A fitted finite volume method for stochastic optimal control problems in finance

Christelle Dleuna Nyoumbi; Antoine Tambue; Christelle Dleuna Nyoumbi; Antoine Tambue

doi:10.3934/math.2021186

AIMS Mathematics

2021, Volume 6, Issue 4: 3053-3079. doi: 10.3934/math.2021186

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Research article

A fitted finite volume method for stochastic optimal control problems in finance

Christelle Dleuna Nyoumbi ¹,
Antoine Tambue ^{2,3
,
,}

1.
Institut de Mathématiques et de Sciences Physiques de l'Université d'Abomey-Calavi, BP 613, Porto-Novo, Benin
2.
Department of Computer science, Electrical engineering and Mathematical sciences, Western Norway University of Applied Sciences, Inndalsveien 28, 5063 Bergen
3.
Center for Research in Computational and Applied Mechanics (CERECAM), Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa

Received: 13 October 2020 Accepted: 28 December 2020 Published: 12 January 2021
MSC : 65M75

In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems in one and two dimensional domain. The computational challenge is due to the nature of the HJB equation, which may be a second-order degenerate partial differential equation coupled with optimization. For such problems, standard scheme such as finite difference losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. In the work, we discretize the HJB equation using the fitted finite volume method, which has for main feature to tackle the degeneracy of the equation. The time discretisation is performed using the Implicit Euler method, which is unconditionally stable. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method.
- stochastic optimal control,
- dynamic programming,
- HJB equations,
- finite volume method,
- finite difference method,
- degenerate parabolic operator,
- proper operator,
- viscosity solutions
Citation: Christelle Dleuna Nyoumbi, Antoine Tambue. A fitted finite volume method for stochastic optimal control problems in finance[J]. AIMS Mathematics, 2021, 6(4): 3053-3079. doi: 10.3934/math.2021186

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Abstract

In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems in one and two dimensional domain. The computational challenge is due to the nature of the HJB equation, which may be a second-order degenerate partial differential equation coupled with optimization. For such problems, standard scheme such as finite difference losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. In the work, we discretize the HJB equation using the fitted finite volume method, which has for main feature to tackle the degeneracy of the equation. The time discretisation is performed using the Implicit Euler method, which is unconditionally stable. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method.

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