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Optimal Controls of Stochastic Differential Equations with Jumps and Random Coefficients: Stochastic Hamilton–Jacobi–Bellman Equations with Jumps

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Abstract

We study the stochastic Hamilton–Jacobi–Bellman (HJB) equation with jump, which arises from a non-Markovian optimal control problem with a recursive utility cost functional. The solution to the equation is a predictable triplet of random fields. We show that the value function of the control problem, under some regularity assumptions, is the solution to the stochastic HJB equation; and a classical solution to this equation is the value function and characterizes the optimal control. With some additional assumptions on the coefficients, an existence and uniqueness result in the sense of Sobolev space is shown by recasting the stochastic HJB equation as a backward stochastic evolution equation in Hilbert spaces with the Brownian motion and Poisson jump.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Huzhou University, Zhejiang, 313000, China

    Qingxin Meng

  2. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Yuchao Dong

  3. School of Risk and Actuarial Studies and CEPAR, University of New South Wales Sydney, Sydney, NSW, 2052, Australia

    Yang Shen

  4. Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Shanjian Tang

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  1. Qingxin Meng
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  2. Yuchao Dong
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  3. Yang Shen
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  4. Shanjian Tang
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Correspondence to Yuchao Dong.

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Q. Meng was supported by the National Natural Science Foundation of China (No. 11871121) and the Natural Science Foundation of Zhejiang Province (No. LY21A010001). Y. Shen was supported by the Australian Research Council (No. DE200101266). S. Tang was supported by the National Science Foundation of China (Grant No. 11631004) and the National Key R &D Program of China (Grant No. 2018YFA0703903)

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Meng, Q., Dong, Y., Shen, Y. et al. Optimal Controls of Stochastic Differential Equations with Jumps and Random Coefficients: Stochastic Hamilton–Jacobi–Bellman Equations with Jumps. Appl Math Optim 87, 3 (2023). https://doi.org/10.1007/s00245-022-09914-8

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