Hamilton-Jacobi-Bellman Equation Arising from Optimal Portfolio Selection Problem
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Sep 21, 2024
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Abstract
The Hamilton-Jacobi-Bellman equation arising from the optimal portfolio selection problem is studied by means of the maximal monotone operator method. The existence and uniqueness of a solution to the Cauchy problem for the nonlinear parabolic partial integral differential equation in an abstract setting are investigated by using the Banach fixed-point theorem, the Fourier transform, and the monotone operators technique.
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Sevcovic, D., & Udeani, C.
(2024).
Hamilton-Jacobi-Bellman Equation Arising from Optimal Portfolio Selection Problem.
Proceedings Of The Conference Algoritmy, , 21 - 25.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2140/1023
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References
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[9] Kilianová, S., Ševčovič, D.: Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization. Kybernetika 54 , 1167-1183 (2018).
[10] Kilianová, S., Ševčovič, D.: Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation. Japan Journal of Industrial and Applied Mathematics, (36(2), 497–517 (2019).
[11] Klatte, D.: On the Lipschitz behavior of optimal solutions in parametric problems of quadratic optimization and linear complementarity. Optimization: A Journal of Mathematical Programming and Operations Research, 16(6), 819–831 (1985).
[12] Macová, Z, Ševčovič, D.: Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management. Int. J. Numer. Anal. Model., 7(4), 619–638 (2010).
[13] Showalter, R.: Monotone operators in Banach space and nonlinear partial differential equations. American Mathematical Soc., 49, (2013).
[14] Udeani, C., Ševčovič, D.: Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem. Japan Journal of Industrial and Applied Mathematics, 5, 1–21 (2021).
[2] Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces. Springer Science & Business Media, (2010).
[3] Bertsekas, D. P.: Dynamic programming and stochastic control. (Academic Press, New York (1976).
[4] Ishimura, N., Ševčovič, D.: On traveling wave solutions to a Hamilton-Jacobi-Bellman equation with inequality constraints. Japan J. Ind. Appl. Math., 30(1), 51–67 (2013).
[5] Milgrom, P., Segal, I.: Envelope theorems for arbitrary choice sets. Econometrica, 70(2), 583–601 (2002).
[6] Kilianová, S., Melicherčı́k, I., Ševčovič, D.: Dynamic Accumulation Model for the Second Pillar of the Slovak Pension System, Finance a uver - Czech Journal of Economics and Finance, 56, 506–521 (2006).
[7] Kilianová, S., Ševčovič, D.: A Transformation Method for Solving the Hamilton-Jacobi-Bellman Equation for a Constrained Dynamic Stochastic Optimal Allocation Problem. ANZIAM Journal, 55, 14–38, (2013).
[8] Kilianová, S., Trnovská, M.: Robust Portfolio Optimization via solution to the Hamilton-Jacobi-Bellman Equation. Int. Journal of Computer Mathematics, (93), 725–734 (2016).
[9] Kilianová, S., Ševčovič, D.: Expected Utility Maximization and Conditional Value-at-Risk Deviation-based Sharpe Ratio in Dynamic Stochastic Portfolio Optimization. Kybernetika 54 , 1167-1183 (2018).
[10] Kilianová, S., Ševčovič, D.: Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation. Japan Journal of Industrial and Applied Mathematics, (36(2), 497–517 (2019).
[11] Klatte, D.: On the Lipschitz behavior of optimal solutions in parametric problems of quadratic optimization and linear complementarity. Optimization: A Journal of Mathematical Programming and Operations Research, 16(6), 819–831 (1985).
[12] Macová, Z, Ševčovič, D.: Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management. Int. J. Numer. Anal. Model., 7(4), 619–638 (2010).
[13] Showalter, R.: Monotone operators in Banach space and nonlinear partial differential equations. American Mathematical Soc., 49, (2013).
[14] Udeani, C., Ševčovič, D.: Application of maximal monotone operator method for solving Hamilton–Jacobi–Bellman equation arising from optimal portfolio selection problem. Japan Journal of Industrial and Applied Mathematics, 5, 1–21 (2021).