In this note, we present a new numerical method for solving backward
stochastic differential equations. Our method can be viewed as an analogue of
the classical finite element method solving deterministic partial differential
equations.Comment: 4 page

Wang, Penghui

Zhang, Xu

English

arXiv

Penghui Wang

Xu Zhang

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 349 (2011) 901–903Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comProbability Theory/Numerical AnalysisNumerical solutions of backward stochastic differential equations:A finite transposition methodSolutions numériques des équations différentielles stochastiques rétrogrades :« A finite transposition method »Penghui Wang a, Xu Zhang b,ca School of Mathematics, Shandong University, Jinan 250100, Chinab Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, Chinac Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, Chinaa r t i c l e i n f o a b s t r a c tArticle history:Received 7 June 2011Accepted 8 July 2011Available online 30 July 2011Presented by Philippe G. CiarletIn this Note, we present a new numerical method for solving backward stochasticdifferential equations. Our method can be viewed as an analogue of the classical finiteelement method solving deterministic partial differential equations.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éDans cette Note, nous présentons une nouvelle méthode pour résoudre numériquement leséquations différentielles stochastiques rétrogrades. Notre méthode ressemble à la méthodedes éléments finis qui permet de résoudre numériquement les équations aux dérivéespartielles déterministes.© 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLinear and nonlinear Backward Stochastic Differential Equations (BSDEs in short) were introduced in [1] and [9], respec-tively. It is well-known that BSDE plays crucial roles in Stochastic Control, Mathematical Finance etc. Clearly, for applications,it deserves to develop effective numerical methods for BSDEs.Let T > 0 and (Ω, F ,F,P) be a complete filtered probability space with F = {Ft}t∈[0,T ] , on which a 1-dimensionalstandard Brownian motion {w(t)}t∈[0,T ] is defined. We denote by L2Ft (Ω;Rn) (n ∈ N) the Hilbert space consisting of all Ft -measurable (Rn-valued) square integrable random variables; by L2F(Ω; Lr(0, T ;Rn)) (1  r  ∞) the Banach space consistingof all Rn-valued {Ft}-adapted processes X(·) such that E|X(·)|2Lr(0,T ;Rn) < ∞; and by L2F(Ω; D([0, T ];Rn)) the Banach spaceconsisting of all Rn-valued {Ft}-adapted càdlàg processes X(·) such that E(|X(·)|2L∞F(0,T ;Rn)) < ∞. For yT ∈ L2FT (Ω;Rn) andf (·, · ,·) satisfies f (·,0,0) ∈ L2(Ω; L1(0, T ;Rn)) and the usual globally Lipschitz condition, we consider the following BSDE{dy(t) = f (t, y(t), Y (t))dt + Y (t)dw(t) in [0, T ],y(T ) = yT . (1)E-mail addresses: phwang@sdu.edu.cn (P. Wang), xuzhang@amss.ac.cn (X. Zhang).1631-073X/$ – see front matter © 2011 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2011.07.011902 P. Wang, X. Zhang / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 901–903Various numerical methods have been developed to solve Eq. (1), say in [2,7,10,12] and the references therein. These meth-ods use essentially the strong form of (1), which holds true only if F is the natural filtration generated by the Brownianmotion. Also, it seems that the previous methods need to compute the conditional expectation, which is in general not easyto be furnished numerically.In this Note, we shall present a new numerical method solving BSDEs from the viewpoint of transposition solutionintroduced in [6], as recalled below.Definition 1.1. A couple (y(·), Y (·)) ∈ L2F(Ω; D([0, T ];Rn)) × L2F(Ω; L2(0, T ;Rn)) is called a transposition solution of BSDE(1), if for any t ∈ [0, T ], (u(·), v(·), η) ∈ L2F(Ω; L1(t, T ;Rn)) × L2F(Ω; L2(t, T ;Rn)) × L2Ft (Ω;Rn), the following variationalequation holdsE〈z(T ), yT〉 − E〈η, y(t)〉 = ET∫t〈z(τ ), f(τ , y(τ ), Y (τ ))〉dτ + ET∫t〈u(τ ), y(τ )〉dτ + ET∫t〈v(τ ), Y (τ )〉dτ , (2)where z(τ ) = η + ∫ τt u(s)ds + ∫ τt v(s)dw(s).We refer to [6] for the well-posedness of Eq. (1) in the sense of transposition solution. It is easy to see that, if this equa-tion admits a strong solution (say under the assumption of natural filtration), it coincides with the transposition solution.Our numerical schemes for solving Eq. (1) can be described as follows.1) Take a suitable finite-dimensional subspace Hm of L2F(Ω; L2(0, T ;Rn)), where m is a natural number;2) If the solution of (1) exists, then it should satisfy the variational equation (2) for any u, v ∈ Hm . By taking u and v tobe the orthonormal basis of Hm , we obtain a system of approximating equations;3) If the solution of the system of approximating equations exists, then we find a class of numerical solutions of (1);4) Finally, we show the convergence of the above numerical solutions.Clearly, the above procedure is, in spirit, very close to that of the classical finite element method solving deterministicPDEs (e.g., [3]). Therefore, our method to solve BSDEs can be viewed as a stochastic version of the finite element-typemethod. Nevertheless, the notion of “stochastic finite element method” has already been used for other purpose, say [4,5,8]and references therein for solving random PDEs. Note also that our method is quite different from that in these references,and therefore instead we call it a finite transposition method.There are at least two reasons for us to develop this new numerical approach for BSDE. The first one is that, we cansolve the BSDE with general filtration. The second is that, in our approach, we do not need to compute the conditionalexpectation.We refer to [11] for the details of the proofs of the results in this Note and other results in this context.2. Numerical schemes and convergenceFor simplicity, we consider only the following linear BSDE (with f (·) ∈ L2(Ω; L1(0, T ;Rn))){dy(t) = f (t)dt + Y (t)dw(t), t ∈ [0, T ),y(T ) = yT . (3)Assume that L2FT (Ω;Rn) is a separable Hilbert space. For any N ∈ N, write RN = {t | t = 2N T ,  = 0, . . . ,2N }. For anyk ∈ {0, . . . ,2N − 1}, define a sequence of simple processes {eki(·,·)}Mk,Ni=1 byeki(t,ω) ={χ[tk,tk+1)(t)hki(ω), 0  k < 2N − 1,χ[tk,T ](t)hki(ω), k = 2N − 1,(4)where {M0,N , M1,N , . . . , M2N −1,N} is an increasing sequence of integers. Since L2Ft (Ω;Rn) ⊆ L2Fs (Ω;Rn) (for any 0  t < s T ), we may assume that {hki} satisfy the following:1) For any fixed k ∈ {0, . . . ,2N − 1}, {hki}Mk,Ni=0 is an orthogonal set in L2Ftk (Ω;Rn), with the norm |hki |L2Ftk (Ω;Rn) =√2NT ,and hence |eki|L2F(Ω;L2(0,T ;Rn)) = 1;2) If 0  k < l  2N − 1, then {hki}Mk,Ni=0 ⊂ {hli}Ml,Ni=0 ; and3) If s0 = k02N0 T for some N0 ∈ N and k0 ∈ {0, . . . ,2N0 − 1}, then s0 = 2l−N0 k0T /2l ∈ Rl for any l  N0. For l  N0, writekl = 2l−N0 k0. Then, {hk j i}Mk j , ji=0 ⊂ {hkl i}Mkl ,li=0 for N0  j < l, and⋃∞l=N {hkl i}Mkl ,li=1 is an orthogonal basis of L2F (Ω;Rn).0 s0P. Wang, X. Zhang / C. R. Acad. Sci. Paris, Ser. I 349 (2011) 901–903 903Denote by HN the subspace of L2F(Ω; L2(0, T ;Rn)) spanned by {e0i}M0,Ni=1 , . . . , {e2N−1,i}M2N −1,Ni=1 . This is the finite elementsubspace that we will employ below. Replace L2F(Ω; L2(0, T ;Rn)) by HN , then the desired numerical scheme follows bytrying to find yN , Y N ∈ HN such that (2) (with f (·, y(·, Y (·)) replacing by f (·)) holds for η = 0 and for all u, v ∈ H N .To find the yN in HN , suppose yN = ∑2N −1k=0 ∑Mk,Ni=0 αkieki . Choosing u = eki , v = 0 and η = 0, we get zki(t) = ∫ t0 eki(τ )dτ ,and hence E〈zki(T ), yT 〉 = E∫ T0 〈zki(τ ), f (τ )〉dτ +∑l, j αl jE∫ T0 〈eki(τ ), elj(τ )〉dτ . Since {eki} is an orthonormal basis of HN ,it follows that E∫ T0 〈elj(τ ), eki(τ )〉dτ = δklδi j . Therefore,αki = TN E〈hki, yT 〉 − ET∫0〈(τ ∧ tk+1 − τ ∧ tk)hki, f (τ )〉dτ . (5)Similarly, suppose Y N = ∑2N −1k=0 ∑Mk,Ni=0 βkieki . By taking u = 0, η = 0 and v = eki to get a corresponding zki(t) =∫ t0 eki(τ )dwτ , we find thatβki = E〈(w(tk+1) − w(tk))hki, yT〉 − ET∫0〈(w(τ ∧ tk+1) − w(τ ∧ tk))hki, f (τ )〉dτ .We now show the convergence of the sequence {(yN , Y N )} of numerical solutions constructed above.Theorem 2.1. Let (y, Y ) be the transposition solution of (3). Then yN and Y N are projections of y (viewing as an element ofL2F(Ω; L2(0, T ;Rn))) and Y onto HN , respectively.By the construction of H N , it is clear that HN ⊂ HM provided N < M . Moreover, ⋃∞N=1 HN is dense in L2F(Ω; L2(0, T ;Rn)).Hence,ET∫0∣∣yN(τ ) − y(τ )∣∣2 dτ + ET∫0∣∣Y N(τ ) − Y (τ )∣∣2 dτ → 0, as N → ∞. (6)Furthermore, starting from (6), we can show the following convergent result.Theorem 2.2. As N → ∞, (yN , Y N ) tends to (y, Y ) in L2F(Ω; D([0, T ],Rn)) × L2F(Ω; L2(0, T ;Rn)).AcknowledgementsThis work is partially supported by the NSF of China under grants 10831007, 60821091 and 60974035, the NationalBasic Research Program of China (973 Program) under grant 2011CB808002, and by Innovation Foundation of ShandongUniversity. The second author thanks Dr. Qi Zhang for stimulating discussions.References[1] J.-M. Bismut, Analyse convexe et probabilitiés, PhD thesis, Faculté des Sciences de Paris, Paris, France, 1973.[2] B. Bouchard, R. Elie, N. Touzi, Discrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs, Radon Ser. Comp. Appl. Math. 8(2009) 1–34.[3] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.[4] R. Ghanem, P. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, 1991.[5] M. Kleiber, T.D. Hien, The Stochastic Finite Element Method: Basic Perturbation Technique and Computer Implementation, John Wiley, 1992.[6] Q. Lü, X. Zhang, Well-posedness of backward stochastic differential equations with general filtration, preprint, see http://arxiv.org/abs/1010.0026v3.[7] J. Ma, P. Protter, J. San Martin, S. Rorres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab. 12 (2000) 302–316.[8] A. Nouy, Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations, Arch. Comput.Methods Eng. 16 (2009) 251–285.[9] E. Pardoux, S. Peng, Adapted solution of backward stochastic equation, Systems Control Lett. 14 (1990) 55–61.[10] S. Peng, M. Xu, Numerical algorithms for 1-d backward stochastic differential equations: convergence and simulations, ESAIM: M2AN 45 (2011) 335–360.[11] P. Wang, X. Zhang, Numerical analysis on backward stochastic differential equations by a finite transposition method, preprint.[12] J. Zhang, A numerical scheme for BSDEs, Ann. Appl. Probab. 14 (2004) 459–488.

Numerical solutions of backward stochastic differential equations: A finite transposition method

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Numerical Solutions of Backward Stochastic Differential Equations: A Finite Transposition Method

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