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Adapted solution of a backward stochastic differential equation. (English) Zbl 0692.93064
Summary: Let \(\{W_ t\); \(t\in [0,1]\}\) be a standard \(k\)-dimensional Wiener process defined on a probability space (\(\Omega\),\({\mathcal F},P)\), and let \(\{\) \({\mathcal F}_ t\}\) denote its natural filtration. Given a \({\mathcal F}_ 1\) measurable d-dimensional random vector X, we look for an adapted pair of processes \(\{\) x(t), y(t); \(t\in [0,1]\}\) with values in \({\mathbb{R}}^ d\) and \({\mathbb{R}}^{d\times k}\) respectively, which solves an equation of the form: \[ x(t)+\int^{1}_{t}f(s,x(s),y(s))ds+\int^{1}_{t}[g(s,x(s))+y(s)]dW_ s=X. \] A linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process. We also generalize our results to the following equation: \[ x(t)+\int^{1}_{t}f(s,x(s),y(s))ds+\int^{1}_{t}g(s,x(s),y(s))dW_ s=X \] under rather restrictive assumptions on g.

MSC:
93E03 Stochastic systems in control theory (general)
93E20 Optimal stochastic control
34F05 Ordinary differential equations and systems with randomness
49K45 Optimality conditions for problems involving randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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References:
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