Numerical scheme for semilinear Stochastic PDEs via ... - CiteSeerX

Numerical scheme for semilinear Stochastic PDEs

arXiv:1302.0440v4 [math.PR] 18 Dec 2013

via Backward Doubly Stochastic Differential Equations Achref BACHOUCH Universit´ e du Maine Institut du Risque et de l’Assurance Laboratoire Manceau de Math´ ematiques e-mail: [email protected]

Mohamed Anis BEN LASMAR University of Tunis El Manar Laboratoire de Mod´ elisation Math´ ematique et Num´ erique dans les Sciences de l’Ing´ enieur, ENIT e-mail: [email protected]

Anis MATOUSSI Universit´ e du Maine Institut du Risque et de l’Assurance Laboratoire Manceau de Math´ ematiques e-mail: [email protected]

Mohamed MNIF University of Tunis El Manar Laboratoire de Mod´ elisation Math´ ematique et Num´ erique dans les Sciences de l’Ing´ enieur, ENIT e-mail: [email protected]

Key words : Backward Doubly Stochastic Differential Equations, Forward-Backward System, Stochastic PDE, Malliavin calculus, Euler scheme, Monte Carlo method. MSC Classification (2000) : 60H10, 65C30. Abstract: In this paper, we investigate a numerical probabilistic method for the solution of a class of semilinear stochastic partial differential equations (SPDEs in short). Our numerical scheme is based on discrete time approximation for solutions of systems of a decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, we prove the convergence and the rate of convergence of our numerical scheme. The proof is based on a generalization of the result on the path regularity of the backward equation. ∗ Research

partly supported by the Chair Financial Risks of the Risk Foundation sponsored by Soci´ et´ e G´ en´ erale, the Chair Derivatives of the Future sponsored by the F´ ed´ eration Bancaire Fran¸caise, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon † This work was partially supported by the research project MATPYL of the F´ ed´ eration de Math´ ematiques des Pays de la Loire 1

A. Bachouch, M.A. Ben Lasmar, A. Matoussi, M. Mnif/

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1. Introduction Stochastic partial differential equations (SPDEs) combine the features of partial differential equations and Itˆ o equations. Such equations play important roles in many applied fields such as the filtering of partially observable diffusion processes, genetic population and other areas. We study the following stochastic partial differential equation (in short SPDE) for a system-valued of predictable random field ut (x) = u (t, x) , satisfying the following equation: ←− dut (x) + Lut (x) +f (t, x, ut (x), ∇ut σ(x)) dt + g(t, x, ut (x), ∇ut σ(x)) · dB t = 0,

(1.1)

over the time interval [0, T ], with a given final condition uT = Φ and non-linear deterministic coefficients f and g. Lu = Lu1 , · · · , Luk is a second order differential operator and σ is the ←− diffusion coefficient. The differential term with dB t refers to the backward stochastic integral with respect to a l-dimensional brownian motion on Ω, F , P, (Bt )t≥0 . We use the backward stochastic integral in the SPDE because we will employ the framework of Backward Doubly Stochastic Differential Equation (BDSDE) introduced first by Pardoux and Peng [29]. They gave a probabilistic representation for the classical solution ut (x) of the SPDE (1.1) (written in the integral form) in term of the following class of BDSDE’s: Z T Z T Z T ←−− Yst,x = Φ(XTt,x ) + f (r, Xrt,x , Yrt,x , Zrt,x )dr + g(r, Xrt,x , Yrt,x , Zrt,x )dBr − Zrt,x dWr , (1.2) s

(Xst,x )t≤s≤T

s

s

where is a diffusion process starting from x at time t driven by the finite dimensional brownian motion (Wt )t≥0 and with infinitesimal generator L. More precisely, under some regularity assumptions on the final condition Φ and coefficients f and g , they have proved that ut (x) = Ytt,x and ∇ut σ(x) = Ztt,x , ∀(t, x) ∈ [0, T ] × Rd . Then, Bally and Matoussi [5] (see also [25] ) showed that the same representation remains true in the case when the final condition (respectively the coefficients f and g) is only measurable in x (resp. are jointly measurable in (t, x) and Lispchitz in u and ∇u). In this paper, weak Sobolev solution of the equation (1.1) has been considered, and the approach was based on stochastic flow technics (see also [19, 20]). Moreover their results were generalized in [25] in the case of a larger class of SPDE’s (1.1) driven by a Kunita-Ito non-linear noise (see [19, 20, 21] for more details). In particular, the Kunita-Ito non-linear noise covers a class of infinite dimensional space-time colored-white noise (see [13], [31], [16]). Generally, the explicit resolution of semi-linear SPDEs is not possible, so it is then necessary to resort to numerical methods. The first approach used to solve numerically nonlinear SPDEs is analytic methods, based on timespace discretization of the SPDEs. The discretization on space can be achieved either by finite differences, or finite elements and spectral Galerkin methods. But most numerical works on SPDEs have concentrated on the Euler finite-difference scheme. Gyongi and Nualart [14] have proved that these schemes converge, and Gyongy [15] determined the order of convergence. J.B. Walsh [32] investigated schemes based on the finite elements methods. He studied the rate of convergence of these schemes for parabolic SPDEs, including the Forward and Backward Euler and the CrankNicholson schemes. He found substantially similar rate of convergence to those found for finite difference schemes. The spectral Galerkin approximation was used by Jentzen and Kloeden [16]. They based their method on Taylor expansions derived for the solution of the SPDE, under some


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regularity conditions. Lototsky, Mikulevicius and Rozovskii in 1997 [22] used the spectral approach for the numerical estimation of the conditional distribution solution of a linear SPDE known as the Zakai equation. Further developments on spectral methods can be found in Lototsky [23]. Milstein and Tretyakov [24] solved a linear Stochastic Partial Differential Equation by using the method of characteristics (the averaging over the characteristic formula). They proposed a numerical scheme based on Monte Carlo technique. Layer methods for linear and semilinear SPDEs are constructed. Picard [30] considered a filtering problem where the observation is a function of a diffusion corrupted by an independent white noise. He estimated the error caused by a discretization of the time interval. He obtained some approximations of the optimal filter which can be computed with Monte-Carlo methods. Crisan [8] studied a particle approximation for a class of nonlinear stochastic partial differential equations. Very interesting results have been obtained by Gyongy and Krylov [13] where they considered a symmetric finite difference scheme for a class of linear SPDE driven by infinite dimensional brownian motion. They have proved that the approximation error is proportional to k 2 where k is the discretization step in space and by the Richardson acceleration method they have even got the error proportional to k 4 . The other alternative for resolving numerically SPDEs is the probabilistic approach by using Monte Carlo methods. These latter methods are tractable especially when the dimension of the state process is very large unlike the finite difference method. Furthermore, their parallel nature provides another advantage to the probabilistic approach: each processor of a parallel computer can be assigned the task of making a random trial and doing the calculus independently. The probabilistic approach requires weaker assumptions on the SPDE’s coefficients. In the deterministic PDE’s case i.e. g ≡ 0, the numerical approximation of the BSDE has already been studied in the literature by Bally [3], Zhang [33], Bouchard and Touzi [6], Gobet, Lemor and Warin[12] and Bouchard and Elie [7].Zhang [33] proposed a discrete-time numerical approximation, by step processes, for a class of decoupled FBSDEs with possible path-dependent terminal values. He proved an L2 -type regularity of the BSDE’s solution, the convergence of his scheme and he derived its rate of convergence. Bouchard and Touzi [6] suggested a similar numerical scheme for decoupled FBSDEs. The conditional expectations involved in their discretization scheme were computed by using the kernel regression estimation. Therefore, they used the Malliavin approach and the Monte carlo method for its computation. Crisan, Manolarakis and Touzi [9] proposed an improvement on the Malliavin weights. Gobet, Lemor and Warin in [12] proposed an explicit numerical scheme. In the case when g 6= 0 and it does not depend on the control variable z, Aman [1] proposed a numerical scheme following the idea used by Bouchard and Touzi [6] and obtained a convergence of order h of the square of the L2 - error (h is the discretization step in time). Aboura [2] studied the same numerical scheme under the same kind of hypothesis, but following Gobet et al. [11]. He obtained a convergence of order h in time and used the regression Monte Carlo method to implement his scheme, following always [11]. In our work, we extend the approach of Bouchard-Touzi-Zhang in the general case when g depends also on the control variable z. We wish to emphasize that this generalization is not obvious because of the strong impact of the backward stochastic integral term on the numerical approximation scheme. It is known that in the associated Stochastic PDE’s (1.1), the term g(u, ∇u) leads to a second order perturbation type which explains the contraction condition assumed on g with


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respect to the variable z (see [29], [27]). Our scheme is implicit in Y and explicit in Z. We prove the convergence of our numerical scheme and we give the rate of convergence. The square of the L2 - error has an upper bound of order the discretization step in time. As a consequence, we get a numerical scheme for the weak solution of the associated semi linear SPDE. We give also a rate of convergence result for the later weak solution. Then, we propose a numerical scheme based on iterative regression functions which are approximated by projections on vector space of functions with coefficients evaluated using Monte Carlo simulations. Finally, we present some numerical tests. Compared to the deterministic numerical method developed by Gyongy and Krylov [13], the probabilistic approach could tackle the semilinear SPDE which could be degenerate and needs less regularity conditions on the coefficients than the finite difference scheme. However, the rate of convergence obtained (as the classical Monte Carlo method) is clearly slower than the results obtained by finite difference and finite element schemes, but of course more available in higher dimension. This paper is organized as follows. In section 2 we introduce preliminaries and assumptions and we describe the approximation scheme for the BDSDE. In section 3 we show an upper bound result for the time discretization errore. In section 4 we give a Malliavin regularity result for the solution of our Forward-Backward Doubly SDE’s. Then, we show a L2 -regularity result for the Z-component of the solution of the BDSDE (1.2) which is crucial to obtain the rate of convergence of our numerical scheme. Section 5 is devoted to the numerical scheme of the SPDE’s weak solution. In section 6, we test statistically the convergence of this scheme by using a path dependent algorithm based on the regression Monte Carlo Method. Finally, we give some technical results in the Appendix 2. Preliminaries and notations 2.1.

Forward Backward Doubly Stochastic Differential Equation

Let {Ws , 0 ≤ s ≤ T } and {Bs , 0 ≤ s ≤ T } be two mutually independent standard brownian motion processes, with values respectively in Rd and in Rl where T > 0 is a fixed horizon time, on the probability space (Ω, F , P ). We shall work on the product space Ω := ΩW × ΩB , where ΩW is the set of continuous functions from [0, T ] into Rd and ΩB is the set of continuous functions from [0, T ] into Rl . We fix t ∈ [0, T ]. For each s ∈ [t, T ], we define W B Fst := Ft,s ∨ Fs,T W B W where Ft,s = σ{Wr − Wt , t ≤ r ≤ s}, and Fs,T = σ{Br − Bs , s ≤ r ≤ T }. We take F W = F0,T , B B W B F = F0,T and F = F ∨ F . We define the probability measures PW on (ΩW , F W ) and PB on (ΩB , F B ). We then define the probability measure P := PW ⊗ PB on (Ω, F W × F B ). Without loss of generality, we assume that F W and F B are complete. Note that the collection {Fst , s ∈ [t, T ]} is neither increasing nor decreasing, and it does not constitute a filtration. Afterthat, we introduce the following spaces: • Cbk (Rp , Rq ) (respectively Cb∞ (Rp , Rq )) denotes the set of functions of class C k from Rp to Rq whose partial derivatives of order less or equal to k are bounded (respectively the set of functions


5

of class C ∞ from Rp to Rq whose partial derivatives are bounded). For any m ∈ N, we introduce the following notations: • H2m ([0, T ]) denotes the set of (classes of dP ×dt a.e. equal) Rm -valued jointly measurable processes {ψu ; u ∈ [0, T ]} satisfying : RT (i) ||ψ||2H2 ([0,T ]) := E[ 0 |ψu |2 du] < ∞, m (ii) ψu is Fu -measurable, for a.e. u ∈ [0, T ]. • S2m ([0, T ]) denotes similary the set of Rm -valued continuous processes satisfying : (i) ||ψ||2S2 ([0,T ]) := E[sup0≤u≤T |ψu |2 ] < ∞, m (ii) ψu is Fu -measurable, for any u ∈ [0, T ]. • S the set of random variables F of the form: F = fˆ(W (h1 ), . . . , W (hm1 ), B(k1 ), . . . , B(km2 )) with fˆ ∈ Cb∞ (Rm1 +m2 , R), h1 , . . . , hm1 ∈ L2 ([0, T ], Rd), k1 , . . . , km2 ∈ L2 ([0, T ], Rl ), where W (hi ) :=

Z

T

hi (s)dWs ,

B(kj ) :=

0

Z

T

←−− kj (s)dBs .

0

For any random variable F ∈ S, we define its Malliavin derivative (Ds F )s with respect to the brownian motion W by Ds F :=

m1 X i=1

ˆ ∇i f W (h1 ), . . . , W (hm1 ); B(k1 ), . . . , B(km2 ) hi (s),

where ∇i fˆ is the derivative of fˆ with respect to its i-th argument. We define a norm on S by:

2

kF k1,2 := E[F ] + E k.k1,2

T

Z

0

|Ds F |2 ds

12

.

is then a Sobolev space. • D1,2 , S • Sk2 ([0, T ], D1,2 ) is the set of processes Y = (Yu , 0 ≤ u ≤ T ) such that Y ∈ S2k ([0, T ]), Yui ∈ D1,2 , 1 ≤ i ≤ k, 0 ≤ u ≤ T and kY k1,2 := {E[

Z

T

0

|Yu |2 du] + E[

T

Z

0

T

Z

0

1

||Dθ Yu ||2 dudθ]} 2 < ∞.

• M2k×d ([0, T ], D1,2 ) is the set of processes Z = (Zu , 0 ≤ u ≤ T ) such that Z ∈ H2k×d ([0, T ]), Zui,j ∈ D1,2 ,1 ≤ i ≤ k, 1 ≤ j ≤ d, 0 ≤ u ≤ T and kZk1,2 := {E[

Z

0

T

kZu k2 du] + E[

Z

0

T

Z

0

T

1

||Dθ Zu ||2 dudθ]} 2 < ∞.

• B 2 ([0, T ], D1,2) := Sk2 ([0, T ], D1,2 ) × M2k×d ([0, T ], D1,2). We define also for a given t ∈ [0, T ]: • L2 ([t, T ], D1,2 ) is the set of progressively measurable processes (vs )t≤s≤T such that : (i) v(s, .) ∈ D1,2 , for a.e. s ∈ [t, T ], (ii) (s, w) −→ Dv(s, w) ∈ L2 ([t, T ] × Ω), RT RT RT (iii) E[ t |vs |2 ds] + E[ t t |Du vs |2 duds] < ∞. • L2 ([t, T ], D1,2 × D1,2 ) := L2 ([t, T ], D1,2 ) × L2 ([t, T ], D1,2 ).


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For all (t, x) ∈ [0, T ] × Rd , let (Xst,x )s be the unique strong solution of the following stochastic differential equation: dXst,x = b(Xst,x )ds + σ(Xst,x )dWs ,

s ∈ [t, T ],

Xst,x = x,

0 ≤ s ≤ t,

(2.1)

where b and σ are two functions on Rd with values respectively in Rd and Rd×d . We will omit the dependance of the forward process X in the initial condition if it starts at time t = 0. We consider the following BDSDE: For all t ≤ s ≤ T , ( ←−− dYst,x = −f (s, Xst,x, Yst,x , Zst,x )ds − g(s, Xst,x , Yst,x , Zst,x )dBs + Zst,x dWs , (2.2) YTt,x = Φ(XTt,x ), where f and Φ are two functions respectively on [0, T ] × Rd × Rk × Rk×d and Rd with values in Rk and g is a function on [0, T ] × Rd × Rk × Rk×d with values in Rk×l . We note that the integral with respect to (Bs , t ≤ s ≤ T ) is a ”backward Itˆ o integral” (see Kunita [21] and Nualart and Pardoux [27] for the definition) and the integral with respect to (Ws , t ≤ s ≤ T ) is a standard forward Itˆ o integral. P Finally, for each real matrix A, we denote by kAk its Frobenius norm defined by kAk = ( i,j a2i,j )1/2 . P For a vector x, |x| stands for its Euclidean norm defined by |x| = ( i |xi |2 )1/2 . The following assumptions will be needed in our work: Assumption (H1) There exist a positive constant K such that

|b(x) − b(x′ )| + kσ(x) − σ(x′ )k ≤ K|x − x′ |, ∀x, x′ ∈ Rd . Assumption (H2) There exist two constants K > 0 and 0 ≤ α < 1 such that for any (t1 , x1 , y1 , z1 ), (t2 , x2 , y2 , z2 ) ∈ [0, T ] × Rd × Rk × Rk×d , (i) (ii) (iii) (iv)

p |t1 − t2 | + |x1 − x2 | + |y1 − y2 | + kz1 − z2 k , kg(t1 , x1 , y1 , z1 ) − g(t2 , x2 , y2 , z2 )k2 ≤ K |t1 − t2 | + |x1 − x2 |2 + |y1 − y2 |2 + α2 kz1 − z2 k2 ,

|f (t1 , x1 , y1 , z1 ) − f (t2 , x2 , y2 , z2 )| ≤ K |Φ(x1 ) − Φ(x2 )| ≤ K|x1 − x2 |,

sup (|f (t, 0, 0, 0)| + ||g(t, 0, 0, 0)||) ≤ K.

0≤t≤T

Assumption (H3) (i) (ii) and

b ∈ Cb2 (Rd , Rd ) and σ ∈ Cb2 (Rd , Rd×d )

Φ ∈ Cb2 (Rd , Rk ), f ∈ Cb2 ([0, T ] × Rd × Rk × Rd×k , Rk ) g ∈ Cb2 ([0, T ] × Rd × Rk × Rd×k , Rk×l ).

Pardoux and Peng [29] proved that there exists a unique solution (Y, Z) ∈ S2k ([t, T ])×H2k×d ([t, T ]) to the BDSDE (2.2). Remark 2.1 Pardoux and Peng [29] assumed the contraction condition 0 ≤ α < 1 to prove the existence and the uniqueness results for the BDSDE’s solution.


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From [10], [29] and [18], the standard estimates for the solution of the Forward-Backward Doubly SDE (2.1)-(2.2) hold and we remind the following theorem : Theorem 2.1 Under Assumptions (H1) and (H2), there exist, for any p ≥ 2, two positive constants C and Cp and an integer q such that : E[ sup |Xst,x|2 ] ≤ C(1 + |x|2 ),

(2.3)

t≤s≤T

E

h

sup |Yst,x |p +

t≤s≤T

Z

t

T

kZst,xk2 ds

p/2 i

≤ Cp (1 + |x|q ),

(2.4)

2.2. Numerical Scheme for decoupled Forward-BDSDE In order to approximate the solution of the BDSDE (2.2), we introduce the following discretized version. Let π : t0 = 0 < t1 < . . . < tN = T,

(2.5)

be a partition of the time interval [0, T ]. For simplicity we take an equidistant partition of [0, T ] T i.e. h = N and tn = nh, 0 ≤ n ≤ N . Throughout the rest, we will use the notations ∆Wn = Wtn+1 − Wtn and ∆Bn = Btn+1 − Btn , for n = 1, . . . , N . The forward component X will be approximated by the classical Euler scheme : ( XtN0 = Xt0 , (2.6) XtNn = XtNn−1 + b(XtNn−1 )(tn − tn−1 ) + σ(XtNn−1 )(Wtn − Wtn−1 ), for n = 1, . . . , N. It is known that as N goes to infinity, one has

sup E|Xtn − XtNn |2 → 0.

0≤n≤N

Quite naturally, the solution (Y, Z) of (2.2) is approximated by (Y N , Z N ) defined by : = Φ(XTN ) and ZtNN = 0, YtN N

(2.7)

N ] + hEtn [f (tn , ΘN = Etn [YtN YtN n )] + Etn [g(tn+1 , Θn+1 )∆Bn ], n+1 n # "

(2.8)

and for n = N − 1, . . . , 0, we set

∗ ∆Wn∗ + g(tn+1 , ΘN hZtNn = Etn YtN n+1 )∆Bn ∆Wn , n+1

(2.9)

where N N N ΘN n := (Xtn , Ytn , Ztn ), ∀n = 0, . . . , N.

∗ denotes the transposition operator and Etn denotes the conditional expectation over the σ-algebra Ft0n . For all n = 0, . . . , N − 1, we define the pair of processes (YtN , ZtN )tn ≤t t, Dθ Ur dWri , θ ≤ t.

For backward Itˆ o integral, and since the Malliavin derivative is with respect to the brownian motion W, we have the following result : R T ←−− Lemma 4.2 Let U ∈ H21 ([t, T ]) and Ii (U ) = t Ur dBri , i = 1, . . . , l. Then for each θ ∈ [0, T ] we have Uθ ∈ D1,2 if and only if Ii (U ) ∈ D1,2 , i = 1, . . . , l and for all θ ∈ [0, T ], we have Z

Dθ Ii (U ) =

T

←−− Dθ Ur dBri , θ > t,

T

←−− Dθ Ur dBri , θ ≤ t.

θ

Z

Dθ Ii (U ) =

t

For later use, we need to prove the a priori estimates for the solution of the BDSDE (see [10] for similar estimates for a standard BSDE). Proposition 4.1 Let (φ1 , f 1 , g 1 ) and (φ2 , f 2 , g 2 ) be two standard parameters of the BDSDE (2.2) and (Y 1 , Z 1 ) and (Y 2 , Z 2 ) the associated solutions. Assume that Assumption (H2) holds. For s ∈ [t, T ], set δYs := Ys1 −Ys2 , δ2 fs := f 1 (s, Xs , Ys2 , Zs2 )−f 2 (s, Xs , Ys2 , Zs2 ) and δ2 gs := g 1 (s, Xs , Ys2 , Zs2 )− g 2 (s, Xs , Ys2 , Zs2 ). Then, we have ||δY ||2S2 ([t,T ]) + ||δZ||2H2

d×k ([t,T ])

d

≤ CE[|δYT |2 +

Z

T t

|δ2 fs |2 ds +

T

Z

+kδ2 gs k2 ds],

t

(4.4)

where C is a positive constant depending only on K, T and α. Proof. The proof of this result is classical in the BSDE’s theory (see El Karoui et al.[10]) so we omit it. ✷ Now, we study the differentiability in the Malliavin sense of the solution of the BDSDE which is technical. To our knowledge, it does not exist in the literature. We have to precise that Pardoux and Peng [29] have skipped details considering that it was just an easy extension of the work on standard BSDEs [28]. We show that the derivative is a solution of a linear BDSDE as Peng and Pardoux [28]have given for the standard BSDE’s, see also El Karoui Peng and Quenez ([10], Proposition 5.3)). Proposition 4.2 Assume that (H1)-(H3) hold. For any t ∈ [0, T ] and x ∈ Rd , let {(Ys , Zs ), t ≤ s ≤ T } denotes the unique solution of the BDSDE: Ys =

Φ(XTt,x )

+

Z

s

T

f (r, Xrt,x , Yr , Zr )dr

+

Z

s

T

←−− g(r, Xrt,x , Yr , Zr )dBr

−

Z

s

T

Zr dWr , t ≤ s ≤ T.

Then, (Y, Z) ∈ B 2 ([t, T ], D1,2 ) and {Dθ Ys , Dθ Zs ; t ≤ s, θ ≤ T } is given by: (i) Dθ Ys = 0, Dθ Zs = 0 for all t ≤ s < θ ≤ T


19

(ii) for any fixed θ ∈ [t, T ], θ ≤ s ≤ T and 1 ≤ i ≤ d, a version of (Dθi Ys , Dθi Zs ) is the unique solution of the BDSDE: Z T ∇x f (r, Xrt,x, Yr , Zr )Dθi Xrt,x dr Dθi Ys = ∇Φ(XTt,x )Dθi XTt,x + s

+

Z

T

s

+

j=1

l Z T X

n=1

+

d X ∇zj f (r, Xrt,x , Yr , Zr )Dθi Zrj dr ∇y f (r, Xrt,x, Yr , Zr )Dθi Yr +

s

l Z X

n=1

s

T

←−− ∇x g n (r, Xrt,x , Yr , Zr )Dθi Xrt,x + ∇y g n (r, Xrt,x , Yr , Zr )Dθi Yr dBrn d ←−− Z X ∇zj g n (r, Xrt,x , Yr , Zr )Dθi Zrj dBrn −

s

j=1

T

d X

Dθi Zrj dWrj ,

(4.5)

j=1

where (z j )1≤j≤d denotes the j-th column of the matrix z, (g n )1≤n≤l denotes the n-th column of the matrix g and B = (B 1 , . . . , B l ). Proof. See Appendix. ✷ The second order differentiability in the Malliavin sense of the solution of the BDSDE will be given in Appendix. 4.3. Representation results for BDSDEs In this subsection, we will prove a representation result of (Z, DZ) which will be useful to prove the rate of convergence of our numerical scheme. Proposition 4.3 Assume that (H1)-(H3) hold. Then : For t ≤ s ≤ T , we have Ds Ys = Zs ,

(4.6)

and kZk2S2

k×d ([t,T ])

≤ C(1 + |x|2 ).

(4.7)

For l1 , l2 ≤ d, t ≤ s ≤ T , we have Dsl2 Dtl1 Ys = Dtl2 Zsl1 ,

(4.8)

and kDsl1 Zk2S2

k×d ([t,T ])

≤ C(1 + |x|4 ).

(4.9)

Proof. To simplify the notations, we restrict ourselves to the case k = d = 1. 1. Notice that for t ≤ s Z s Z s Z s ←−− Ys = Yt − f (r, Σr )dr − g(r, Σr )dBr + Zr dWr , t

t

t

(Xrt,x , Yr , Zr ).

where Σr := It follows from Lemma 4.1 and Lemma 4.2 that, for t < θ ≤ s Z s Dθ Ys = Zθ − ∇x f (r, Σr )Dθ Xr + ∇y f (r, Σr )Dθ Yr + ∇z f (r, Σr )Dθ Zr dr Z s θ ←−− Z s Dθ Zr dWr . ∇x g(r, Σr )Dθ Xr + ∇y g(r, Σr )Dθ Yr + ∇z g(r, Σr )Dθ Zr dBr + − θ

θ


20

Then by taking θ = s, it follows that equality (4.6) holds. From (7.1), we deduce that (4.7) holds. 2. Notice that for θ ≤ t ≤ s Z s ∇x f (r, Σr )Dθ Xr + ∇y f (r, Σr )Dθ Yr + ∇z f (r, Σr )Dθ Zr dr Dθ Ys = Dθ Yt − t Z s ←−− Z s − ∇x g(r, Σr )Dθ Xr + ∇y g(r, Σr )Dθ Yr + ∇z g(r, Σr )Dθ Zr dBr + Dθ Zr dWr . t

t

It follows from Lemma 4.1 and Lemma 4.2 that, for θ ≤ t < v ≤ s Z s Z s Dv Dθ Ys = Dθ Zv − Dv (Σr )∗ [Hf ](r, Σr )Dθ (Σr )dr − ∇f (r, Σr )Dv Dθ (Σr )dr v v Z s Z s ←−− ←−− − Dv (Σr )∗ [Hg](r, Σr )Dθ (Σr )dBr − ∇g(r, Σr )Dv Dθ (Σr )dBr v Zv s + Dv Dθ Zr dWr . v

Then by taking v = s and t = θ, it follows that equality (4.8) holds. We have from estimate (2.4) and inequality (4.3), that for each v ≤ T and θ ≤ T E[ sup |Dv Dθ Ys |2 ] + E[ t≤s≤T

Z

t

T

|Dv Dθ Zs |2 ds] ≤ C(1 + |x|4 ).

and then by taking v = s and t = θ we deduce that (4.9) holds.

(4.10) ✷

4.4. Path regularity In this subsection, we extend the result of Zhang [33] which concerns the L2 -regularity of the martingale integrand Z. Such result is crucial to derive the rate of convergence of our numerical scheme. We start with the following proposition which gives an upper bound for i i h h and E ||Zu − Zs ||2 , t ≤ s ≤ u ≤ T. E sup |Yr − Ys |2 r∈[s,u]

Proposition 4.4 Assume that (H1)-(H3) hold. Then for t ≤ s ≤ u ≤ T , we have i h ≤ C(1 + |x|2 )|u − s|, E sup |Yr − Ys |2

(4.11)

r∈[s,u]

i h E ||Zu − Zs ||2

≤ C(1 + |x|2 )|u − s|.

Proof. To simplify the notations, we restrict ourselves to the case k = d = l = 1. (i) Plugging inequality (4.7) in the estimate (7.12), the result (4.11) holds. (ii) From Proposition 4.3, we have h i E |Zu − Zs |2 ≤ CE[|Du Yu − Ds Yu |2 ] + CE[|Ds Yu − Ds Ys |2 ].

(4.12)

(4.13)


21

From the definition of the BDSDE (4.5), we have Du Yu − Ds Yu = ∇Φ(XT )(Du XT − Ds XT ) + Z

+

Z

+

Z

+

T u T u T u

Z

T

u

∇x f (r, Σr )(Du Xr − Ds Xr ) dr

∇y f (r, Σr )(Du Yr − Ds Yr ) + ∇z f (r, Σr )(Du Zr − Dt Zr ) dr

←−− ∇x g(r, Σr )(Du Xr − Ds Xr ) + ∇y g(r, Σr )(Du Yr − Ds Yr ) dBr ←−− Z ∇z g(r, Σr )(Du Zr − Ds Zr ) dBr −

T

u

(Du Zr − Ds Zr )dWr .

Applying the generalized Itˆ o’s formula (see [29], Lemma 1.3), we obtain

−

|Du YT − Ds YT |2 − |Du Yu − Ds Yu |2 = Z T Z 2 ∇x f (r, Σr )(Du Xr − Ds Xr )(Du Yr − Ds Yr )dr − 2

−

2

−

2

− − + − +

u Z T

u Z T u T

Z

2

u T

Z

2

u T

2 Z

Z

u T

u T

Z

u

T

u

∇y f (r, Σr )(Du Yr − Ds Yr )2 dr

∇z f (r, Σr )(Du Zr − Du Zr )(Du Yr − Ds Yr )dr ←−− ∇x g(r, Σr )(Du Xr − Ds Xr )(Du Yr − Ds Yr )dBr ←−− ∇y g(r, Σr )(Du Yr − Ds Yr )2 dBr ←−− ∇z g(r, Σr )(Du Zr − Ds Zr )(Du Yr − Ds Yr )dBr (Du Zr − Ds Zr )(Du Yr − Ds Yr )dWr

∇x g(r, Σr )(Du Xr − Ds Xr ) + ∇y g(r, Σr )(Du Yr − Ds Yr ) + ∇z g(r, Σr )(Du Zr − Ds Zr 2 dr |Du Zr − Ds Zr |2 dr.

From inequalities (7.1) and (4.1), using the Burkholder-Davis-Gundy’s inequality and Assumption (H2), the stochastic integrals which appear in the last equation disappear when we take the


22

expectation. By Young inequality, we obtain, for ǫ′ > 0 Z T E[|Du Yu − Ds Yu |2 ] + E[ |Du Zr − Ds Zr |2 ]dr ≤ E[|∇Φ(XT )(Du XT − Ds XT )|2 ] u

+ +

2E[ 2E[

Z

T

u Z T u T

+

2E[

Z

+ +

∇y f (r, Σr )(Du Yr − Ds Yr )2 dr] ∇z f (r, Σr )(Du Zr − Du Zr )(Du Yr − Ds Yr )dr]

u

+

∇x f (r, Σr )(Du Xr − Ds Xr )(Du Yr − Ds Yr )dr]

1 )E[ ǫ′

C(1 +

Z

T

u Z T

∇x g(r, Σr )2 |Du Xr − Ds Xr |2 dr]

1 )E[ ∇y g(r, Σr )2 |Du Yr − Ds Yr |2 dr] ǫ′ u Z T (1 + ǫ′ )E[ ∇z g(r, Σr )2 |Du Zr − Ds Zr |2 dr].

C(1 +

u

Hence by using Assumption (H2) and Young inequality, we have for ǫ, ǫ′ > 0 and C > 0, Z T E[|Du Yu − Ds Yu |2 ] + E[ |Du Zr − Ds Zr |2 dr] ≤ K 2 E[|Du XT − Ds XT |2 ] u

+ + + +

2KE[

Z

T

u T

|Du Xr − Ds Xr |2 dr] + 4KE[

Z

T

u

|Du Yr − Ds Yr |2 dr]

Z T K 2 KǫE[ |Du Yr − Ds Yr | dr] + E[ |Du Zr − Ds Zr |2 dr] ǫ u u Z T Z T 1 1 CK 2 (1 + ′ )E[ |Du Xr − Ds Xr |2 dr] + CK 2 (1 + ′ )E[ |Du Yr − Ds Yr |2 dr] ǫ ǫ u u Z T (1 + ǫ′ )α2 E[ |Du Zr − Ds Zr |2 dr]. Z

u

Then, we obtain 2

E[|Du Yu − Ds Yu | ] + E[ + + +

K(2 + KC(1 +

1 ))E[ ǫ′

Z

Z

T u

T

u

|Du Zr − Ds Zr |2 dr] ≤ K 2 E[|Du XT − Ds XT |2 ]

|Du Xr − Ds Xr |2 dr]

Z T 1 ) + (4 + ǫ)K)E[ |Du Yr − Ds Yr |2 dr] ǫ′ u Z T K ((1 + ǫ′ )α2 + )E[ |Du Zr − Ds Zr |2 dr]. ǫ u

(K 2 C(1 +

For ǫ large enough and ǫ′ small enough, we have (1 + ǫ′ )α2 + Kǫ < 1. From inequality (4.2), we deduce that Z T E[|Du Yu − Ds Yu |2 ] ≤ C (1 + |x|2 )|u − s| + E[ |Du Yr − Ds Yr |2 dr] , u

where C is a positive constant. From Gronwall’s lemma we have

E[|Du Yu − Ds Yu |2 ] ≤ C(1 + |x|2 )|u − s|.

(4.14)


23

Since (Ds Yu )s≤u≤T satisfies the BDSDE (4.5), inequalities (7.12)-(4.7) hold for (Ds Yu , Ds Zu )s≤u≤T and yield E[|Ds Yu − Ds Ys |2 ] ≤ C(1 + |x|2 )|u − s|. Plugging (4.14) and (4.15) into (4.13), we obtain (4.12).

(4.15) ✷

The following theorem states the rate of convergence of our numerical scheme. Theorem 4.1 Under Assumptions (H1)-(H3), there exists a positive constant C (depending only on T , K, α, |b(0)|, ||σ(0)||, |f (t, 0, 0, 0)| and ||g(t, 0, 0, 0)||) such that ErrorN (Y, Z) ≤ Ch(1 + |x|2 ).

(4.16)

Proof. From the definition (3.1), Z¯tn is the best approximation of (Zt )tn ≤t k + 1 + d2 , the coefficients are m-times continuously differentiable in x. When they used a


28

symmetric finite difference scheme and d = 2, the L2 -error is proportional to h2 where h is the discretization step in space and by the Richardson acceleration, the error is proportional to h4 . Compared to their work, our scheme is more general. It converges in the non linear case. Our √ convergence is of order h where h is the discretization step in time. However, in our work, the rate of convergence does not depend on the space dimension d. Remark 5.2 If we assume more regularity conditions on the coefficients and the final condition as in Pardoux and Peng [29], namely, Φ ∈ Cb3 (Rd , Rk ), f ∈ Cb3 ([0, T ] × Rd × Rk × Rd×k , Rk ) and g ∈ Cb3 ([0, T ] × Rd × Rk × Rd×k , Rk×l ). If (Yst,x , Zst,x )t≤s≤T is the solution the BDSDE (1.2). Then, ut (x) = Ytt,x , ∀(t, x) ∈ [0, T ] × Rd is the unique classical solution of the SPDE (1.1) in the integral sense (see [29]). Therefore, we can obtain a stronger result. In fact, the estimation on the error (5.7) obtained in the previous theorem can be replaced by: 2 E[ sup |uN t (x) − u(t, x)| ] + E[ 0≤t≤T

Z

0

T

||vtN (x) − v(t, x)||2 dt] ≤ Ch.

This last equation gives an estimation which holds for all x ∈ Rd and which is not only almost sure anymore. For the Monte Carlo method, we estimate the solution for one point x at time t, and by varying x and t we obtain the solution u(t, x) on the whole domain. 6. Implementation and numerical tests In this part, we are interested in implementing our numerical scheme. Our aim is only to test statically its convergence. Further analysis of the convergence of the used method and of the error bounds will be accomplished in a future work. 6.1. Notations and algorithm We use a path-dependent algorithm, for every fixed path of the brownian motion B, we approximate by a regression method the solution of the associated PDE. Then, we replace the conditional expectations which appear in (6.1) and (6.2) by L2 (Ω, P) projections on the function basis approximating L2 (Ω, Ftn ). We compute ZtNn in an explicit manner and we use I Picard iterations in a implicit way. Actually, we proceed as in [12], except that in our case the to compute YtN n and ZtNn are measurable functions of (XtNn , (∆Bi )n≤i≤N −1 ). So, each solution given solutions YtN n by our algorithm depends on the fixed path of B. 6.1.1. Numerical scheme For each fixed path of B, the solution of (2.1)-(2.2) is approximated by (Y N , Z N ) defined by the following algorithm, given in the multidimensional case. For 0 ≤ n ≤ N − 1: ∀j1 ∈ {1, . . . , k}, l i h X N N N N N N N )∆B , Z , Y g (X ) + , Z , Y (X + hf Y = E YtN n,j , j1 ,j j1 tn tn+1 tn+1 tn+1 tn tn tn tn+1 ,j1 n ,j1 j=1

(6.1)


29

∀j1 ∈ {1, . . . , k} and ∀j2 ∈ {1, . . . , d} l i h X , ZtNn+1 )∆Bn,j ∆Wn,j2 . gj1 ,j (XtNn+1 , YtN + ∆W hZtNn ,j1 ,j2 = Etn YtN n,j2 n+1 n+1 ,j1

(6.2)

j=1

We stress that at each discretization time, the solution of the algorithm depends on the fixed path of the brownian motion B. 6.1.2. Vector spaces of functions At every tn , we select k(d + 1) deterministic functions bases (pi,n (.))1≤i≤k(d+1) and we look for and ZtNn which will be denoted respectively by ynN and znN , in the vector space approximations of YtN n spanned by the basis (pj1 ,n (.))1≤j1 ≤k (respectively (pj1 ,j2 ,n (.))1≤j1 ≤k,1≤j2 ≤d ). Each basis pi,n (.) is considered as a vector of functions of dimension Li,n . In other words, Pi,n (.) = {α.pi,n (.), α ∈ RLi,n }. As an example, we cite the hypercube basis (HC) used in [12]. In this case, pi,n (.) does not depend nor on i neither on n and its dimension is simply denoted by L. A domain D ⊂ Rd centered on Qd X0 = x, that is D = i=1 (xi − a, xi + a], can be partitionned on small hypercubes of edge δ. Then, S D = i1,...,i Di1,...,id where Di1 ,...,id = (xi − a + i1 δ, xi − a + i1 δ] × . . . × (xi − a + id δ, xi − a + id δ]. d Finally we define pi,n (.) as the indicator functions of this set of hypercubes. 6.1.3. Monte Carlo simulations To compute the projection coefficients α, we will use M independent Monte Carlo simulations of and ∆Wnm , m = 1, . . . ,M . XtnN and ∆Wn which will be respectively denoted by XtN,m n 6.1.4. Description of the algorithm N,m N,m,I )) and (zN )=0. → Initialization: For n = N , take (yN ) = (Φ(XtN,m N → Iteration: For n = N − 1, . . . , 0: • We approximate (6.2) by computing for all j1 ∈ {1, . . . , k} and j2 ∈ {1, . . . , d}

αM j1 ,j2 ,n

=

arginf α

+

l X

M m 1 X N,M,I N,m ∆Wn,j 2 yn+1,j1 (Xtn+1 ) M m=1 h

2 ∆Bn,j ∆W m n,j2 N,M ) ), zn+1 (XtN,m ,y N,M,I(XtN,m gj1 ,j XtN,m − α.pm j1 ,j2 ,n . n+1 n+1 n+1 n+1 h j=1

N,M Then we set zn,j (.) = (αM j1 ,j2 ,n .pj1 ,j2 ,n (.)), j1 ∈ {1, . . . , k}, j2 ∈ {1, . . . , d}. 1 ,j2 • We use I Picard iterations to obtain an approximation of Ytn in (6.1): · For i = 0: ∀j1 ∈ {1, . . . , k}, αM,0 j1 ,n = 0. · For i = 1, . . . , I: We approximate (6.1) by calculating αM,i j1 ,n , ∀j1 ∈ {1, . . . , k}, as the minimizer


of: M 1 X N,M,I N,m N,m N,m N,m yn+1,j1 (Xtn+1 )+ hfj1 Xtn ,ynN,M,i−1 (Xtn ),znN,M(Xtn ) M m=1

+

l X j=1

2 N,M gj1 ,j XtN,m ) ∆Bn,j −α.pm ),zn+1 (XtN,m ,y N,M,I(XtN,m j1 ,k . n+1 n+1 n+1 n+1

Finally, we define ynN,M,I (.) as:

N,M,I yn,j (.) = (αM,I j1 ,n .pj1 ,n (.)), ∀j1 ∈ {1, . . . , k}. 1

6.2. One-dimensional case (Case when d = k = l = 1) 6.2.1. Function bases We use the basis (HC) defined above. So we set: d1 = min Xtmn , n,m

d2 − d1 δ h defined by Dj = d + (j − 1)δ, d + jδ , ∀j.

d2 = max Xtmn and L = n,m

where δ is the edge of the hypercubes (Dj )1≤j≤L We take at each time tn

), j = 1, . . . , L ) = 1[d+(j−1)δ,d+jδ) (XtN,m 1Dj (XtN,m n n and (pm i,n (.)) =

s n

o M ),1 ≤ j ≤ L , i = 0, 1. 1Dj(XtN,m n card(Dj )

Card(Dj ) denotes the number of simulations of XtNn which are in our cube Dj . This system is orthonormal with respect to the empirical scalar product defined by < ψ1 , ψ2 >n,M :=

M 1 X ). )ψ2(XtN,m ψ1(XtN,m n n M m=1

In this case, the solutions of our least squares problems are given by: αM 1,n

αM,i 0,n

M n ∆Wnm 1 X N,M,I ) ) yn+1 (XtN,m p1,n (XtN,m n+1 n M m=1 h ∆B m ∆W m o N,m N,M, N,m N,M,I n n ) (X ), z (X , y + g XtN,m , t t n+1 n+1 n+1 n+1 n+1 h M n 1 X N,m N,m N,m N,M,i−1 N,m N,M,I N,M = ) ), z (X , y (X ) + hf X (X ) y p0,n (XtN,m n tn n tn tn tn+1 n+1 n M m=1 o N,M N,M,I ) ∆Bnm . ), zn+1 (XtN,m , yn+1 (XtN,m + g XtN,m n+1 n+1 n+1

=

30


31

Remark 6.1 We note that for each value of M , N and δ, we launch the algorithm 50 times and 0,x,N,M,I we denote by (Y0,m )1≤m′ ≤50 the set of collected values. Then we calculate the empirical mean ′ 0,x,N,M,I

Y0

and the empirical standard deviation σ N,M,I defined by: v u 50 50 u1 X 1 X 0,x,N,M,I 0,x,N,M,I 0,x,N,M,I 2 0,x,N,M,I N,M,I Y0,m′ |Y0,m and σ =t −Y 0 Y0 = | . ′ 50 ′ 49 ′ m =1

(6.3)

m =1

We also note before starting the numerical examples that our algorithm converges after at most three Picard iterations. Finally, we stress that (6.3) gives us an approximation of u(0, x) the solution of the SPDE (1.1) at time t = 0. 6.2.2. Case when f and g are linear in y and independent of z   

dXt = Xt (µdt + σdWt ), Φ(x) = −x + K, f (y) = a0 y, g(y) = b0 y

and we set K = 115, r = 0.01, R = 0.06, X0 = 100, µ = 0.05, σ = 0.2, T = 0.25, d1 = 60, d2 = 200, a0 and b0 are fixed constants. Let Yexplicit be the solution of our BDSDE in this particular case. By the integration by parts formula, we get 1 2

t,x B Yt,explicit = E[Φ(XTt,x )ea0 (T −t)+b0 (BT −Bt )− 2 b0 (T −t) /Ft,T ].

At t=0, we have 0,x Y0,explicit

1 2

=

B ] E[Φ(XT0,x )e(a0 − 2 b0 )T +b0 BT /F0,T

=

e(a0 − 2 b0 )T +b0 BT E[Φ(XT0,x )].

1 2

0,x,N,M,I

Then, we define Y 0 as the numerical approximation of the solution of the BDSDE in this case (computed by our algorithm) and σ N,M,I as its standard deviation. In the other hand, we 0,x compute the solution Y0,explicit in this linear case by using the explicit formula of the expectation 0,x of Φ(XT ), as follows

1 2

1 2

0,x Yexplicit = e(a0 − 2 b0 )T +b0 BT E[Φ(XT0,x )] = e(a0 − 2 b0 )T +b0 BT (K − xeµT ).

For a0 = 0.5, b0 = 0.5 and δ = 1 M 0,x N=20, Yexplicit = 13.724

100 1000 5000

0,x,N,M,I

Y0

0,x,N,M,I

(σ N,M,I )

13.910(1.178) 13.792(0.309) 13.847(0.117)

0,x |Yexplicit −Y 0

0,x Yexplicit

0.013 0.004 0.008

|


32

For a0 = 0.5, b0 = 0.5 and δ = 0.5 M 0,x N=30, Yexplicit = 14.115

100 1000 5000

0,x,N,M,I

Y0

0,x,N,M,I

(σ N,M,I )

0,x |Yexplicit −Y 0

|

0,x Yexplicit

14.246(1.045) 14.195(0.337) 14.236(0.129)

0.009 0.005 0.008

6.2.3. Comparison of numerical approximations of the solutions of the FBDSDE and the FBSDE: the general case Now we take       

Φ(x) = −x + K,

f (t, x, y, z) = −θz − ry + (y − σz )− (R − r),

g1 (t, x, y, z) = 0.1z + 0.5y + log(x)

and we set θ = (µ − r)/σ, K = 115, X0 = 100, µ = 0.05, σ = 0.2, r = 0.01, R = 0.06, δ = 1, N = 20, T = 0.25 and we fix d1 = 60 and d2 = 200 as in [11]. The functions g1 ,g2 and g3 tooken in the following are examples of the function g. They are sufficiently regular and Lipschitz on [60, 200] × R × R and could be extended to regular Lipschitz functions on R3 . In this case, Assumptions (H1)-(H3) are satisfied. t,x,N,M,I ) and the BSDE’s one We compare the numerical solution of our BDSDE (noted again Y t 0,x,N,M (noted here by Y t,BSDE ), without g and B. When t is close to maturity

When t = 0

0,x,N,M

(σ N,M,I ) Y t19 15.452(0.948) 15.534(0.409) 15.464(0.240) 15.484(0.097) 15.501(0.058)

0,x,N,M

(σ N,M,I ) Y t15 17.894(1.096) 17.774(0.429) 17.607(0.270) 17.623(0.104) 17.627(0.064)

M 128 512 2048 8192 32768

Y t19 ,BSDE (σ N,M ) 13.748(0.879) 13.827(0.384) 13.762(0.223) 13.781(0.091) 13.796(0.054)

M 128 512 2048 8192 32768

Y t15 ,BSDE (σ N,M ) 14.168(0.905) 14.113(0.388) 13.988(0.226) 13.985(0.093) 13.994(0.055)

0,x,N,M,I

0,x,N,M,I


M 128 512 2048 8192 32768

0,x,N,M

Y 0,BSDE (σ N,M ) 15.431(1.005) 15.029(0.428) 14.763(0.243) 14.718(0.098) 14.715(0.060)

0,x,N,M,I

Y0

(σ N,M,I ) 13.571(1.146) 13.173(0.500) 12.885(0.280) 12.825(0.106) 12.804(0.064)

For g2 (y, z) = 0.1z + 0.5y 0,x,N,M,I

M 128 512 2048 8192 32768

(σ N,M,I ) Y t19 14.767(0.949) 14.850(0.410) 14.781(0.240) 14.801(0.097) 14.818(0.058)

M 128 512 2048 8192 32768

Y t15 (σ N,M,I ) 16.267(1.093) 16.166(0.428) 16.007(0.270) 16.024(0.104) 16.029(0.064)

M 128 512 2048 8192 32768

Y0

M 128 512 2048 8192 32768

Y t19 (σ N,M,I ) 15.452(0.948) 15.534(0.409) 15.464(0.240) 15.484(0.097) 15.501(0.058)

M 128 512 2048 8192 32768

(σ N,M,I ) Y t15 18.253(1.068) 18.166(0.453) 18.010(0.266) 18.006(0.109) 18.017(0.065)

0,x,N,,M,I

When t = 0 0,x,N,M,I

(σ N,M,I ) 13.821(0.063) 14.555(1.132) 14.176(0.495) 13.899(0.277) 13.842(0.105)

For g3 (x, y) = logx + 0.5y: When t is close to maturity 0,x,N,M,I

0,x,N,M,I

33


34

When t = 0 M 128 512 2048 8192 32768

0,x,N,M,I

Y0

(σ N,M,I ) 12.071(0.054) 12.075(0.088) 12.122(0.218) 12.384(0.381) 12.791(0.903)

In the previous tables, we test our algorithm for different examples of the function g (g1 and g2 are dependent in z, g3 is independent of z). We see the convergence of the BDSDE’s solution when we increase the number of simulations M . In figure 1, we study statically the main result of this paper. So, we fix all the parameters (δ = 1, and M = 2000 ) and we draw the map of the BDSDE’s solution, for the function g1 , with respect to the number of time discretization steps N . The solution is computed for five different paths of the brownian motion B. We can examine there the convergence of our scheme.

The approximated solution at time t=0 for each path of B

20 19.5 19 18.5 18 17.5 17 16.5 16 15.5 15 10

15

20 25 The Number of the time discretization steps

30

35

Figure 1. The BDSDE’s solution with respect to the number of time discretization steps for five different parhs of B. The figure is obtained for M = 2000 and δ = 1.

We see on Figure 2 the impact of the function g on the solution; we variate N , M and δ as in [12], by taking these quantities as follows: First we fix d1 = 40 and d2 = 180 (which means that x ∈ [d1 , d2 ] = [40, 180] and in this case our assumptions (H1)-(H3) are satisfied). Let j ∈ N, we √ √ √ take αM = 3, β = 1, N = 2( 2)(j−1) , M = 2( 2)αM (j−1) and δ = 50/( 2)(j−1)(β+1)/2 . Then, we draw the map of each solution at t = 0 with respect to j.


35

25

The approximation of the solution Y at time t=0

BSDE BDSDE for g1 BDSDE for g2 20

15

10

5

0

−5 1

2

3

4

5 The parameter j

6

7

8

9

Figure 2. Comparison of the BSDE’s solution and the BDSDE’s one: The solution of the BSDE is with circle markers, the solution of the BDSDE for g1 (x, y, z) = 0.1z + 0.5y + log(x) is with star markers and the one for g2 (y, z) = 0.1z + 0.5y is with cross markers. Confidence intervals are with dotted lines.

7. Appendix 7.1. Proof of Proposition 4.2. To simplify the notations, we restrict ourselves to the case k = d = l = 1. (Dθ Y, Dθ Z) is well defined and from inequalities (2.4) and (4.1), we deduce that for each θ ≤ T Z T E[ sup |Dθ Ys |2 ] + E[ |Dθ Zs |2 ds] ≤ C(1 + |x|2 ). (7.1) t≤s≤T

t

We define recursively the sequence (Y m , Z m ) as follows. First we set (Y 0 , Z 0 ) = (0, 0). Then, given (Y m−1 , Z m−1 ), we define (Y m , Z m ) as the unique solution in S2k ([t, T ]) × H2k×d ([t, T ]) of Z T Z T Z T ←−− Ysm = Φ(XTt,x ) + f (r, Xrt,x, Yrm−1 , Zrm−1 )dr + g(r, Xrt,x , Yrm−1 , Zrm−1 )dBr − Zrm dWr . s

s

s

We recursively show that (Y , Z ) ∈ B ([t, T ], D ). Suppose that (Y , Z ) ∈ B ([t, T ], D1,2 ) and let us show that (Y m+1 , Z m+1 ) ∈ B 2 ([t, T ], D1,2 ). RT 1,2 From the induction assumption, we have Φ(XT ) + s f (r, Σm . r )dr ∈ D R ←−− T m 1,2 1,2 We have g(r, Σr ) ∈ D for all r ∈ [t, T ]. From Lemma 4.2, we have t g(r, Σm . then r )dBr ∈ D Z T Z T ←−− W B 1,2 Ysm+1 = E Φ(XTt,x ) + f (r, Σm g(r, Σm , r )dr + r )dBr |Ft,s ∨ Ft,T ∈ D m

m

2

s

where Hence

Σm r Z

t

:=

T

1,2

m

m

2

s

(Xrt,x , Yrm , Zrm ).

Zrm+1 dWr

=

Φ(XTt,x )

+

Z

t

T

f (r, Σm r )dr

+

Z

t

T

←−− m+1 g(r, Σm ∈ D1,2 . r )dBr − Yt


36

It follows from Lemma 4.1 that Z m+1 ∈ M2k×d ([t, T ], D1,2 ) and we have Dθ Ysm+1 = Dθ Zsm+1 = 0 for t ≤ s ≤ θ and for θ ≤ s ≤ T Dθ Ysm+1 = ∇Φ(XTt,x )Dθ XTt,x Z T m m m m dr ∇x f (r, Σm + r )Dθ Xr + ∇y f (r, Σr )Dθ Yr + ∇z f (r, Σr )Dθ Zr

(7.2)

s

+

T

Z

s

−

Z

T

s

−− m m m m ← dBr ∇x g(r, Σm )D X + ∇ g(r, Σ )D Y + ∇ g(r, Σ )D Z θ r y θ z θ r r r r r

Dθ Zrm+1 dWr .

From inequality (2.4), we deduce that for each θ ≤ T |Dθ Ysm+1 |2 ]

E[ sup t≤s≤T

+ E[

Z

T t

|Dθ Zsm+1 |2 ds] ≤ C(1 + |x|2 ).

It is known that inequality (2.4) holds for (Y m+1 , Z m+1 ) and so we deduce that kY m+1 k1,2 + kZ m+1 k1,2 < ∞, which shows that (Y m+1 , Z m+1 ) ∈ B 2 ([t, T ], D1,2 ). Using the contraction mapping argument as in El Karoui Peng and Quenez [10], we deduce that (Y m+1 , Z m+1 ) converges to (Y, Z) in S2 ([t, T ]) × H2 ([t, T ]). We will show that (Dθ Y m , Dθ Z m ) converges to (Y θ , Z θ ) in L2 (Ω × [t, T ] × [t, T ], dP ⊗ dt ⊗ dt), where Ysθ = Zsθ = 0 for all t ≤ s ≤ θ and (Ysθ , Zsθ , θ ≤ s ≤ T ) is the solution of the BDSDE. Ysθ

= ∇Φ(XTt,x )Dθ XTt,x Z T + ∇x f (r, Σr )Dθ Xr + ∇y f (r, Σr )Yrθ + ∇z f (r, Σr )Zrθ dr

(7.3)

s

+

Z

T

s

−

Z

T

s

←−− ∇x g(r, Σr )Dθ Xr + ∇y g(r, Σr )Yrθ + ∇z g(r, Σr )Zrθ dBr

Zrθ dWr .

From equations (7.2) and (7.3), we have Dθ Ysm+1

−

Ysθ

=

Z

T s

m +∇y f (r, Σm r )Dθ Yr

+

Z

T

s

+

Z

T

s

−

Z

s

T

t,x (∇x f (r, Σm r ) − ∇x f (r, Σr ))Dθ Xr

m θ − ∇y f (r, Σr )Yrθ + ∇z f (r, Σm )D Z − ∇ f (r, Σ )Z θ z r r r r dr

−− t,x m m θ ← (∇x g(r, Σm r ) − ∇x g(r, Σr ))Dθ Xr + ∇y g(r, Σr )Dθ Yr − ∇y g(r, Σr )Yr dBr −− m θ ← ∇z g(r, Σm r )Dθ Zr − ∇z g(r, Σr )Zr dBr

(Dθ Zrm+1 − Zrθ )dWr .


37

From Proposition 4.1, we have E[ sup |Dθ Ysm+1 − Ysθ |2 ] + E[ θ≤s≤T

T

hZ

Z

s

T

|Dθ Zrm+1 − Zrθ |2 dr]

(7.4)

t,x m θ θ (∇x f (r, Σm r ) − ∇x f (r, Σr ))Dθ Xr + ∇y f (r, Σr )Yr − ∇y f (r, Σr )Yr s 2 i θ θ +∇z f (r, Σm r )Zr − ∇z f (r, Σr )Zr dr hZ T m θ θ +CE (∇x g(r, Σm r ) − ∇x g(r, Σr ))Dθ Xr + ∇y g(r, Σr )Yr − ∇y g(r, Σr )Yr ≤ CE

s

2 i θ θ +∇z g(r, Σm )Z − ∇ g(r, Σ )Z z r r r r dr .

Therefore, we obtain Z E[

T

t

≤ CE[

Z

Z T

t

T t

Z

t

|Dθ Ysm+1 T

−

Ysθ |2 dsdθ]

m 2 | drdθ] + CE[ |δr,θ

Z

+ E[

Z

t

T t

Z

t

T

T

Z

t

T

|Dθ Zsm+1 − Zsθ |2 dsdθ]

(7.5)

2 |ρm r,θ | drdθ],

where m δr,θ

=

t,x m θ θ (∇x f (r, Σm r ) − ∇x f (r, Σr ))Dθ Xr + ∇y f (r, Σr )Yr − ∇y f (r, Σr )Yr

+

θ θ ∇z f (r, Σm r )Zr − ∇z f (r, Σr )Zr ,

=

t,x m θ θ (∇x g(r, Σm r ) − ∇x g(r, Σr ))Dθ Xr + ∇y g(r, Σr )Yr − ∇y g(r, Σr )Yr

(7.6)

and ρm r,θ

+

θ θ ∇z g(r, Σm r )Zr − ∇z g(r, Σr )Zr .

(7.7)

RT RT m 2 RT m )t≤r,θ≤T , we have E[ t t |δr,θ From the definition of (δr,θ | drdθ] ≤ C t (Am (θ, t, T )+Bm (θ, t, T ))dθ, where hZ T i t,x 2 Am (θ, t, T ) = E |(∇x f (r, Σm r ) − ∇x f (r, Σr ))Dθ Xr | dr t

Bm (θ, t, T ) =

E

hZ

T

t

+

E

hZ

T

t

i θ 2 (∇y f (r, Σr ) − ∇y f (r, Σm ))Y dr r r

i θ 2 (∇z f (r, Σr ) − ∇z f (r, Σm ))Z r r dr

Moreover, since ∇x f is bounded and continuous with respect to (x, y, z), it follows by the dominated convergence theorem and inequality (2.3) that Z T lim Am (θ, t, T )dθ = 0. (7.8) m→∞

t

Furthermore, since ∇y f and ∇z f are bounded and continuous with respect to (x, y, z), it follows, also, by the dominated convergence theorem and inequality (2.4) that Z T Bm (θ, t, T )dθ = 0. (7.9) lim m→∞

t


38

RT RT m 2 RT ′ ′ From the definition of (ρm r,θ )s≤r,θ≤T , we have E[ t t |ρr,θ | drdθ] ≤ C t (Am (θ, t, T )+Bm (θ, t, T ))dθ with hZ T i t,x 2 A′m (θ, t, T ) = E |(∇x g(r, Σm r ) − ∇x g(r, Σr ))Dθ Xr | dr t

′ Bm (θ, t, T ) =

+

E

hZ

T

t

E

hZ

t

T

θ 2 (∇y g(r, Σr ) − ∇y g(r, Σm dr r ))Yr

i θ 2 (∇z g(r, Σr ) − ∇z g(r, Σm r ))Zr dr .

Similarly as shown above, since ∇y g and ∇z g are bounded and continuous with respect to (x, y, z) we can show that: Z T Z T ′ ′ lim Am (θ, t, T )dθ = lim Bm (θ, t, T )dθ = 0. (7.10) m→∞

m→∞

t

t

Plugging (7.8), (7.9) and (7.10) into inequality (7.5), we deduce that lim E[

m→∞

Z

T

t

Z

T

t

|Dθ Ysm+1

−

Ysθ |2 dsdθ]

+ E[

Z

T t

Z

t

T

|Dθ Zsm+1 − Zsθ |2 dsdθ] = 0.

It follows that (Y m , Z m ) converges to (Y, Z) in L2 ([t, T ], D1,2 × D1,2 ) and a version of (Dθ Y, Dθ Z) is given by (Y θ , Z θ ) which is the desired result. ✷ 7.2.

Second order Malliavin derivative of the solution of BDSDE’s

We apply know similar computation to get the second order Malliavin derivatives representation of the solution of BDSDE ’s, so we will omit the proof. Proposition 7.1 Assume that assumptions (H2) and (H3) hold. We fix t ∈ [0, T ]. Then for each t ≤ θ ≤ T , (Dθ Y, Dθ Z) belongs to B 2 ([t, T ], D1,2 ). For each t ≤ v ≤ T and 1 ≤ i, j ≤ d, Dvj Dθi Ys = Dvj Dθi Zsn = 0, 1 ≤ n ≤ d, if s < θ ∨ v, and a version of (Dvj Dθi Ys , Dvj Dθi Zs )v∨θ≤s≤T is the unique solution of the equation: Dvj Dθi Ys = T1 (Φ) + T2 (f ) + T3 (g) + T4 (W ), where T1 (Φ) =

k X

n1 =1

∇((∇Φ)n1 (XTt,x ))Dvj XTt,x (Dθi XTt,x )n1 + ∇Φ(XTt,x )Dvj Dθi XTt,x ,


T2 (f ) =

s

+ +

n1 =1

∇x f (r, Xrt,x , Yr , Zr )Dvj Dθi Xrt,x dr Z

T

s

+ +

k X ∇x ((∇x f )n1 (r, Xrt,x , Yr , Zr ))Dvj Xrt,x (Dθi Xrt,x )n1

T

Z

k X

n1 =1

d Z X d X

n2 =1

T3 (g) =

l Z X

n3 =1

+

l X

n3 =1

+

Z

l Z X

n3 =1

+

l X

n3 =1

+

s

Z

s

k X

n1 =1 T

s

d X

←−−− ∇x ((∇x g n3 )n1 (r, Xrt,x , Yr , Zr ))Dvj Xrt,x (Dθi Xrt,x )n1 dBrn3

←−−− ∇y ((∇y g n3 )n1 (r, Xrt,x , Yr , Zr ))Dvj Yr (Dθi Yr )n1 dBrn3

←−−− ∇y g n3 (r, Xrt,x , Yr , Zr )Dvj Dθi Yr dBrn3

n3 =1 n2 =1

+

s

←−−− ∇x g n3 (r, Xrt,x , Yr , Zr )Dvj Dθi Xrt,x dBrn3

T s

∇zn2 ((∇zn2 f )n1 (r, Xrt,x , Yr , Zr ))Dvj Zrn2 (Dθi Zrn2 )n1 dr

∇zn2 f (r, Xrt,x , Yr , Zr )Dvj Dθi Zrn2 dr,

n1 =1 T

k X

n1 =1 T

Z

k X

T s

T

l d Z X X l X

∇y ((∇y f )n1 (r, Xrt,x , Yr , Zr ))Dvj Yr (Dθi Yr )n1

∇y f (r, Xrt,x , Yr , Zr )Dvj Dθi Yr dr n2 =1

+

39

n3 =1 n2 =1

T

s

Z

n1 =1 T

s

k X

←−−− ∇zn2 ((∇zn2 g n3 )n1 (r, Xrt,x , Yr , Zr ))Dvj Zrn2 (Dθi Zrn2 )n1 dBrn3

←−−− ∇zn2 g n3 (r, Xrt,x , Yr , Zr )Dvj Dθi Zrn2 dBrn3 ,

T4 (W ) = −

d Z X

n2 =1

s

T

Dvj Dθi Zrn2 dWrn2 ,

(z j )1≤j≤d denotes the j-th column of the matrix z, (g n3 )1≤n3 ≤l denotes the n3 -th column of the matrix g, B = (B 1 , . . . , B l ), (Dθi Xrt,x )n1 is the n1 -th component of the vector (Dθi Xrt,x), (Dθi Yr )n1 is the n1 -th component of the vector (Dθi Yr ) and (Dθi Zrn2 )n1 is the n1 -th component of the vector (Dθi Zrn2 ). 7.3. Some estimates on the solution of the FBDSDE Lemma 7.1 Let (b1 , σ 1 ) and (b2 , σ 2 ) be the standard parameters of the SDE (2.1) with initial condition x1 (resp. x2 ). We assume that (H1) holds. Put δXs = Xs1 − Xs2 , δbs = (b1 − b2 )(Xs1 ) and δσs = (σ 1 − σ 2 )(Xs1 ).Then ||X 1 ||S2d ≤ C(1 + |x|2 ).


For all s1 , s2 ∈ [0, T ], we have h E sup

s1 ≤u≤s2

40

i |Xu1 − Xs11 | ≤ C(1 + |x|2 )|s2 − s1 |,

and for all s1 ≤ s ≤ s2 , we have Z ||δX||S2d ([s1 ,s2 ]) ≤ C |x1 − x2 |2 + |s2 − s1 | + E

s2

s1

|δbs |2 + |δσs |2 ds] ,

where C is a generic constant depending only on K, T , (b1 (0), σ 1 (0)) and (b2 (0), σ 2 (0)). Lemma 7.2 Let (X t,x , Y t,x , Z t,x ) be the solution of the FBDSDE (2.1)-(2.2). We assume that Assumptions (H1) and (H2) hold. Then, we have ||Y t,x ||S2d + ||Z t,x ||H2d×k ≤ C(1 + |x|2 ), and for all s′ , s ∈ [t, T ], s′ ≤ s, we have i h 2 2 ≤ C (1 + |x|2 )|s − s′ | + ||Z t,x ||Mk×d E sup |Yut,x − Yst,x ′ | [s′ ,s] .

(7.11)

(7.12)

s′ ≤u≤s

Proof. The technics used to prove these estimates are classical in the BSDE’s theory (see El Karoui et al.[10]) so we omit it. ✷

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