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Large time behavior of solutions to semi-linear equations with quadratic growth in the gradient

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  • Robertson, Scott
  • Xing, Hao

Abstract

This paper studies the large time behavior of solutions to semi-linear Cauchy problems with quadratic nonlinearity in gradients. The Cauchy problem considered has a general state space and may degenerate on the boundary of the state space. Two types of large time behavior are obtained: i) pointwise convergence of the solution and its gradient; ii) convergence of solutions to associated backward stochastic di�erential equations. When the state space is Rd or the space of positive de�nite matrices, both types of convergence are obtained under growth conditions on coe�cients. These large time convergence results have direct applications in risk sensitive control and long term portfolio choice problems.

Suggested Citation

  • Robertson, Scott & Xing, Hao, 2015. "Large time behavior of solutions to semi-linear equations with quadratic growth in the gradient," LSE Research Online Documents on Economics 60578, London School of Economics and Political Science, LSE Library.
    • Handle: RePEc:ehl:lserod:60578
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    File URL: http://eprints.lse.ac.uk/60578/
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    References listed on IDEAS

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    Cited by:

    1. Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2016. "Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1932-1973.
    2. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    3. Hyungbin Park & Heejun Yeo, 2022. "Dynamic and static fund separations and their stability for long-term optimal investments," Papers 2212.00391, arXiv.org, revised Mar 2023.
    4. Tim Leung & Hyungbin Park & Heejun Yeo, 2023. "Robust Long-Term Growth Rate of Expected Utility for Leveraged ETFs," Papers 2310.02084, arXiv.org.

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    More about this item

    Keywords

    semilinear equation; quadratic growth gradient; large time behaviour; ergodic equation;
    All these keywords.

    JEL classification:

    • F3 - International Economics - - International Finance
    • G3 - Financial Economics - - Corporate Finance and Governance

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