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Zbigniew Palmowski

Publications and source records attributed to Zbigniew Palmowski.

At least 37 records · Page 2Linked to original sources
arXiv2023

Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions

In this work, we consider moments of exponential functionals of Lévy processes on a deterministic horizon. We derive two convolutional identities regarding these moments. The first one relates the complex moments of the exponential functional of a general Lévy process up to a deterministic time to those of the dual Lévy process. The second convolutional identity links the complex moments of the exponential functional of a Lévy process, which is not a compound Poisson process, to those of the exponential functionals of its ascending/descending ladder heights on a random horizon determined by the respective local times. As a consequence, we derive a universal expression for the half-negative moment of the exponential functional of any symmetric Lévy process, which resembles in its universality the passage time of symmetric random walks. The $(n-1/2)^{th}$, $n\geq 0$ moments are also discussed. On the other hand, under extremely mild conditions, we obtain a series expansion for the complex moments (including those with negative real part) of the exponential functionals of subordinators. This significantly extends previous results and offers neat expressions for the negative real moments. In a special case, it turns out that the Riemann zeta function is the minus first moment of the exponential functional of the Gamma subordinator indexed in time.

math.PR↗
arXiv2023

Exact asymptotics of ruin probabilities with linear Hawkes arrivals

In this paper we determine bounds and exact asymptotics of the ruin probability for risk process with arrivals given by a linear marked Hawkes process. We consider the light-tailed and heavy-tailed case of the claim sizes. Main technique is based on the principle of one big jump, exponential change of measure, and renewal arguments.

math.PR↗
arXiv2023

Asymptototic Expected Utility of Dividend Payments in a Classical Collective Risk Process

We find the asymptotics of the value function maximizing the expected utility of discounted dividend payments of an insurance company whose reserves are modeled as a classical Cramér risk process, with exponentially distributed claims, when the initial reserves tend to infinity. We focus on the power and logarithmic utility functions. We perform some numerical analysis as well.

math.OC↗
arXiv2023

Ruin probability for the quota share model with~phase-type distributed claims

In this paper, we generalise the results presented in the literature for the ruin probability for the insurer--reinsurer model under a pro-rata reinsurance contract. We consider claim amounts that are described by a phase-type distribution that includes exponential, mixture of exponential, Erlang, and mixture of Erlang distributions. We derive the ruin probability formulas with the use of change-of-measure technique and present important special cases. We illustrate the usefulness of the introduced model by fitting it to the real-world loss data. With the use of statistical tests and graphical tools, we show that the mixture of Erlangs is well-fitted to the data and is superior to other considered distributions. This justifies the fact that the presented results can be useful in the context of risk assessment of co-operating insurance companies.

q-fin.MF↗
arXiv2023

Random walk on a quadrant: mapping to a one-dimensional level-dependent Quasi-Birth-and-Death process (LD-QBD)

We consider a neighbourhood random walk on a quadrant, $\{(X_1(t),X_2(t),φ(t)):t\geq 0\}$, with state space \begin{eqnarray*} \mathcal{S}&=&\{(n,m,i):n,m=0,1,2,\ldots;i=1,2,\ldots,k(n,m)\}. \end{eqnarray*} Assuming start in state $(n,m,i)$, the process spends exponentially distributed amount of time in $(n,m,i)$ according to some parameter $λ_i^{(n,m)}$. Upon leaving state $(n,m,i)$ the process moves to some state $(n^{'},m^{'},j)$ with $j\in\{1,\ldots,k(n^{'},m^{'})\}$ and $n^{'}\in\{n-1,n,n+1\}$, $m^{'}\in\{m-1,m,m+1\}$, according to some probabilities $(p_{n;a}^{m;b})_{i,j}$ with $a,b\in\{+,-,0\}$. We transform this process into a one-dimensional LD-QBD $\{(Z(t),χ(t)):t\geq 0\}$ with level variable $Z(t)$ and phase variable $χ(t)$. Using this transform we find its transient and stationary analysis using matrix-analytic methods, as well as the distribution at first hitting times.

math.PR↗
arXiv2023

Extreme positions of regularly varying branching random walk in random and time-inhomogeneous environment

In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the $n$-th generation) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Frechét random variable. Furthermore, we derive the weak limit of the point processes composed of appropriately scaled positions at the $n$-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes. Hence, an analog of the predictions by Brunet and Derrida (2011) holds. We have obtained an explicit description of the limit point process. This description captures the influence of the environment in the joint asymptotic behavior of the extreme positions. We show that the law of the clusters in the limit depends on the time-reversed environment. The asymptotic distribution of the normalized rightmost position is derived as a consequence. This approach (based on weak convergence of extremal processes) can not be adapted when the genealogical structure is given through a supercritical Branching Process in a time-Inhomogeneous Environment (BPIE) (due to lack of structural regularity in the genealogical structure). We provide a simple example where the point processes do not converge weakly. The tightness of the point processes holds in this example though the point processes do not (weakly) converge (due to having different subsequential weak limits). This phenomenon is not yet known in the literature.

math.PR↗
arXiv2022

Last passage American cancellable option in Lévy models

We derive the explicit price of the perpetual American put option cancelled at the last passage time of the underlying above some fixed level. We assume the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first epoch when asset price process drops below an optimal threshold. We perform numerical analysis as well considering classical Black-Scholes models and the model where logarithm of the asset price has additional exponential downward shocks. The proof is based on some martingale arguments and fluctuation theory of Lévy processes.

q-fin.MF↗
arXiv2022

Gerber-Shiu Theory for Discrete Risk Processes in a Regime Switching Environment

In this paper we develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a Markovian (regime switching) environment. In particular, by expressing the Gerber-Shiu function in terms of potential measures of an upward (downward) skip-free discrete-time and discrete-space Markov Additive Process (MAP), we derive closed form expressions for the Gerber-Shiu function in terms of the so-called (discrete) $\boldsymbol{W}_v$ and $\boldsymbol{Z}_v$ scale matrices, which were introduced in arXiv:2008.06697. We show that the discrete scale matrices allow for a unified approach for identifying the Gerber-Shiu function as well as the value function of the associated constant dividend barrier problems.

math.PR↗
arXiv2022

Multivariate Lévy-type drift change detection and mortality modeling

In this paper we give a solution to the quickest drift change detection problem for a multivariate Lévy process consisting of both continuous (Gaussian) and jump components in the Bayesian approach. We do it for a general 0-modified continuous prior distribution of the change point. Classically, our criterion of optimality is based on a probability of false alarm and an expected delay of the detection, which is then reformulated in terms of a posterior probability of the change point. We find a generator of the posterior probability, which in case of general prior distribution is inhomogeneous in time. The main solving technique uses the optimal stopping theory and is based on solving a certain free-boundary problem. We also construct a Generelized Shiryaev-Roberts statistic, which can be used for applications. The paper is supplemented by two examples, one of which is further used to analyze Polish life tables (after proper calibration) and detect the drift change in the correlated force of mortality of men and women jointly.

math.PR↗
arXiv2022

On Busy Periods of the Critical GI/G/1 Queue and BRAVO

We study critical GI/G/1 queues under finite second moment assumptions. We show that the busy period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a BRAVO phenomenon (Balancing Reduces Asymptotic Variance of Outputs) for the work-output process (namely the busy-time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy-period. In the current paper we strengthen the previous results by reducing to assumptions to existence of $2+ε$ moments.

math.PR↗
arXiv2022

Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set $A$ not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.

math.PR↗
arXiv2022

Matrix-Analytic Methods for the analysis of Stochastic Fluid-Fluid Models

Stochastic fluid-fluid models (SFFMs) offer powerful modeling ability for a wide range of real-life systems of significance. The existing theoretical framework for this class of models is in terms of operator-analytic methods. For the first time, we establish matrix-analytic methods for the efficient analysis of SFFMs. We illustrate the theory with numerical examples.

math.PR↗
arXiv2022

First exit time for a discrete time parallel queue

We consider a discrete time parallel queue, which is two-queue network, where at each time-slot there is a the same batch arrival to both queues and at each queue there is a random service available. The service law at each time-slot for each queue is different. Let $(Q_n^1, Q_n^2)$ be the queue length at $n$th time-slot. We present several open questions related to the steady-state of this queue.

math.PR↗
arXiv2021

Pricing Perpetual American put options with asset-dependent discounting

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as \begin{equation*} V^ω_{\text{A}^{\text{Put}}}(s) = \sup_{τ\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^τω(S_w) dw} (K-S_τ)^{+}], \end{equation*} where $\mathcal{T}$ is a family of stopping times, $ω$ is a discount function and $\mathbb{E}$ is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process $S_t$ is a geometric Lévy process with negative exponential jumps, i.e. $S_t = s e^{ζt + σB_t - \sum_{i=1}^{N_t} Y_i}$. The asset-dependent discounting is reflected in the $ω$ function, so this approach is a generalisation of the classic case when $ω$ is constant. It turns out that under certain conditions on the $ω$ function, the value function $V^ω_{\text{A}^{\text{Put}}}(s)$ is convex and can be represented in a closed form; see Al-Hadad and Palmowski (2021). We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of $ω$ such that $V^ω_{\text{A}^{\text{Put}}}(s)$ takes a simplified form.

q-fin.MF↗
arXiv2021

Unified approach for solving exit problems for additive-increase and multiplicative-decrease processes

We analyse an additive-increase and multiplicative-decrease (aka growth-collapse) process that grows linearly in time and that experiences downward jumps at Poisson epochs that are (deterministically) proportional to its present position. This process is used for example in modelling of Transmission Control Protocol (TCP) and can be viewed as a particular example of the so-called shot noise model, a basic tool in modeling earthquakes, avalanches and neuron firings. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All the eight Laplace transforms can be described in terms of two so-called scale functions $Z_{\uparrow}$ and $L_{\uparrow}$. Here $Z_{\uparrow}$ is described in terms of multiple explicit sums, and $L_{\uparrow}$ in terms of an explicit recursion formula. All other Laplace transforms can be obtained from $Z_{\uparrow}$ and $L_{\uparrow}$ by taking limits, derivatives, integrals and combinations of these.

math.PR↗
arXiv2021

How much we gain by surplus-dependent premiums -- asymptotic analysis of ruin probability

In this paper, we build on the techniques developed in Albrecher et al. (2013), to generate initial-boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and an exponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations developed in Fedoryuk (1993), we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are {\it linearly} dependent on reserves, representing for instance returns on risk-free investments of the insurance capital, we firstly derive explicit formulas for the ruin probabilities, from which we can easily determine their asymptotics, only to match the ones obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve own by the insurance company.

math.PR↗