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We propose a recursive algorithm for the numerical computation of the optimal value function $\inf _{t\le \tau \le T} \mathbb {E} \left [\sup _{0\le s\le T } Y_{s} / Y_{\tau } \left | {\mathcal F}_{t}\right .\right ] $ over the stopping times τ with respect to the filtration of a geometric Brownian motion Y with Markovian regime switching. This method allows us to determine the boundary functions of the optimal stopping set when no associated Volterra integral equation is available. It applies in particular when regime-switching drifts have mixed signs, in which case the boundary functions may not be monotone. [ABSTRACT FROM AUTHOR]

We provide closed-form expressions for bond prices in interest rate models based on compact Lie groups. Our approach uses a Doob transform technique and PDE solutions by the Mathieu periodic functions. As a by-product, we derive formulas for bond option prices as well as new identities for the Laplace transform of periodic functionals of Brownian motion and Brownian diffusion processes. [ABSTRACT FROM AUTHOR]