Zbigniew Palmowski

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• PhD
• Professor at Wrocław University of Science and Technology

While stochastic resetting (or total resetting) is a less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In...

In this article, we consider a Branching Random Walk (BRW) 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 o...

In this paper, we adopt the least squares Monte Carlo (LSMC) method to price time-capped American options. The aforementioned cap can be an independent random variable or dependent on asset price at random time. We allow various time caps. In particular, we give an algorithm for pricing the American options capped by the first drawdown epoch. We fo...

While stochastic resetting (or total resetting) is less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In th...

We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a L\'evy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\mathbf{Y}$ between these two consecutive moments, $c \in (0, 1)$. We focus o...

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative L\'evy processes using the binomial expansion approach. More specifically, we study the distributions of such excursions and related quantities such as the joint distribution of the shortest and longest negative excursion and t...

In this work, we consider moments of exponential functionals of Lévy processes on a deter-ministic 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 convolu...

In this paper we consider a multidimensional random walk killed on leaving a right circular cone with a distribution of increments belonging to the normal domain of attraction of an $\alpha$-stable and rotationally-invariant law with $\alpha \in (0,2)\setminus \{1\}$. Based on Bogdan et al. (2018) describing the tail behaviour of the exit time of $...

In this paper, we study the concept of the speed of recovery of a Lévy risk process, which is the duration between the last recovery time and the ruin time. Our results improve the existing literature in which, for some cases, only the Laplace transforms are known. Additionally, we study the speed of recovery under Poissonian observation scheme. As...

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 e...

In this paper, we consider (upward skip-free) discrete-time and discrete-space Markov additive chains (MACs) and develop the theory for the so-called W~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidem...

In this paper we consider the following optimal stopping problem where the process \(S_t\) is a jump-diffusion process, \({\mathcal {T}}\) is a family of stopping times while g and \(\omega \) are fixed payoff function and discount function, respectively. In a financial market context, if \(g(s)=(K-s)^+\) or \(g(s)=(s-K)^+\) and \({\mathbb {E}}\) i...

In this work, we consider moments of exponential functionals of L\'{e}vy 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 L\'{e}vy process up to a deterministic time to those of the dual L\'{e}vy process. The sec...

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction...

In this paper, we solve exit problems for a level-dependent L\'evy processes which is exponentially killed with a killing intensity that depends on the present state of the process. Moreover, we analyse the respective resolvents. All identities are given in terms of new generalisations of scale functions (counterparts of the scale function from the...

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction...

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 as well as for a random post-change drift parameter. Classically, o...

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.

We derive the subexponential tail asymptotics for the distribution of the maximum of a compound renewal process with linear component and of a L\'evy process, both with negative drift, over random time horizon $\tau$ that does not depend on the future increments of the process. Our asymptotic results are uniform over the whole class of such random...

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 also perfo...

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 Cram\'er risk process, with exponentially distributed claims, when the initial reserves tend to infinity. We focus on the power and logarithmic utility functions. We perform...

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ( $i=1,2,\dots$ ) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature; that is, if the surplus process just before the i th arrival is at level u , t...

We consider a neighbourhood random walk on a quadrant, $\{(X_1(t),X_2(t),\varphi(t)):t\geq 0\}$, 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 $\...

We perform the sensitivity analysis of a level-dependent QBD with a particular focus on applications in modelling healthcare systems.

We derive the explicit price of the perpetual American put option canceled at the last-passage time of the underlying above some fixed level. We assume that the asset process is governed by a geometric spectrally negative Lévy process. We show that the optimal exercise time is the first moment when the asset price process drops below an optimal thr...

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 L\'evy process. We show that the optimal exercise time is the first epoch when asset price process drops below an optimal threshold....

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems...

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...

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 e...

Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$ , $X_0=x$ , killed at some terminal time T , where $Y_t$ is a Markov process having only jumps of length smaller than $\delta$ , and $Z_t$ is a compound Poisson process with jumps of length bigger than $\delta$ , for some fixed $\delta>0$ . Under the assumptions that the summan...

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...

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...

Motivated by a seminal paper of Kesten et al. ( Ann. Probab. , 3(1) , 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters A n , n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, t...

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 qu...

We study the all-time supremum of the perturbed branching random walk, known to be the endogenous solution to the high-order Lindley equation: W=DmaxY,max1≤i≤N(Wi+Xi),where the {Wi} are independent copies of W, independent of the random vector (Y,N,{Xi}) taking values in R×N×R∞. Under Kesten assumptions, this solution satisfies P(W>t)∼He−αt,t→∞,whe...

In this paper we give a few expressions and asymptotics of ruin probabilities for a Markov modulated risk process for various regimes of a time horizon, initial reserves and a claim size distribution. We also consider a few versions of the ruin time.

In this paper we analyse the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, only transient and stationary analyses of SFMs have been considered in the literature. The limiting conditional distribution gives useful insights into what happens...

In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asympt...

This paper presents an analysis of the stochastic recursion $$W_{i+1} = [V_iW_i+Y_i]^+$$ W i + 1 = [ V i W i + Y i ] + that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model’s stability condition. Writing $$Y_i=B_i-A_i$$ Y i = B i - A i , for independent sequences of non...

In this paper we give few expressions and asymptotics of ruin probabilities for a Markov modulated risk process for various regimes of a time horizon, initial reserves and a claim size distribution. We also consider few versions of the ruin time.

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 VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a...

We consider the Lévy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case, the stopping region that characterizes the optimal stopping time is either a half‐line or an interval. The objective of this paper is to obtain explicit expressions of the st...

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^{\omega}_{\text{A}^{\text{Put}}}(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} (K-S_\tau)^{+}], \end{...

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 par...

In this article, we consider a Branching Random Walk (BRW) on the real line where the underlying genealogical structure is given through a supercritical branching process in i.i.d. environment and satisfies Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the...

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 sol...

We consider a dual risk model with constant expense rate and i.i.d. exponentially distributed gains $C_i$ ($i=1,2,\dots$) that arrive according to a renewal process with general interarrival times. We add to this classical dual risk model the proportional gain feature, that is, if the surplus process just before the $i$th arrival is at level $u$, t...

This article considers an optimal dividend distribution problem for an insurance company where the dividends are paid in a foreign currency. In the absence of dividend payments, our risk process follows a spectrally negative Lévy process. We assume that the exchange rate is described by a an exponentially Lévy process, possibly containing the same...

In this paper we analyze the distributional properties of a busy period in an on-off fluid queue and the a first passage time in a fluid queue driven by a finite state Markov process. In particular, we show that in Anick-Mitra-Sondhi model the first passage time has a IFR distribution and the busy period has a DFR distribution.

In this paper, I analyze the distributional properties of the busy period in an on-off fluid queue and the first passage time in a fluid queue driven by a finite state Markov process. In particular, I show that the first passage time has a IFR distribution and the busy period in the Anick-Mitra-Sondhi model has a DFR distribution.

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...

The contagion dynamics can emerge in social networks when repeated activation is allowed. An interesting example of this phenomenon is retweet cascades where users allow to re-share content posted by other people with public accounts. To model this type of behaviour we use a Hawkes self-exciting process. To do it properly though one needs to calibr...

In this note we prove that the speed of convergence of the workload of a L\'evy-driven queue to the quasi-stationary distribution is of order $1/t$. We identify also the Laplace transform of the measure giving this speed and provide some examples.

This paper revisits the optimal capital structure model with endogenous bankruptcy, first studied by Leland (J. Finance 49:1213–1252, 1994) and Leland and Toft (J. Finance 51:987–1019, 1996). Unlike in the standard case where shareholders continuously observe the asset value and bankruptcy is executed instantaneously without delay, the information...

In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman in 1997 to price a π-option. This method is based on the simulated price tree that comes from discretization and replication of possible trajectories of the underlying asset’s price. As a result, this algorithm produces the lower and the upper bounds that converge t...

In this paper we develop the theory of the so-called $\mathbf{W}$ and $\mathbf{Z}$ scale matrices for (upwards skip-free) discrete-time and discrete-space Markov additive processes, along the lines of the analogous theory for Markov additive processes in continuous-time. In particular, we provide their probabilistic construction, identify the form...

Motivated by a seminal paper of Kesten et al. (1975) we consider a branching process with a geometric offspring distribution with i.i.d. random environmental parameters $A_n$, $n\ge 1$ and size -1 immigration in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution $F$ of...

In this paper, using Bayesian approach, we solve the quickest drift change detection problem for a multidimensional L\'evy process consisting of both a continuous gaussian part and a jump component. We allow a general a priori distribution of the change point as well as random post-change drift parameter. Classically, our optimality criterion is ba...

In this paper we consider the following optimal stopping problem $$V^{\omega}_{{\rm A}}(s) = \sup_{\tau\in\mathcal{T}} \mathbb{E}_{s}[e^{-\int_0^\tau \omega(S_w) dw} g(S_\tau)],$$ where the process $S_t$ is a jump-diffusion process, $\mathcal{T}$ is a family of stopping times and $g$ and $\omega$ are fixed payoff function and discounting function,...

In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman (1997) to price a $\pi$-option. This method is based on the simulated price tree that comes from discretization and replication of possible trajectories of the underlying asset's price. As a result this algorithm produces lower and upper bounds that converge to the...

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with...

In this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of ne...

This paper describes an application of dynamic programming to determine the optimal strategy for assigning grapes to pressing tanks in one of the largest Portuguese wineries. To date, linear programming has been employed to generate proposed solutions to analogous problems, but this approach lacks robustness and may, in fact, result in severe losse...

This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at fixed rate when the asset price is above a pre-specified level and continu...

We consider the distributional fixed-point equation: $$R \stackrel{\mathcal{D}}{=} Q \vee \left( \bigvee_{i=1}^N C_i R_i \right),$$ where the $\{R_i\}$ are i.i.d.~copies of $R$, independent of the vector $(Q, N, \{C_i\})$, where $N \in \mathbb{N}$, $Q, \{C_i\} \geq 0$ and $P(Q > 0) > 0$. By setting $W = \log R$, $X_i = \log C_i$, $Y = \log Q$ it is...

We consider the L\'evy model of the perpetual American call and put options with a negative discount rate under Poisson observations. Similar to the continuous observation case as in De Donno et al. [24], the stopping region that characterizes the optimal stopping time is either a half-line or an interval. The objective of this paper is to obtain e...

In this paper, we analyse some equity-linked contracts that are related to drawdown and drawup events based on assets governed by a geometric spectrally negative L\'evy process. Drawdown and drawup refer to the differences between the historical maximum and minimum of the asset price and its current value, respectively. We consider four contracts....

This paper presents an analysis of the stochastic recursion $W_{i+1} = [V_iW_i+Y_i]^+$ that can be interpreted as an autoregressive process of order 1, reflected at 0. We start our exposition by a discussion of the model's stability condition. Writing $Y_i=B_i-A_i$, for independent sequences of non-negative i.i.d.\ random variables $\{A_i\}_{i\in N...

We consider in this paper a risk reserve process where the claims and gains arrive according to two independent Poisson processes. While the gain sizes are phase-type distributed, we assume instead that the claim sizes are phase-type perturbed by a heavy-tailed component; that is, the claim size distribution is formally chosen to be phase-type with...

This paper considers an optimal dividend distribution problem for an insurance company where the dividends are paid in a foreign currency. In the absence of dividend payments, our risk process follows a spectrally negative L\'evy process. We assume that the exchange rate is described by a an exponentially L\'evy process, possibly containing the sam...

Let $X_t^\sharp$ be a multivariate process of the form $X_t =Y_t - Z_t$, $X_0=x$, killed at some stopping time $T$, where $Y_t$ is a Markov process having only jumps of the length smaller than $\delta$, and $Z_t$ is a compound Poisson process with jumps of length bigger than $\delta$ for some fixed $\delta>0$. Under heavy-tailed assumptions we inve...

In this paper we provide the analysis of the limiting conditional distribution (Yaglom limit) for stochastic fluid models (SFMs), a key class of models in the theory of matrix-analytic methods. So far, transient and stationary analysis of the SFMs only has been considered in the literature. The limiting conditional distribution gives useful insight...

Motivated by seminal paper of Kozlov et al. Kesten et al. (1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability A and immigration equals 1 in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is logA−1(1−A) is re...

Motivated by seminal paper of Kozlov et al.(1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability $A$ and immigration equals $1$ in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is $\log A^{-1} (1-A)$ is regul...

This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at fixed rate when the asset price is above a pre-specified level and continu...

We revisit the optimal capital structure model with endogenous bankruptcy first studied by Leland \cite{Leland94} and Leland and Toft \cite{Leland96}. Differently from the standard case, where shareholders observe continuously the asset value and bankruptcy is executed instantaneously without delay, we assume that the information of the asset value...

We study a portfolio selection problem in a continuous-time Itô–Markov additive market with prices of financial assets described by Markov additive processes that combine Lévy processes and regime switching models. Thus, the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. For this reason, the mar...

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 gr...

In this paper, we consider a classical risk model refracted at given level. We give an explicit expression for the joint density of the ruin time and the cumulative number of claims counted up to ruin time. The proof is based on solving some integro-differential equations and employing the Lagrange’s Expansion Theorem.

For a multivariate Lévy process satisfying the Cramér moment condition and having a drift vector with at least one negative component, we derive the exact asymptotics of the probability of ever hitting the positive orthant that is being translated to infinity along a fixed vector with positive components. This problem is motivated by the multivaria...

This paper concerns the dual risk model, dual to the risk model for insurance applications, where premiums are surplus-dependent. In such a model premiums are regarded as costs, while claims refer to profits. We calculate the mean of the cumulative discounted dividends paid until ruin, if the barrier strategy is applied. We formulate associated Ham...

In this paper we apply the dynamic programming optimization to find an optimal strategy for assigning grapes to pressing tanks in one of the biggest Portuguese wineries. Up to now, the proposed solutions were obtained with the use of linear programming. Such approach though lacks of robustness and can lead to severe losses in case of sudden situati...

In this paper we solve the exit problems for an one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the present state of the environment. Moreover, we analyze respective resolvents. All identities are given in terms of new ge...

In this paper we consider the Parisian ruin probabilities for the dual risk model in a discrete-time setting. By exploiting the strong Markov property of the risk process we derive a recursive expression for the fnite-time Parisian ruin probability, in terms of classic discrete-time dual ruin probabilities. Moreover, we obtain an explicit expressio...

We study a portfolio selection problem in a continuous-time It\^o-Markov additive market with prices of financial assets described by Markov additive processes which combine L\'evy processes and regime switching models. Thus the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. For this reason the...