Abstract
For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.
Highlights
In 1973, Bismut [1] introduced the linear BSDEs
There are a large number of documents about the constant variable step size schemes. is implies that the theory of implementable numerical methods of BSDEs is booming. e variable step size numerical methods play a vital role in the field of numerical methods of stochastic differential equations while they are not seen in the field of numerical theory of BSDEs. us, for this motivation, this paper is to provide novel high-order nonlinear discretization schemes called variable step size Adams scheme (14) by utilizing Ito–Taylor expansion
We focus on the least squares Monte Carlo (LSMC) method
Summary
In 1973, Bismut [1] introduced the linear BSDEs. Until 1990, the well-posedness result of nonlinear BSDEs was rigorously proved by Pardoux and Peng [2,3,4]. Us, the main purpose of this paper is to design a new numerical scheme to solve the following BSDE: T. t t where T > 0 denotes a fixed terminal time and W is a d-dimensional Brownian motion defined on a filtered complete probability space (Ω, F, (Ft)0≤t≤T, P); Φ(WT): Rd ⟶ Rm is a given terminal condition of BSDE, and f(t, y, z): [0, T] × Rm × Rm×d ⟶ Rm is the generator function. (i) We derive the variable step size Adams scheme for BSDEs by means of Ito–Taylor expansion This scheme is a novel high-order nonlinear time-discretization scheme.
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