Abstract
For solving the regime switching utility maximization, Fu et al. (Eur J Oper Res 233:184–192, 2014) derive a framework that reduce the coupled Hamilton–Jacobi–Bellman (HJB) equations into a sequence of decoupled HJB equations through introducing a functional operator. The aim of this paper is to develop the iterative finite difference methods (FDMs) with iteration policy to the sequence of decoupled HJB equations derived by Fu et al. (2014). The convergence of the approach is proved and in the proof a number of difficulties are overcome, which are caused by the errors from the iterative FDMs and the policy iterations. Numerical comparisons are made to show that it takes less time to solve the sequence of decoupled HJB equations than the coupled ones.
Highlights
The utility maximization is a kind of stochastic control problems
We study the iterative finite difference methods (FDMs) with policy iterations for solving the sequence of decoupled HJB equations in [16] and prove the convergence
We extend the finite difference methods (FDMs) with policy iterations to the HJB system arising from regime switching utility maximization problems
Summary
The utility maximization is a kind of stochastic control problems. The dynamic programming approach is often applied to the optimal value function and the so-called HJB equation is derived (see the books [25,34] for the stochastic control and its applications). Since the HJB equation is a fully nonlinear PDE, the closed-form classical solution cannot be found except for some simple cases: a Black-Scholes complete market model with particular utility functions, see [6,7]. For constrained market models it has to use numerical methods to solve.
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