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Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising from optimal portfolio selection problem

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  • Daniel Sevcovic
  • Cyril Izuchukwu Udeani

Abstract

In this paper, we investigate a fully nonlinear evolutionary Hamilton-Jacobi-Bellman (HJB) parabolic equation utilizing the monotone operator technique. We consider the HJB equation arising from portfolio optimization selection, where the goal is to maximize the conditional expected value of the terminal utility of the portfolio. The fully nonlinear HJB equation is transformed into a quasilinear parabolic equation using the so-called Riccati transformation method. The transformed parabolic equation can be viewed as the porous media type of equation with source term. Under some assumptions, we obtain that the diffusion function to the quasilinear parabolic equation is globally Lipschitz continuous, which is a crucial requirement for solving the Cauchy problem. We employ Banach's fixed point theorem to obtain the existence and uniqueness of a solution to the general form of the transformed parabolic equation in a suitable Sobolev space in an abstract setting. Some financial applications of the proposed result are presented in one-dimensional space.

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  • Daniel Sevcovic & Cyril Izuchukwu Udeani, 2021. "Application of maximal monotone operator method for solving Hamilton-Jacobi-Bellman equation arising from optimal portfolio selection problem," Papers 2104.06115, arXiv.org.
    • Handle: RePEc:arx:papers:2104.06115
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    References listed on IDEAS

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    1. Sona Kilianova & Daniel Sevcovic, 2013. "Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem," Papers 1307.3672, arXiv.org, revised Jul 2013.
    2. Salvatore Federico & Paul Gassiat & Fausto Gozzi, 2015. "Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation," Finance and Stochastics, Springer, vol. 19(2), pages 415-448, April.
    3. Sona Kilianova & Daniel Sevcovic, 2019. "Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation," Papers 1903.10065, arXiv.org.
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    Cited by:

    1. Daniel Sevcovic & Cyril Izuchukwu Udeani, 2023. "Hamilton-Jacobi-Bellman Equation Arising from Optimal Portfolio Selection Problem," Papers 2308.02627, arXiv.org.
    2. Jose Cruz & Maria Grossinho & Daniel Sevcovic & Cyril Izuchukwu Udeani, 2022. "Linear and Nonlinear Partial Integro-Differential Equations arising from Finance," Papers 2207.11568, arXiv.org.

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