Spectral Decomposition of Optimal Asset-Liability Management
21 Pages Posted: 6 Oct 2008 Last revised: 16 Mar 2009
Date Written: January 31, 2008
Abstract
This paper concerns optimal asset-liability management when the assets and the liabilities are modeled by means of correlated geometric Brownian motions as suggested in Gerber and Shiu (2003). In a first part, we apply singular stochastic control techniques to derive a free boundary equation for the optimal value creation as a growth of liabilities or as dividend payment to shareholders. We provide analytical solutions to the HJB optimality equation in a rather general context. In a second part, we study the convergence of the cash flows to the optimal value creation using spectral methods. For particular cases, we also provide a series expansion for the probabilities of bankruptcy in finite time.
Keywords: asset-liability management, HJB principle, local time, spectral theory
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