Appl Math Optim (2011) 64:155–169
Sufﬁcient Stochastic Maximum Principle
in a Regime-Switching Diffusion Model
Published online: 21 January 2011
© Springer Science+Business Media, LLC 2011
Abstract We prove a sufﬁcient stochastic maximum principle for the optimal control
of a regime-switching diffusion model. We show the connection to dynamic program-
ming and we apply the result to a quadratic loss minimization problem, which can be
used to solve a mean-variance portfolio selection problem.
Keywords Sufﬁcient maximum principle · Regime-switching · Optimal control ·
Mean-variance portfolio selection
The aim of this paper is to prove a sufﬁcient stochastic maximum principle for opti-
mal control within a regime-switching diffusion model. This extends the result of ,
which is in a jump-diffusion setting. To prove this, we follow the method in . As
in their paper, we show the connection to dynamic programming and show how to
apply the result to a quadratic loss minimization problem.
An early maximum principle for a diffusion model is in , where a necessary
maximum principle is derived in a model which is somewhat structurally similar to
our own and, as we also ﬁnd in our set-up, this results in jumps in the adjoint variables
of the Hamiltonian.
For a hidden Markovian regime-switching diffusion model, Elliott et al.  apply,
though do not state explicitly, a sufﬁcient maximum principle to a mean-variance
portfolio selection problem. However, their model is not the same as the one we
consider and hence they do not obtain jumps in the adjoint variables.
In Sect. 2 we detail the regime-switching diffusion model and in Sect. 3 we set out
the control problem. The sufﬁcient stochastic maximum principle is given in Sect. 4.
C. Donnelly (
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, UK