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Generalized solutions to nonlinear stochastic differential equations with vector--valued impulsive controls

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  • We develop a notion of generalized solution to a stochastic differential equation depending in a nonlinear way on a vector--valued stochastic control process $\{U_t\},$ merely of bounded variation, and on its derivative. Our results rely on the concept of Lipschitz continuous graph completion of $\{U_t\}$ and the generalized solution turns out to coincide a.e. with the limit of classical solutions to (1). In the linear case our notion of solution is equivalent to the usual one in distributional sense. We prove that the generalized solution does not depend on the particular graph-completion of the control process $\{U_t\}$ both for vector-valued controls under a suitable commutativity condition and for scalar controls.
    Mathematics Subject Classification: Primary: 49E05, 49N25; Secondary: 93E03.


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