Document Zbl 1017.60067

Representation theorems for backward stochastic differential equations. (English) Zbl 1017.60067

Ann. Appl. Probab. 12, No. 4, 1390-1418 (2002).

Let \(W\) be a \(d\)-dimensional Brownian motion. The authors investigate the backward stochastic differential equation (BSDE) \[ dY_t= f(t, X_t, Y_t, Z_t)dt+ Z_t dW_t,\quad t\in [0,T],\;Y_t= \xi, \] driven by a forward equation \[ dX_t= b(t, X_t)dt+ \sigma(X_t) dW_t,\quad t\in [0,T],\;X_T= x, \] with strictly elliptic diffusion coefficient: \(\sigma\sigma^*\geq \varepsilon I\). BSDE’s of this type with terminal condition of the form \(\xi= g(X_T)\) have been studied extensively in the past decade, we refer, e.g., to E. Pardoux and S. Peng [in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 200-217 (1992; Zbl 0766.60079)] and the survey book edited by N. El Karoui and L. Mazliak [“Backward stochastic differential equations” (Harlow, 1997)]. The goal of the authors is twice: Assuming that the terminal condition depends on the path of \(X\) at a finite number of time points, \(\xi= g(X_{t_1},\dots, X_{t_n})\), they establish an explicit representation formula for the solution \((Y,Z)\) of the BSDE. These formulas for \(Y\) and \(Z\) can be thought as a new type of Feynman-Kac formula; the significance of that for \(Z\) lies in the fact that it does not depend on the derivatives of the coefficients of the BSDE. This allows to get it with only Lipschitz assumptions on the coefficients, and enables the authors to prove the pathwise regularity of the process \(Z\). The main device in the authors’ approach is an integration by parts formula for the Skorokhod integral.

Reviewer: Rainer Buckdahn (Brest)

Cited in 6 Reviews

Cited in 66 Documents

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05	Ordinary differential equations and systems with randomness
91B24	Microeconomic theory (price theory and economic markets)

Keywords:

backward stochastic differential equations; Malliavin calculus; viscosity solution; path regularity

Citations:

Zbl 0766.60079; Zbl 0866.00052

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References:

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