A Convolution Method for Numerical Solution of Backward Stochastic Differential Equations

Abstract

We propose a new method for the numerical solution of backward stochastic differential equations (BSDEs) which finds its roots in Fourier analysis. The method consists of an Euler time discretization of the BSDE with certain conditional expectations expressed in terms of Fourier transforms and computed using the fast Fourier transform (FFT). The problem of error control is addressed and a local error analysis is provided. We consider the extension of the method to forward-backward stochastic differential equations (FBSDEs) and reflected FBSDEs. Numerical examples are considered from finance demonstrating the performance of the method. comments of the meeting participants are gratefully acknowledged. 1 C. B. HYNDMAN AND P. OYONO NGOU Convolution method for BSDEs has a unique solution u ∈ C 1,2 then the solution (Y, Z) for the one-dimensional BSDE (1.1) with terminal condition ξ = g(W T ) admits the representation). (1.4) Conversely, the solution of the PDE (1.2) can be interpreted in terms of the solution of the BSDE (1.1). General formulations of the nonlinear Feynman-Kac formula for FBSDEs, quasilinear parabolic PDEs, and viscosity solutions have been studied extensively. Deriving an explicit solution to a nontrivial (F)BSDE is possible only in very few situations, such as [40], [23] and [37]. Thus, numerical methods for BSDEs have been studied extensively. Numerical methods for (F)BSDEs can be classified into three main groups: PDE based methods, spatial discretization based methods, and Monte-Carlo based methods. PDE based methods, which started with the finite difference approach of [16], consider a numerical resolution of the nonlinear parabolic PDE related to the (F)BSDE. The two other methods rely on a time discretization of the (F)BSDE. Spatial discretization based methods (see [12], [2], [15], [13], [38] or [35] among others) use a deterministic space grid. On the other hand, the space discretization is random in Monte-Carlo based methods (for instance, [9], [22], and [3]).In this paper, we propose an alternative spatial discretization method for BSDEs and illustrate its implementation in the one-dimensional case. To the best of our knowledge, the most efficient approach, in terms of speed and accuracy, in this simple case is the binomial method of [35] which has connections with the theoretical work of [10] and [29]. However, our method avoids a notable drawback of the binomial method: the contraction of the space grid leading to the approximation of the Wiener process by means of scaled random walks. Indeed, we use a fixed equidistant space grid, thus allowing an exact simulation of the Wiener process at time nodes. The FFT algorithm, which plays a key role in our method, helps in producing an efficient algorithm. As in [11] and [28] in the context of option pricing under Lévy processes, we employ the FFT algorithm to compute quadratures. The presence of dynamic programming through the Euler scheme is a major similarity between our method and [28]. The method presented in [38] is somewhat similar in t...

Numerical Fourier Method and Second-Order Taylor Scheme for Backward SDEs in Finance

“…The general multiple Itô integral is defined recursively by (see [17], Chapter 5.2) 14) with C > 0 a constant, which does not depend on t, and P (.) any 2(γ w + 1) times continuously differentiable function of polynomial growth.…”

Efficient Numerical Fourier Methods for Coupled Forward-Backward SDEs

“…Accurate results are quickly obtained for small values of N. Furthermore, we observe that the error in the approximations is dominated by the number of time points M, as the error stops decreasing at some point for increasing values of N. Therefore, we are more concerned with the behaviour of the error as the number of time points varies. For all following numerical tests, we will take N = 2 9 to ensure sufficient accuracy.…”

Numerical convergence of backward stochastic differential equation with non-Lipschitz coefficients

“…Ma, Protter and Martin [7] presented the following Lemma 3.1. Suppose thatỸ (n) be the solution of equation (4). Then the jumps ofỸ (n) converge uniformly to zero.…”

“…Hyndman and Ngou [4] proposed a new method for the numerical solution of backward stochastic differential equations which finds its roots in Fourier analysis. They presented the convolution method for the numerical solutions of the backward stochastic differential equations, for implementing the convolution method they proposed the discretization of intermediate quadratures and their association to the discrete Fourier transform which can be efficiently calculated using the fast Fourier transform.…”