arXiv subjects

Noah Beelders

Publications and source records attributed to Noah Beelders.

4 recordsLinked to original sources

arXiv2026

Lévy processes with partially stochastic resetting

In this paper, we solve exit problems for a Lévy process that resets proportionally to its current position at independent Poisson epochs times. This resetting causes an additional (proportional to its current level) downward (upward) jump when the current position of the process is on the positive (negative) domain. Such a process can be expressed as an SDE, whose existence and uniqueness it discussed. All identities are given in terms of new family of scale functions. To obtain the new family of scale functions, we reduce the problem of the LT of the exit times into integral equations that are solve in terms of resolvent series.

math.PR↗

arXiv2026

Poissonian potential measures for refracted-reflected Lévy processes

In this paper we study the potential measures and the Laplace transforms of the occupation times of a refracted-reflected spectrally negative Lévy process when the process is observed at the arrival epochs of two independent Poisson processes. In this case, the rates of observing the underlying process differ in time which deviates from the classical theory of Poissonian observations. Explicit expressions for the so-called Poissonian potential measures and the Poissonian occupation times are derived in terms of (known) scale functions. Other fluctuation identities are also derived.

math.PR↗

arXiv2026

Lévy processes under level-dependent Poissonian switching

In this paper, we derive identities for the upward and downward exit problems and resolvents for a process whose motion changes between two Lévy processes if it is above (or below) a barrier $b$ and coincides with a Poissonian arrival time. This can be expressed in the form of a (hybrid) stochastic differential equation, for which the existence of its solution is also discussed. All identities are given in terms of new generalisations of scale functions (counterparts of the scale functions from the theory of Lévy processes). To illustrate the applicability of our results, the probability of ruin is obtained for a risk process with delays in the dividend payments.

math.PR↗

arXiv2024

Probabilistic Cauchy Functional Equations

In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$ f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2), $$ where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution $μ$ having appropriate support on the real line. The symbol $\stackrel{d}{=}$ denotes equality in distribution. When $μ$ follows an exponential distribution, we provide sufficient (regularity) conditions on the function $f$ to ensure that the unique measurable solution to the above equation is solely linear. Furthermore, we present some partial results in the general case, establishing a connection to integrated Cauchy functional equations.

math.PR↗