Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

Ivanovs, Jevgeņijs

doi:10.1239/jap/1421763333

“…This is because none of the existing proofs employ compound Poisson approximations of Y . See for example [39] which uses martingale arguments, [40] using Itô excursion theory and [41] using potential theory, and we refer to [27,42,43,44] for further discussions of these known results and proofs. Also worth highlighting that several key steps of our proof are not tied to the one-sidedness of our processes.…”

Section: Forward Generatormentioning

confidence: 99%

Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II

Baeumer,

Kovács,

Toniazzi

2021

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We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Lévy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Lévy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified Lévy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified Lévy process by showing continuity of the modifications with respect to the Skorokhod topology.

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“…This is remarkable because before the embedding procedure the approximating processes are not Feller, and the domain of the generators in Table 1 is not smooth. Moreover it is natural to expect that this technique is extendable to fast-forwarded non-recurrent Y , Robin and other mixed boundary conditions, to other Lévy processes such as α-stable processes in 1-dimension [41,42], and possibly to general one-sided Markov additive processes [43] and higher dimensions. These extensions are indeed part of our current investigations.…”

Section: Grünwald Type Approximationsmentioning

confidence: 99%

“…Proof. [of Theorem 4.10] Recall the matrix (63), definition (43) and that h(n+1) = 2. By Lemmata 3.19 and 3.14,…”

Section: Appendixmentioning

confidence: 99%

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Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes I

Baeumer¹,

Kovács²,

Toniazzi³

2020

Preprint

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We connect boundary conditions for one-sided pseudo-differential operators with the generators of modified one-sided Lévy processes. On one hand this allows modellers to use appropriate boundary conditions with confidence when restricting the modelling domain. On the other hand it allows for numerical techniques based on differential equation solvers to obtain fast approximations of densities or other statistical properties of restricted one-sided Lévy processes encountered, for example, in finance. In particular we identify a new nonlocal mass conserving boundary condition by showing it corresponds to fast-forwarding, i.e. removing the time the process spends outside the domain. We treat all combinations of killing, reflecting and fast-forwarding boundary conditions. In Part I we show wellposedness of the backward and forward Cauchy problems with a one-sided pseudo-differential operator with boundary conditions as generator. We do so by showing convergence of Feller semigroups based on grid point approximations of the modified Lévy process. In Part II we show that the limiting Feller semigroup is indeed the semigroup associated with the modified Lévy process by showing continuity of the modifications with respect to the Skorokhod topology.

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“…An alternative approach is to use fluid embedding of PH jumps to arrive at a second order fluid flow model (Markov modulated Brownian motion) and to treat the given problem in that context, see Asmussen et al (2004); Mordecki (2002). Indeed, there is a well-developed theory for such models (Asmussen, 1995;Ivanovs, 2010), as well as for much more general one-sided Markov additive processes (Ivanovs and Palmowski, 2012;Ivanovs, 2014). This, however, requires dealing with matrix calculus, partitioning of phases, and various further complications, not to mention a certain necessary experience.…”

Section: Introductionmentioning

confidence: 99%

On scale functions for Lévy processes with negative phase-type jumps

Ivanovs¹

2020

Preprint

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We provide a novel expression of the scale function for a Lévy processes with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and analyzed. Our numerical examples suggest that this algorithm allows to employ phase-type distributions with a hundred of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as infinite-dimensional phase-type and those with rational transform, can be anticipated.

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Potential Measures of One-Sided Markov Additive Processes with Reflecting and Terminating Barriers

Cited by 15 publications

References 27 publications

Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II

Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes II

Boundary conditions for nonlocal one-sided pseudo-differential operators and the associated stochastic processes I

On scale functions for Lévy processes with negative phase-type jumps

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