First passage times for some classes of fractional time-changed diffusions

Leonenko, Nikolai; Pirozzi, Enrica

doi:10.1080/07362994.2021.1953386

Cited by 3 publications

(2 citation statements)

References 33 publications

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“…The latter leads to a quite interesting representation of f α (s; t), which can be also used to prove some regularity results (as, for instance, done in [49]). Further discussion on the various representations of g α and f α is given, for instance, in the Appendix of [50].…”

Section: Inverse Stable Subordinators and Semi-markov Processesmentioning

confidence: 99%

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

2022

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Several queueing systems in heavy traffic regimes are shown to admit a diffusive approximation in terms of the Reflected Brownian Motion. The latter is defined by solving the Skorokhod reflection problem on the trajectories of a standard Brownian motion. In recent years, fractional queueing systems have been introduced to model a class of queueing systems with heavy-tailed interarrival and service times. In this paper, we consider a subdiffusive approximation for such processes in the heavy traffic regime. To do this, we introduce the Delayed Reflected Brownian Motion by either solving the Skorohod reflection problem on the trajectories of the delayed Brownian motion or by composing the Reflected Brownian Motion with an inverse stable subordinator. The heavy traffic limit is achieved via the continuous mapping theorem. As a further interesting consequence, we obtain a simulation algorithm for the Delayed Reflected Brownian Motion via a continuous-time random walk approximation.

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Section: Inverse Stable Subordinators and Semi-markov Processesmentioning

confidence: 99%

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

2022

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“…Ricciardi et al [12], Linetsky [17]). Furthermore, the FPT problems play a relevant role also in the context of fractional processes (see, for instance, Guo et al [18], Wiese [19], Abundo [20], Leonenko and Pirozzi [21]).…”

Section: Introductionmentioning

confidence: 99%

On the Absorbing Problems for Wiener, Ornstein–Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

Giorno

Nobile

2022

Fractal Fract

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For the Wiener, Ornstein–Uhlenbeck, and Feller processes, we study the transition probability density functions with an absorbing boundary in the zero state. Particular attention is paid to the proportional cases and to the time-homogeneous cases, by obtaining the first-passage time densities through the zero state. A detailed study of the asymptotic average of local time in the presence of an absorbing boundary is carried out for the time-homogeneous cases. Some relationships between the transition probability density functions in the presence of an absorbing boundary in the zero state and between the first-passage time densities through zero for Wiener, Ornstein–Uhlenbeck, and Feller processes are proven. Moreover, some asymptotic results between the first-passage time densities through zero state are derived. Various numerical computations are performed to illustrate the role played by parameters.

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On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy

Pirozzi

2022

Mathematics

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The paper deals with a fractional time-changed stochastic risk model, including stochastic premiums, dividends and also a stochastic initial surplus as a capital derived from a previous investment. The inverse of a ν-stable subordinator is used for the time-change. The submartingale property is assumed to guarantee the net-profit condition. The long-range dependence behavior is proven. The infinite-horizon ruin probability, a specialized version of the Gerber–Shiu function, is considered and investigated. In particular, we prove that the distribution function of the infinite-horizon ruin time satisfies an integral-differential equation. The case of the dividends paid according to a multi-layer dividend strategy is also considered.

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First passage times for some classes of fractional time-changed diffusions

Cited by 3 publications

References 33 publications

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

Skorokhod Reflection Problem for Delayed Brownian Motion with Applications to Fractional Queues

On the Absorbing Problems for Wiener, Ornstein–Uhlenbeck, and Feller Diffusion Processes: Similarities and Differences

On a Fractional Stochastic Risk Model with a Random Initial Surplus and a Multi-Layer Strategy

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