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Stochastic targets with mixed diffusion processes and viscosity solutions

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  • Bouchard, Bruno

Abstract

Let Zt,z[nu] be a -valued mixed diffusion process controlled by [nu] with initial condition Zt,z[nu](t)=z. In this paper, we characterize the set of initial conditions such that Zt,z[nu] 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.

Suggested Citation

  • Bouchard, Bruno, 2002. "Stochastic targets with mixed diffusion processes and viscosity solutions," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 273-302, October.
    • Handle: RePEc:eee:spapps:v:101:y:2002:i:2:p:273-302
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    References listed on IDEAS

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    1. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
    2. repec:dau:papers:123456789/1533 is not listed on IDEAS
    3. Touzi, Nizar, 2000. "Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 305-328, August.
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    Cited by:

    1. Marc Abeille & Bruno Bouchard & Lorenzo Croissant, 2023. "Diffusive Limit Approximation of Pure-Jump Optimal Stochastic Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 147-176, January.
    2. Erhan Bayraktar & Jiaqi Li, 2016. "Stochastic Perron for Stochastic Target Problems," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 1026-1054, September.

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