An efficient third-order scheme for BSDEs based on nonequidistant difference scheme
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An efficient third-order scheme for BSDEs based on nonequidistant difference scheme Chol-Kyu Pak1
· Mun-Chol Kim1 · Chang-Ho Rim1
Received: 7 August 2018 / Accepted: 9 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we propose an efficient third-order numerical scheme for backward stochastic differential equations(BSDEs). We use 3-point Gauss-Hermite quadrature rule for approximation of the conditional expectation and avoid spatial interpolation by setting up a fully nested spatial grid and using the approximation of derivatives based on nonequidistant sample points. As a result, the overall computational complexity is reduced significantly. Several examples show that the proposed scheme is of third order and very efficient. Keywords Backward stochastic differential equations · Gauss-Hermite quadrature · Third-order scheme Mathematics Subject Classification (2010) 60H35 · 65C20 · 60H10
1 Introduction Let (, F , P) be a probability space, T > 0 a finite time and {Ft }0≤t≤T a filtration satisfying the usual conditions. Let (, F , P, {Ft }0≤t≤T ) be a complete filtered probability space on which a standard d-dimensional Brownian motion Wt = (Wt1 , Wt2 , · · · , Wtd )T is defined and F0 contains all the P-null sets of F . The general form of backward stochastic differential equation (BSDE) is T T yt = ξ + f (s, ys , zs )ds − zs dW s, t ∈ [0, T ] (1.1) t
t
where the generator f : [0, T ] × Rm × Rm×d → Rm is {Ft }-adapted for each (y, z) and the terminal variable ξ is a FT -measurable and square integrable random variable. A process (yt , zt ) : [0, T ] × → Rm × Rm×d is called an L2 -solution of Chol-Kyu Pak
[email protected] 1
Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea
Numerical Algorithms
the BSDE (1.1) if it satisfies the equation (1.1) while it is {Ft }-adapted and square integrable . In 1990, Pardoux and Peng first proved in [7] the existence and uniqueness of the solution of general nonlinear BSDEs and afterwards there has been very active research in this field with many applications [3]. In this paper, we assume that the terminal condition is a function of WT , i.e., ξ = ϕ(WT ) and the BSDE (1.1) has a unique solution (yt , zt ). It was shown in [8] that the solution (yt , zt ) of (1.1) can be represented as yt = u(t, Wt ), zt = ∇x u(t, Wt ), ∀t ∈ [0, T )
(1.2)
where u(t, x) is the solution of the parabolic partial differential equation ∂u 1 ∂ 2 u + f (t, u, ∇x u) = 0 + ∂t 2 ∂xi2 d
(1.3)
i=1
with the terminal condition u(T , x) = ϕ(x), and ∇x u is the gradient of u with respect to the spatial variable x. The smoothness of u depends on f and ϕ. Although BSDEs and their extensions such as FBSDEs have very important applications in many fields such as mathematical finance and stochastic control, it is well known that it is difficult to obtain the analytical solutions except some special cases and there have been many works on numerical methods [1, 2, 4–6, 9–11, 13, 14, 16]. Amo
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