A numerical scheme for BSDEs. (English) Zbl 1056.60067
The author considers the decoupled system of forward-backward stochastic differential equations
\[
\begin{aligned} X_t&=x + \int_0^t b(s,X_s) \,ds + \int_0^t \sigma(s,X_s)\,dW_s,\\ Y_t&= \Phi(X)+\int_t^Tf(s,X_s,Y_s,Z_s)\,ds-\int_t^TZ_s\,dW_s, \end{aligned} \tag{1}
\]
where \(b,\,\sigma\) and \(f\) are deterministic functions and \(W\) is a standard Brownian motion. The deterministic functional \(\Phi\) is assumed to be a so-called \(L^\infty\)-Lipschitz or a \(L^1\)-Lipschitz functional or takes the form \(g(X_T)\). Based on results of J. Ma and J. Zhang [Probab. Theory Relat. Fields 122, No. 2, 163–190 (2002; Zbl 1014.60060)] he proves \(L^2\)-regularity of the martingale integrand \(Z\). A numerical scheme is proposed, consisting of the Euler method for the forward diffusion \(X\) and an approximation via step processes for the pair \((Y,Z)\). Convergence in the mean-square sense of the method is proved and convergence rates are obtained. The latter depend in particular on the above stated smoothness properties of \(\Phi\).
Reviewer: Evelyn Buckwar (Berlin)
MSC:
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H20 | Stochastic integral equations |
| 34F05 | Ordinary differential equations and systems with randomness |
Keywords:
backward stochastic differential equations; Euler method; convergence in mean-square; \(L^2\) regularityCitations:
Zbl 1014.60060References:
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