On the existence or non-existence of solutions for certain backward stochastic differential equations
Let \(W\) be a standard \(d\)-dimensional Wiener process, defined on a stochastic basis \((\Omega,\mathcal F, (\mathcal F_{t}),P)\), where \((\mathcal F_{t})\) is the Brownian filtration. Denote by \(\mathcal P\) the predictable \(\sigma\)-algebra, let \(f: [0,1]\times\Omega\times\mathbb R \times\mathbb R^{d}\to\mathbb R\) be a \(\mathcal P\otimes\mathcal B(\mathbb R) \otimes\mathcal B(\mathbb R^{d})\)-measurable mapping and \(\xi\) a bounded \(\mathcal F_1\)-measurable random variable. Backward stochastic differential equations of the form \[ Y_{t} = \xi + \int^{1}_{t} f(s,\omega,Y_{s},Z_{s}) ds - \int^1_{t} Z_{s} dW_{s}, \quad 0\leq t\leq 1, \tag{1} \] are studied, where \(Y\) is to be an adapted process and \(Z\) a predictable one. It is supposed that \(f(s,\omega,\cdot,\cdot)\) is a continuous function for every \((s,\omega)\) and there exist continuous nonnegative functions \(G\), \(F\), \(R\) such that \(R(0)=0\), \(R\) is subquadratic, \(G\) is increasing on \(\mathbb R_{+}\) and decreasing on \(\mathbb R_{-}\), and \(|f(s,\omega,y,z)|\leq G(y) + F(y)R(z)\). Then it is shown that there exists a (maximal) local solution \((Y,Z)\) to (1), defined on \(\mathopen]t^*,1\mathclose]\cap [0,1]\), \(t^\ast\) being given by an explicit formula involving only the function \(G\) and constants \(b\leq 0\leq a\) such that \(b\leq\xi\leq a\) \(P\)-almost surely. Moreover, sufficient conditions for non-explosion are given in the particular case \(|f(s,\omega,y,z)|\leq G(y) + A|z|^2\).
backward stochastic differential equations
scientific article; zbMATH DE number 1786697