Spectral decomposition of optimal asset-liability management

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. Geometric Brownian motion models for assets and liabilities: from pension funding to optimal dividends. North American Actuarial Journal 7(3), 37-51]. 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 Hamilton-Jacobi-Bellman (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.