Multidimensional backward stochastic differential equations with uniformly continuous coeffi\-cients
Given an \(m\)-dimensional Brownian motion \(W=(W_t)_{t\in [0,1]}\) and a square-integrable \(d\)-dimen\-sional random variable \(\xi\), measurable with respect to the \(\sigma\)-field generated by \(W\), the author studies the existence and the uniqueness of the solution for the backward stochastic differential equation (for short: BSDE) \(dY_t=-f(t,Y_t,Z_t)dt+Z_tdW_t\), \(t\in[0,1]\), \(Y_1=\xi\). Nonlinear BSDEs of this general type were introduced by \textit{E. Pardoux} and \textit{S. G. Peng} [Syst. Control Lett. 14, No. 1, 55--61 (1990; Zbl 0692.93064)]. Since then, motivated by the multitude of applications, many authors have worked on this subject and given improvements of the initial Lipschitz assumptions made by Pardoux and Peng on the driving coefficient \(f\). While in the one-dimensional case (\(d=1\)) the comparison theorem for BSDEs can be used to show that one has the existence of a solution under only continuity of the driver \(f\) [\textit{J.-P. Lepeltier} and \textit{J. San Martín}, Stochastics Stochastics Rep. 63, No. 3/4, 227--240 (1998; Zbl 0910.60046)], one does not have such a comparison result in higher dimensions. Among the recent works generalizing the assumptions on \(f\) in higher dimensions, one should cite that of \textit{S. Hamadène, J.-P. Lepeltier} and \textit{S. Peng} [in: Backward stochastic differential equations. Pitman Res. Notes Math. Ser. 364, 111--128 (1997; Zbl 0892.60062)] where the randomness stems from a driving diffusion process, and also the more recent paper by \textit{K. Bahlali, E. H. Essaky, M. Hassani} and \textit{E. Pardoux} [C. R., Math., Acad. Sci. Paris 335, No. 9, 757--762 (2002; Zbl 1017.60068)], who showed the existence, uniqueness and stability of the solutions for a continuous, locally monotone coefficient \(f\) that must not be locally Lipschitz in the variable \(y\) or in the variable \(z\). The author of the present paper, submitted in 2000, supposes that \((y,z)\rightarrow f(t,\omega,y,z)\) is uniformly continuous and the continuity modulus \(\Phi\) w.r.t the \(y\)-variable shall be such that \(\int_{\gamma-}\Phi(u)\,du=+\infty\). This assumption is comparable with that made by Bahlali, Essaky, Hassani and Pardoux on the \(y\)- and \(z\)-variables of \(f\). The price the author of the present paper has to pay for the absence of such an assumption w.r.t \(z\) is the hypothesis that the \(i\)th component of \(f\) does only depend on the \(i\)th row of \(z\).
backward stochastic differential equation
scientific article; zbMATH DE number 2075980