Penalty and penalty-like methods for nonlinear HJB PDEs

Christina C. Christara and Ruining Wu

Applied Mathematics and Computation, 2022, vol. 425, issue C

Abstract: There are numerous financial problems that can be posed as optimal control problems, leading to Hamilton–Jacobi–Bellman or Hamilton–Jacobi–Bellman–Issacs equations. We reformulate these problems as nonlinear PDEs, involving max and/or min terms of the unknown function, and/or its first and second spatial derivatives. We suggest efficient numerical methods for handling the nonlinearity in the PDE through an adaptation of the discrete penalty method Forsyth and Vetzal(2002)[1] that gives rise to tridiagonal penalty matrices. We formulate a penalty-like method for the use with European exercise rights, and extend this to American exercise rights resulting in a double-penalty method. We also use our findings to improve the policy iteration algorithms described in Forsyth and Labahn(2007)[2]. Numerical results are provided showing clear second-order convergence, and where applicable, we prove the convergence of our algorithms.

Keywords: Partial differential equations; Black–Scholes; Nonlinear iteration; Finite differences; Crank–Nicolson; Control problem; Hamilton–Jacobi–Bellman (HJB) equation; Transaction costs; Stock borrowing fees; Penalty methods (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0096300322001011
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:425:y:2022:i:c:s0096300322001011

DOI: 10.1016/j.amc.2022.127015

Access Statistics for this article

Applied Mathematics and Computation is currently edited by Theodore Simos

More articles in Applied Mathematics and Computation from Elsevier
Bibliographic data for series maintained by Catherine Liu ().