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Stochastic targets with mixed diffusion processes and viscosity solutions. (English) Zbl 1075.60557

Summary: Let \(Z^v_{t,z}\) be a \(\mathbb R^{d+1}\)-valued mixed diffusion process controlled by \(v\) with initial condition \(Z^v_{t,z} (t) = z\). In this paper, we characterize the set of initial conditions such that \(Z^v_{t,z}\) can be driven above a given stochastic target at time \(T\) by proving that the corresponding value function is a discontinuous viscosity solution of a variational partial differential equation. As applications of our main result, we study two examples: a problem of optimal insurance under self-protection and a problem of option hedging under jumping stochastic volatility where the underlying stock pays a random dividend at a fixed date.

MSC:

60J60 Diffusion processes
60J75 Jump processes (MSC2010)
35R60 PDEs with randomness, stochastic partial differential equations
49J20 Existence theories for optimal control problems involving partial differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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References:

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