Overview
This study investigates a specific class of Markov additive processes (MAPs) where the additive components are real-valued and exhibit no positive jumps. The research focuses on characterizing their associated scale operators and subsequently applying these characterizations to address two distinct problems: the two-sided exit problem and the characterization of potential measures for processes terminated when their additive components depart from a specified interval.
Research Context
The core subject of this paper is Markov additive processes (MAPs). These processes are characterized by $\mathbb{R}$-valued additive components. A specific constraint is imposed on these additive components: they possess no positive jumps. Furthermore, the modulators of these MAPs are defined as Feller processes. A key property of these modulators is that they exhibit only a finite number of jumps within any given bounded time interval. Similarly, the additive components themselves are also restricted to having only finitely many jumps over any bounded time interval.
Approach
The research methodology hinges on the characterization of scale operators specifically designed for the described Cramér-Lundberg type Markov additive processes. The derivation and application of these scale operators leverage several theoretical constructs and properties. These include:
- Properties associated with analogues of local times.
- Properties of exit systems, which are obtained through their pairing with suitable kernels.
- Results pertaining to the extension of $C_0$-semigroups to $C_0$-groups.
Findings
The primary finding of this research is the characterization of scale operators for the specified class of Markov additive processes. This characterization serves as a foundational result, enabling subsequent applications. Utilizing these characterized scale operators, the study addresses two problems:
- Solution to the Two-Sided Exit Problem: The scale operators are employed to solve the two-sided exit problem. This problem concerns the determination of the first entrance time into one half-line, conditional on this event occurring before the process reaches the other half-line.
- Characterization of Potential Measures: The scale operators are also used to characterize the potential measures of processes. This characterization applies specifically to processes that are considered 'killed' when their additive components exit a defined interval.
Why This Matters
This work provides explicit characterizations for scale operators in a specific class of Markov additive processes, which are fundamental mathematical objects in probability theory. The derived operators offer a framework for solving specific boundary problems, such as the two-sided exit problem, and for analyzing the behavior of processes constrained to intervals, by characterizing their potential measures. These are foundational tools for theoretical analysis of such stochastic processes.