petroleum industries. And these thoughts were at that time central to the elaboration of Gaia. This quick overview enables us to acknowledge that Lovelock was not a philosopher or a poet trying to resurrect a romantic view of nature. He was not even a theoretician, but a chemist and engineer, with a hard core ethos of a practical scientist. His problems and arguments, he did not find in books 22 , but in the chemical compounds he smelled and measured. In the early 1970's, the foundational decade of the Gaia hypothesis, Lovelock was not creating a mathematical model of the genetics of a population, sorting Drosophila in the lab, making ethological observations of chimpanzees in central Africa, dancing and making invocations with druids: he was measuring chemical compounds across the Atlantic, in the stratosphere, and in the English and Irish countryside.

The institutional context in which Gaia developed and spread: Earth sciences and environmental counterculture

The community to which Gaia was explicitly addressed was originally that of Earth scientists, mostly geochemists. At NASA's meetings in the 1960's, Lovelock had not only met Lynn Margulis but also Lars Sillén (1916 -1970), a Swedish chemist who had a decisive influence on oceanography and geochemistry, and Heinrich 'Dick' Holland (1927 -2012), a top-notch contemporary geochemist of Earth's atmosphere and oceans. Dawkins can nag that Gaia reminds him of naive views of a "balance of nature", but he entirely missed that Lovelock had no interest whatsoever in plant and animal demographythe epicenter of the balance of nature tradition ever since Linnaeus. [START_REF] Frank | Changing concepts of the balance of nature[END_REF]

If

Lovelock was interested in demography, it was that of atmospheric gases. And, back in the early 1970's, the unravelling of the long term chemical history of Earth's atmosphere and oceans was in its infancy.

Three central and pioneering figures of these geochemical researches, Robert Garrels, Abraham Lerman and Fred Mackenzie, in a 1976 famous paper entitled "Controls of Atmospheric O2 and CO2: Past, Present, and Future" [START_REF] Garrels | Controls of atmospheric O2 and CO2: past, present, and future[END_REF] presented a conclusion "in agreement with Lovelock and Margulis's "Gaia" hypothesis (1974)". This conclusion is that of a paper that used box models to describe geochemical processes to estimate the long term evolution of atmospheric O2 and CO2 -not of two pages of abstract argument over group selection and altruism, as was Dawkins' argument. This is not to say that all geochemists embraced Gaia enthusiastically. In the 1970's, Gaian papers were not cited that much.

Yet in the 1980's, Gaia became a major topic. Not only because Lovelock changed his audience and published a book for the general public, but in part thanks to the crucial editorial work of the famous climatologist Stephen Schneider. Though wary and cautious about the meaning of the Gaia hypothesis, Schneider wanted Gaia to be subject to scientific debate. [START_REF] Stephen | A goddess of the Earth: The debate on the Gaia Hypothesis[END_REF] In 1988, he organized, with Penelope Boston, the first international scientific conference on Gaia as a Chapman conference of the American Geophysical Union (AGU). In 2000, he co-organized the second such conference. In the early 2000's, he managed a space for the discussion of Gaia in his journal Climatic Change.

If Schneider carried out this work, it is because he did not want to leave Gaia to American and environmental counterculture. In the Daisyworld model sparked over one hundred of papers proposing variants of the original model. [START_REF] Wood | What good are abstract and what-if models? lessons from the Gaïa hypothesis[END_REF] The two other major papers of the 1980's were not abstract mathematical models but specific empirical mechanisms. The first one, published in Nature by Lovelock and Michael Whitfield in 1982 [START_REF] Lovelock | Life span of the biosphere[END_REF] , proposed a mechanism by which life, through its influence on rock weathering, may have counteracted the long term increase of solar luminosity over Earth's long history, and thus maintained Earth habitable. This paper was an important stepping stone in the study of Earth's chemical and climatic history.

Finally, in 1987, the paper exposing what would soon be known as the CLAW hypothesis was published in Nature. [START_REF] Charlson | Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate[END_REF] The hypothesis suggested that algae may regulate the climate through a negative feedback loop involving the emission of DMS, which has a crucial role in cloud formation. This paper has been cited more than 3000 times and certain atmospheric scientists have devoted their career to the influence of DMS on contemporary climate.

Lovelock considered these three mechanisms (rock weathering, DMS and Daisyworld) as "arguments" in favor of the Gaia hypothesis:

the empirical examples were for geochemists and climatologists, Daisyworld was for evolutionary biologists. Clearly, for Lovelock, "Gaia" was the name of a hypothesis (and then a theory), that is, a general proposition which can make predictions and can be tested against empirical facts. A particular emphasis on Gaia as a theory or hypothesis was brought by Lovelock in the early 1990's 30 , after the famous critiques made by the geomorphologist James Kirchner in his Popperian lecture. Kirchner distinguished four or five different formulations of the Gaia hypothesis. He then argued that the weaker versions were trivial, and the stronger ones were not testable.

After the 1990's, the development of the Gaia theory continued, focused on the elaboration of new versions of Daisyworld. Timothy Lenton, now a climate and Earth system scientist, who did his PhD under the supervision of Andrew Watson (and Lovelock), pursued this development with Watson and a team of students in England 31 .

Overall, scientists were very much willing to discuss life's influence on rock weathering, DMS influence on climate, and Daisyworld modelling in different specialized journals of (bio)geochemistry, climatic sciences and biological theory. Yet Gaia, as a more general hypothesis or theory, was not discussed in specialized journals, as we expect every run-of-the-mill scientific hypothesis to be. The contexts of publications on Gaia per se were always exceptional: one to three international conferences per decade, a general paper in Nature and Science in some occasions, special issues of Climatic Change, books, but not regular discussion in specialized journals such as, say, Geochemistry, Geophysics, Geosystems. Scientists normally don't write books to discuss and test run-of-the-mill scientific hypotheses. But it is thanks to books that Lovelock was mostly known for Gaia and through which Gaia was criticized. 32

Gaia, the great research program

For Lovelock, Margulis, and other Gaia supporters, Gaia was not only the name of a hypothesis to be tested with empirical arguments and elaborated with mathematical models, but that of a research program or a paradigm, a new discipline, a new way of doing science, of viewing and studying the world. Rather than talking about "Gaian science" they tried several terms such as "geophysiology" or "geognosy." Gaian science or geophysiology were often compared, contrasted or opposed to other names of established scientific disciplines: geochemistry, biogeochemistry, chemical oceanography, microbial ecology, ecology, environmental science. This research program made explicit methodological claims. The first claim was that the influence of living beings on their geological environment must be recognized and taken into account. In the 1970's, this claim was addressed to Earth scientists. They were accused of only considering rocks and chemistry and neglecting the pervasive influence of living beings. Then, beginning in 1983, Lovelock started to criticize the notion of adaptation. He argued that the fit which exists between life and its environment could be the result of life's influence on its geological environment, rather than the result of organisms adapting to their environment.

The second methodological claim was that Earth should be studied "as a system," "as a whole," or with a holistic and not a reductionist perspective. Lovelock and Margulis lamented over the separation between life and Earth sciences, as well as the splitting of the Earth sciences into atmospheric chemistry, climatology, fluid dynamics, study of rocks, etc. On many occasions they used Gaia as the name of a revolutionary way of doing science, that resolved the two centuries long divorce between biology and geology. All these entities of Earth's surface, such as rocks, bacteria, soils, chemical compounds of the oceans and atmospheres, were to be studied into one unified science.

These methodological claims rapidly turned into historiographical claims: for its authors, the emergent "Earth System Science" were To describe and understand the interactive physical, chemical, and biological processes that regulate the total Earth system, the unique environment that it provides for life, the changes that are occurring in this system, and the manner in which they are influenced by human actions. 33 Interestingly, even those sceptical about Gaia as a hypothesis, or wary of some of Lovelock's environmental and political claims, such as

Stephen Schneider and Ann Henderson-Sellers, credited Lovelock with having brought climatologists' attention to the pervasive influence of life on Earth's history. Ecosystem ecologists, such as Eugen Odum, who shared a cybernetic and systemic framework with Gaia theory, also pointed to Gaia's important role in emphasizing life's influence on planetary chemical cycles and in contributing to the emergence of global ecology. 34 And so did the founders of a new field, "geobiology", studying the interactions between the history of life and its environment, a research agenda departing from the one of the paleobiology of the 1970's. 35 Most importantly, major actors of Earth System Science acknowledged Gaia's decisive role in calling scientists' attention to the existence of a new object of study: the "Earth System." [START_REF]Local ecologies and global science discourses and strategies of the International Geosphere-Biosphere Programme[END_REF] Gaia, philosophy of nature and environmental prescriptions Finally, Gaia was also for Lovelock the name of a philosophy of nature:

[Gaia] is an alternative to that pessimistic view which sees nature as a primitive force to be subdued and conquered. It is also an alternative to that equally depressing picture of our planet as a demented spaceship, forever travelling, driverless and purposeless, around an inner circle of the sun. [START_REF] Lovelock | [END_REF] The first sentence opposes our modern view of nature inherited from Bacon and Descartes. 38 The second opposes the metaphor of the spaceship Earth, very popular in the 1970's, conceiving the Earth as a vessel, which environmental problems ought to be managed by experts. 39 Though, again, Lovelock is not anchored in a literary philosophical tradition, there is no doubt that Gaia was to propose a reconception of important and related concepts which were: life, nature, the environment. And certainly his philosophy of nature is not to be found in his explicit reflections over the categories of life and nature, but in the tacit assumptions he made while building models, in what needs to be taken for granted to engage in the research questions he envisaged.

To address Gaia as a philosophy of nature, the relevant attitude is not to "test" it against empirical facts. It is not to argue about the priority of such and such scientific issues or about the way scientific institutions should be organized, as it was for Gaia as a research program. It is to embrace and elaborate Gaia's categories and worldviews or to reject and dismiss them.

Another attitude one can adopt is to make explicit Lovelock's metaphysics, ontology or categories and to contrast them with other ways to think about nature, life, and the world. And here again, Gaia found important echoes. The anthropologist and philosopher Bruno Latour sketched a symmetry between the way Galileo has contributed to overthrow Aristotelian conception of the cosmos and to make us consider alike the physical properties of terrestrial bodies and of celestial bodies, and the way Lovelock has, conversely, managed to render the Earth so peculiar and local in the Solar System, influenced as it is by living entities. 40 For conservative environmentalists such as Edward Goldsmith, Gaia was truly to be thought as an organismic ordered whole: not to follow the "natural rules" of this ordered whole should be seen as something deeply wrong.

But for Lovelock also, ever since the 1970's, Gaia has not only been a grandiose view of life on Earth but also a framework to think about the very concept of pollution, from which he derived many practical and concrete environmental and political prescriptions. In the 1970's, against the (green) current, he opposed to the ban of CFCs responsible for ozone's hole; he has long been actively militating for nuclear energy; and in the recent years, he has been criticized for taking radical positions ranging from the suspension of democracy and human rights to the proposition of geoengineering techniques going through a more or less voluntary reduction of the world's population.

Interestingly, in the three emblematic critiques made by scientists to the Gaia hypothesis -that of Doolittle and Kirchner in the 1980's, and of Tyrrell in his 2013 book -Lovelock's environmental and political prescriptions made in Gaia's name were mentioned as the central nerve and as the reason why it mattered whether Gaia hypothesis was "right or wrong", "true or false". And many supporters of the Gaia hypothesis or research program, and even way of thinking about nature, found difficult to follow Lovelock's "Gaian" environmental prescriptions.

Conclusion

The Gaian literature is vast, with enthusiasts and critiques, talking about rich and complex issues in very different domains. Gaia was not initially addressed to evolutionary biologists, but to geochemists. The very ambiguity of what Gaia meant had an important role in Gaia's pervasive diffusion and is what makes Gaia so rich and interesting.

Lovelock, in the same papers and books, used the word "Gaia" to refer to very different things: a hypothesis, about which you can argue with empirical arguments and mathematical models, dealing with the peculiar influence that life may have had over Earth's history on its to think about nature. Gaia has been central to our contemporary accepted view of the Earth as a planetary system of interrelated entities, teeming with life, but also that of a planet with its stable states which can be overthrown, as is now dramatically pictured by the anthropocene discourse.

  : an independent and practical scientist, a chemist, an engineer Lovelock was born in Hertfordshire on July, 26th, 1919 and grew up in London. His parents owned a small painting shop in Brixton. But Lovelock was more moved by his early and frequents visit to the Science Museum than by the artistic environment which surrounded him. He graduated in chemistry at the University of Manchester in 1941. He started working at the National Institute for Medical Research at Hampstead and obtained a PhD in medicine in the London School of Hygiene and Tropical Medicine. For twenty years, he worked on biochemical and engineering issues related to medical problems. He published an important number of pioneering articles in Nature, some of them being cited hundreds of times, on various issues: transmission of infections, the effect of heat on biological tissues and blood coagulation, cryobiology and resurrection of frozen hamsters.He excelled in the invention of small-scale instruments, usually made to detect chemical substances. His most renown invention, the one to which he owes his invitation from NASA, and in 1997, the Blue Planet Prize, remains the Electron Capture Detector (ECD) which he invented in 1957. This small device enables scientists to detect minute quantities of chemical compounds (with a precision orders of magnitude above what was attainable beforehand).

Figure 1 :

 1 Figure 1: Lovelock's illustration of Gaia, showing the importance of

Figure 2 :

 2 Figure 2: Lovelock and his daughter collecting air samples in the

  1970's, and early 1980's, papers by Lovelock and Margulis, but also Doolittle's famous critique, were published in Stewart Brand's journal, Coevolution Quarterly. This journal was the successor of the famous Whole Earth Catalog, which sold millions of copies and enabled Brand to became a central figure of American counterculture, in which contemporary cyberculture finds its roots. 26 Gaian systemic and cybernetic views of the Earth deeply resonated with the spirit of Coevolution Quarterly. And this journal had a very important role in the diffusion of Gaian papers. In addition, Lovelock (and Margulis) also published on Gaia in the two major environmentalist journals in the UK: Resurgence and The Ecologist, the last one having been founded by Lovelock's friend and sponsor, Edward Goldsmith. The development of the hypothesis and theory Initially many critics complained about the paucity of mechanisms that could account for global homeostasis. In the 1980's, Lovelock and others offered such mechanisms in three major papers. In 1983, Andrew Watson and Lovelock published the computational model Daisyworld, developed specifically to address biologists' critiques and to show that the regulation of a global environmental variable could emerge from the influence of living organisms on their environment. The model depicted a fictive planet in which temperature is regulated by the proportion of population of black and white daisies influencing the climate through their albedo. This

  Gaia's research program with another name. Earth System Science emerged in the 1990's and 2000's with new departments, institutes, centers, chairs and textbooks of Earth System Science out of the institutional work carried out by NASA and then the International Geosphere-Biosphere Program (IGBP) in the 1980's. Earth System Science was not so much conceived as a new scientific discipline, but as an entirely new and revolutionary way of looking at the Earth and of organizing the Earth sciences in an interdisciplinary fashion. Its central aim was:

  geological environment; a research program guiding and imposing the way Earth and life sciences should study Earth's chemistry, climate and living beings; a philosophy of nature challenging our modern conceptions of life and nature. While dismissed as a problematic hypothesis, Gaia has been credited by Earth scientists for its role as fostering new research programs, such as those in the Earth System Science. Certainly the most decisive and revolutionary contribution of the Gaia hypothesis was an ontological one. Gaia has called for the recognition of a new entity: the system composed of the entire life and the geological environment with which it interacts. It was the the recognition of this new entity, which laid the ground for a new research program in the Earth sciences and offered a new framework

30

  James Lovelock, "Hands up for the Gaia hypothesis," Nature, 344 (1990): 100. 31 Timothy M. Lenton and Andrew Watson, Revolutions that made the Earth (Oxford: Oxford University Press, 2011). 32 Toby Tyrrell, On Gaia: A Critical Investigation of the Relationship Between Life and Earth (Princeton: Princeton University Press, 2013). 33 This is an iconic statement of the first report of the IGBP , The International Geosphere-Biosphere Programme: A Study of Global Change. Final Report of the Ad Hoc Planning Group, ICSU 21st General Assembly, Berne, Switzerland 14-19 September, 1986, 3, chaired by the famous swedish meteorologist Bert Bolin, Editor-in-Chief of the journal Tellus which published the iconic Lovelock and Margulis, "Atmospheric homeostasis". 34 Eugene P. Odum, "Great ideas in ecology for the 1990s," BioScience, 42 (1992): 542. 35 For a philosophical and historical overview of paleontology and paleobiology, see David Sepkoski and Michael Ruse, The paleobiological revolution: essays on the growth of modern paleontology (Chicago: The University of Chicago Press, 2009); Derek Turner, Paleontology: A philosophical introduction. (Cambridge: Cambridge University Press, 2011). No equivalent study exists for

Carolyn Merchant, The death of nature: women, ecology and the

Various moving passages of Lovelock's autobiography as well as scientific papers he published reveal his style of science in the 1960's and 1970's. As he recalls, "it was a family ritual at Bowerchalke to measure the haze density using a sun photometer" 18 (see figure 2).

Troubled by an unusual presence of fog, persuaded that it was of anthropic origin, he settled to measure and trace the presence of Chlorofluorocarbons (CFCs), a compound only produced by artificial means, to show that the atmospheric masses had been polluted. In the early 1970's, Lovelock embarked in the marine vessel Shackleton to measure the CFCs over the Atlantic, thanks to his ECD. These first global measure of CFCs [START_REF] James Lovelock | Halogenated hydrocarbons in and over the atlantic[END_REF] were decisive for the imputation of these compounds as the causal agents of ozone destruction by Mario Molina and Franklin Rowland, [START_REF] Molina | Stratospheric sink for chlorofluoromethanes: chlorine atom-catalysed destruction of ozone[END_REF] winners of the Nobel Prize for this discovery.

Intrigued by certain predictions of Molina and Rowland's theory, Lovelock did not propose an alternative theory: he found a meteorological plane flying at stratospheric heights which would enable him to make the chemical measurements he needed. On board of the marine vessel Shakleton, Lovelock also measured the quantities of Dimethylsuphide (DMS). He had earlier realized that this sulfur compound was produced in great quantities by algae as he walked along his cottage in Ireland, identifying algae and measuring their 2 James Lovelock. Gaia: A new look at life on Earth (Oxford: Oxford University Press, 1979), 11. 3 Stephen Jay Gould, The structure of evolutionary theory (Cambridge,