KLI

Indefinite Linear-Quadratic Stochastic Control Problem for Jump-Diffusion Models with Random Coefficients: A Completion of Squares Approach

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Abstract
In this paper, we study the indefinite linear-quadratic (LQ) stochastic optimal control problem for stochastic differential equations (SDEs) with jump diffusions and random coefficients driven by both the Brownian motion and the (compensated) Poisson process. In our problem setup, the coefficients in the SDE and the objective functional are allowed to be random, and the jump-diffusion part of the SDE depends on the state and control variables. Moreover, the cost parameters in the objective functional need not be (positive) definite matrices. Although the solution to this problem can also be obtained through the stochastic maximum principle or the dynamic programming principle, our approach is simple and direct. In particular, by using the Ito-Wentzell’s formula, together with the integro-type stochastic Riccati differential equation (ISRDE) and the backward SDE (BSDE) with jump diffusions, we obtain the equivalent objective functional that is quadratic in control u under the positive definiteness condition, where the approach is known as the completion of squares method. Then the explicit optimal solution, which is linear in state characterized by the ISRDE and the BSDE jump diffusions, and the associated optimal cost are derived by eliminating the quadratic term of u in the equivalent objective functional. We also verify the optimality of the proposed solution via the verification theorem, which requires solving the stochastic HJB equation, a class of stochastic partial differential equations with jump diffusions.
Author(s)
문준정진호
Issued Date
2021
Type
Article
Keyword
stochastic systemsjump diffusionscompletion of squares methodstochastic HJB equation
DOI
10.3390/math9222918
URI
https://oak.ulsan.ac.kr/handle/2021.oak/9603
https://ulsan-primo.hosted.exlibrisgroup.com/primo-explore/fulldisplay?docid=TN_cdi_doaj_primary_oai_doaj_org_article_6c80e97f68b94e3b936ccc2741ba7d87&context=PC&vid=ULSAN&lang=ko_KR&search_scope=default_scope&adaptor=primo_central_multiple_fe&tab=default_tab&query=any,contains,Indefinite%20Linear-Quadratic%20Stochastic%20Control%20Problem%20for%20Jump-Diffusion%20Models%20with%20Random%20Coefficients:%20A%20Completion%20of%20Squares%20Approach&offset=0&pcAvailability=true
Publisher
MATHEMATICS
Location
스위스
Language
영어
ISSN
2227-7390
Citation Volume
9
Citation Number
22
Citation Start Page
2918
Citation End Page
2918
Appears in Collections:
Natural Science > Mathematics
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