Abstract
While numerical approaches to solve financial and actuarial stochastic optimization problems are usually based on dynamic programming, we explore an approach through a stochastic maximum principle formulation followed by the use of least squares regression to determine the optimal control policy. We show that this methodology can be applied to a number of realistic financial and actuarial problems of increasing complexity to highlight potential strengths and applications of this approach. We cast a direct connection between this approach and the stochastic duality approach to stochastic optimization. In particular, we discuss the potential improvements which can derive from this reformulation in terms of numerical precision and in order to provide bounds to control the simulation errors. The critical numerical issue is shown to be the numerical computation of conditional expectations which is performed applying the approach of Longstaff and Schwartz [1].
Highlights
While numerical approaches to solve financial and actuarial stochastic optimization problems are usually based on dynamic programming, we explore an approach through a stochastic maximum principle formulation followed by the use of least squares regression to determine the optimal control policy
While usually numerical methods are formulated within a dynamic programming approach to the optimization problem, we explore the possibility to state a numerical scheme that goes through the solution of the forward-backward
Stochastic differential equation (FBSDE, hereafter) which arises from the maximum principle formulation of the optimal control problem
Summary
While usually numerical methods are formulated within a dynamic programming approach to the optimization problem, we explore the possibility to state a numerical scheme that goes through the solution of the forward-backward. We exemplify situations where the maximum principle approach to portfolio optimization jointly with the simulation approach to the backward stochastic differential equation (BSDE, hereafter) provides an efficient alternative to standard simulation methods based on direct Montecarlo simulation and produces a reliable estimate of the optimal allocation strategies. Our computation approach relies on the least-squares Monte Carlo approach for estimating the conditional expectations made popular in financial mathematics by of Longstaff and Schwartz [1] and it was first applied to BSDEs and analyzed in this setting by Bouchard and Touzi [2], Gobet et al [3] and Lemor et al [4].
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