SIAM Journal on Applied Mathematics
This paper develops a method of upper and lower solutions for a general system of second-order ordinary differential equations with two-point boundary conditions. Our motivation of study stems from a class of financial mathematics problems under regime-switching diffusion models. Two examples are double barrier option valuation and optimal selling rules in asset trading. We establish the existence of a unique C2 solution of the two-point boundary value problem. We construct monotone sequences of upper and lower solutions that are shown to converge to the unique solution of the boundary value problem. This construction provides a feasible numerical method to compute approximate solutions. An important feature of the proposed numerical method is that the unique solution is bracketed by the upper and lower approximate solutions, which provide an interval estimate of the unique solution function. We apply the general results to a regime-switching mean-reverting model and improve related results already reported in the literature. For the mean-reverting model, explicit upper and lower solutions are obtained and numerical integration methods are employed. In another case (Example 3 in section 5) a different regime-switching model is considered, where the general results apply, but only the upper solution is explicitly obtained. In that example, only the sequence of upper solutions is numerically constructed using finite difference methods. Numerical results are reported.
Eloe, Paul W. and Liu, R. H., "Upper and lower solutions for regime-switching diffusions with applications in financial mathematics" (2011). Mathematics Faculty Publications. 118.