Drawdown preceding drawup when b < a

Develop a linear system characterization and an efficient computational algorithm for A(q, x, y) = E_{x,y}[e^{-q τ_a} 1_{ {τ_a < \widehat{τ}_b} } f(Y_{τ_a})], the Laplace transform associated with the drawdown time τ_a preceding the drawup time \widehat{τ}_b, under the continuous-time Markov chain approximation of general time-homogeneous Markov processes, in the case where the drawup threshold b is smaller than the drawdown threshold a (b < a).

Background

In Subsection 3.1, the paper defines A(q, x, y) = E_{x,y}[e^{-q τa} 1{ {τa < \widehat{τ}_b} } f(Y{τ_a})] for a continuous-time Markov chain Y that approximates a general time-homogeneous Markov process. Proposition 3.1 provides a linear system for A(q, x, y) when the drawup threshold b is greater than or equal to the drawdown threshold a (b ≥ a), together with an algorithm (Algorithm 1) and complexity analysis.

The authors explicitly state that handling the case b < a is more complicated and is not addressed in the paper, leaving it as future work. This constitutes an open problem to extend the established framework to the regime where b is smaller than a, which would complete the treatment of drawdown preceding drawup under general Markov models.