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Stochastic Hamiltonians associated with stochastic differential equations and non-smooth final value. (English) Zbl 1150.60398
Summary: For a given Lipschitz continuous function \(\varphi(x):\mathbb R^d\to\mathbb R\) admitting a weak gradient \(\partial_x\varphi(x):\mathbb R^d\to\mathbb R^d\) we associate the random variable \(\varphi(x(T))\), where \(x(t),t\in[0,T]\) is the solution of a stochastic differential system with Lipschitz continuous coefficients and first order continuously differentiable difussion coefficients. It is proved that the random variable \(\varphi(x(T))\) can be represented as a final value \(S(T,x(T))=\varphi(x(T))\) using a continuous function \(S(t,x):[0,T]\times\mathbb R^d\to\mathbb R\) which admits a weak gradient \(\partial_x S(t,x):[0,T]\times\mathbb R^d\to\mathbb R^d\) and \(S(t,x(t)),\;t\in[0,T]\), fulfills a first order stochastic differential equation (the stochastic differential \(d_t[S(t,x(t))]\) equals a stochastic hamiltonian). It can be meaningful for getting feedback optimal control associated with stochastic control problems and for describing an admissible feedback strategy involved in a financial market.
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93C42 Fuzzy control/observation systems