A forward scheme for backward SDEs

Abstract. We design a numerical scheme for solving the Multi step-forward Dynamic Programming (MDP) equation arising from the time-discretization of backward stochastic differential equations. The generator is assumed to be locally Lipschitz, which includes some cases of quadratic drivers. When the large sequence of conditional expectations is computed using empirical leastsquares regressions, under general conditions we establish an upper bound error as the average, rather than the sum, of local regression errors only, suggesting that our error estimation is tight. Despite the nested regression problems, the interdependency errors are justified to be at most of the order of the statistical regression errors (up to logarithmic factor). Finally, we optimize the algorithm parameters, depending on the dimension and on the smoothness of value functions, in the limit as the time mesh size goes to zero and compute the complexity needed to achieve a given accuracy. Keywords. Backward stochastic differential equations, dynamic programming equation, empirical regressions, non-asymptotic error estimates. IntroductionFramework. Let T > 0 be a fixed terminal time and W be a q-dimensional (q ≥ 1) Brownian motion defined on a filtered probability space (Ω, F, P), where the filtration (F t ) 0≤t≤T satisfies the usual hypotheses; the filtration may be larger than that generated by W . Let π := {0 = t 0 < . . . < t N = T } be a time-grid for the interval [0, T ], whose (i + 1)-th time-step t i+1 − t i is denoted by ∆ i , whose mesh size is defined by |π| := max 0≤i

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“…In fact, Equation (1.1) is inspired by the algorithm of [3]. Unlike that work, Picard iterations are not used in our scheme.…”

Section: Introductionmentioning

confidence: 99%

Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

2015

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“…Similarly, we choose ϕ 0 = sin(x) as the initial guess of u(t, x). Then, it is easy to gain the solution of (3.24): 27) step by step, starting from m = 1, where the integration coefficient B m (x) is determined by (3.25). This is rather efficient computationally by means of the computer algebra software such as Mathematica.…”

Section: An Example Of 2fbsdesmentioning

confidence: 99%

“…In 2008, Delarue and Menozzi [21] improved this scheme by introducing a interpolation procedure. Apart from solving the parabolic partial differential equations, some numerical schemes are proposed to directly solve the BSDEs and FBSEDs [22][23][24][25][26][27][28][29]. In 1999, Peng [30] proposed a linear approximation algorithm to solve the Chow's lagrangean one-dimensional BSDEs [31].…”

Section: Introductionmentioning

confidence: 99%

On the homotopy analysis method for backward/forward-backward stochastic differential equations

Zhong

Liao

2017

Numer Algor

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In this paper, an analytic approximation method for highly nonlinear equations, namely the homotopy analysis method (HAM), is employed to solve some backward stochastic differential equations (BSDEs) and forwardbackward stochastic differential equations (FBSDEs), including one with high dimensionality (up to 12 dimensions). By means of the HAM, convergent series solutions can be quickly obtained with high accuracy for a FBSDE in a 6 dimensional case, within less than 1% CPU time used by a currently reported numerical method for the same case [34]. Especially, as dimensionality enlarges, the increase of computational complexity for the HAM is not as dramatic as this numerical method. All of these demonstrate the validity and high efficiency of the HAM for the backward/forward-backward stochastic differential equations in science, engineering and finance.

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“…Since it is typically difficult to obtain analytical solutions, numerical solutions are highly desired in practical applications. Numerical methods for FBSDEs without jumps have been well studied in the literature [3,7,8,10,12,15,[32][33][34]36], nevertheless, there are very few numerical schemes developed for FBSDEs with jumps, and most of those schemes only focused on temporal discretization. For instance, a Picard's iterative method was provided in [18], and numerical schemes of backward SDE were studied in [4,5].…”

Section: Introductionmentioning

confidence: 99%

Second-order schemes for solving decoupled forward backward stochastic differential equations

Zhang

Yang

2014

Sci. China Math.

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We propose new numerical schemes for decoupled forward-backward stochastic differential equations (FBSDEs) with jumps, where the stochastic dynamics are driven by a ddimensional Brownian motion and an independent compensated Poisson random measure. A semi-discrete scheme is developed for discrete time approximation, which is constituted by a classic scheme for the forward SDE [17,25] and a novel scheme for the backward SDE. Under some reasonable regularity conditions, we prove that the semi-discrete scheme can achieve second-order convergence in approximating the FBSDEs of interest; and such convergence rate does not require jump-adapted temporal discretization. Next, to add in spatial discretization, a fully discrete scheme is developed by designing accurate quadrature rules for estimating the involved conditional mathematical expectations. Several numerical examples are given to illustrate the effectiveness and the high accuracy of the proposed schemes.Mathematics subject classification: 60H35, 60H10, 65C20, 65C30

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A forward scheme for backward SDEs

Cited by 166 publications

References 16 publications

Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

On the homotopy analysis method for backward/forward-backward stochastic differential equations

Second-order schemes for solving decoupled forward backward stochastic differential equations

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