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Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data

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Abstract

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+|x|p) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superlinear convex gradient nonlinearity. If the corresponding control problem has a bounded diffusion with respect to the control, then our results apply to a larger class of solutions, namely those growing like O(1+|x|p) at infinity. This latter case encompasses some equations related to backward stochastic differential equations.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura e Applicata, Via Trieste, 63, 35121, Padova, Italy

    Francesca Da Lio

  2. IRMAR, INSA de Rennes, 35708, Rennes, France

    Olivier Ley

Authors
  1. Francesca Da Lio
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  2. Olivier Ley
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Corresponding author

Correspondence to Francesca Da Lio.

Additional information

Part of this work was done while the second author was a visitor at the FIM at the ETH in Zürich in January 2007. He would like to thank the Department of Mathematics for his support. We thank Guy Barles for useful comments on the first version of this paper.

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Da Lio, F., Ley, O. Convex Hamilton-Jacobi Equations Under Superlinear Growth Conditions on Data. Appl Math Optim 63, 309–339 (2011). https://doi.org/10.1007/s00245-010-9122-9

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