Papers by Zbigniew Palmowski
arXiv (Cornell University), Feb 26, 2019
We consider the sample average of a centered random walk in R d with regularly varying step size ... more We consider the sample average of a centered random walk in 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.
Queueing Systems
We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-per... more 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 Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon 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 $$2+\epsilon $$ 2 + ϵ moments.
Abstract. In this note we give, for a spectrally negative Lévy process, a compact formula for the... more Abstract. In this note we give, for a spectrally negative Lévy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero which length exceeds a certain fixed period r. The formula involves only the scale function of the spectrally negative Lévy process and the distribution of the process at time r.
arXiv: Optimization and Control, 2018
In this paper we apply the dynamic programming optimization to find an optimal strategy for assig... more 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 situation change. Therefore, we endow our model with stochasticity, what makes it less vulnerable for changes. The analysis is based on real data, showing that the proposed algorithm is very efficient and can be easily used to support decisions at the winery. Suggested solution could be applied as well to many other factories relying their production on outer supplies.
Probability and Mathematical Statistics, 2020
We consider the classical risk process (the case σ = 0) and the classical risk process perturbed ... more We consider the classical risk process (the case σ = 0) and the classical risk process perturbed by a Brownian motion (the case σ > 0). We analyze the expected NPV describing the mean of the cumulative discounted dividend payments paid up to the Parisian or classical ruin time and further penalized by the number of claims that appeared up to that time. We identify this function for a constant barrier strategy and we find sufficient conditions for this strategy to be optimal. We also analyze a numerical example of exponential claim sizes.
Journal of Optimization Theory and Applications, 2017
This paper concerns the dual risk model, dual to the risk model for insurance applications, where... more 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 Hamilton-Jacobi-Bellman equation and identify sufficient conditions for a barrier strategy to be optimal. Some numerical examples are provided when profits have exponential law. KEYWORDS. Dividends PDMP optimal strategy barrier strategy integro-differential HJB equation Dual model stochastic control Exit problems CONTENTS 1. Introduction 2. The Dual Model 3. The Classical model 4. Optimal Dividend Strategy 4.1. HJB equations 4.2. Barrier Strategies 5. Examples 6. Appendix 6.1. Proof of the Verification Theorem 4 6.2. Additional facts 6.3. Proof of the Theorem 8 6.4. Proof of the Theorem 9 References
arXiv: Probability, 2019
We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying... more 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.
Queueing Systems, 2011
The areas under the workload process and under the queueing process in a single-server queue over... more The areas under the workload process and under the queueing process in a single-server queue over the busy period have many applications not only in queueing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.
In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman (1997) to pri... more In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman (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 to the true price with the increasing depth of the tree. Under specific parametrization, this π-option is related to relative maximum drawdown and can be used in the real-market environment to protect a portfolio against volatile and unexpected price drops. We also provide some numerical analysis.
This paper describes an application of dynamic programming to determine the optimal strategy for ... more 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 losses in cases of sudden changes, which frequently occur in weather-dependent wine factories. Hence, we endowed our model with stochasticity, thereby rendering it less vulnerable to such changes. Our analysis, which is based on real-world data, demonstrates that the proposed algorithm is highly efficient and, after calibration, can be used to support winery's decision-making. The solution proposed herein could also be applied in numerous other contexts where production processes rely on outside supplies.
Abstract. In this paper we analyze discrete time Parisian ruin probability that happens when surp... more Abstract. In this paper we analyze discrete time Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time ζ> 0. We identify expressions for the ruin probabilities with finite and infinite-time horizon. We find also their asymptotics when reserves tends to infinity. Finally, we calculate these probabilities for a few explicit examples.
Let X_t^♯ be a multivariate process of the form X_t =Y_t - Z_t, X_0=x, killed at some terminal ti... more Let X_t^♯ 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 the length smaller than δ, and Z_t is a compound Poisson process with jumps of the length bigger than δ for some fixed δ>0. Under the assumptions that the summands in Z_t are sub-exponential, we investigate the asymptotic behaviour of the potential function u(x)= E^x ∫_0^∞ℓ(X_s^♯)ds. The case of heavy-tailed entries in Z_t corresponds to the case of "big claims" in insurance models and is of practical interest. The main approach is based on fact that u(x) satisfies a certain renewal equation.
We consider the problem of maximizing the expected utility of discounted dividend payments of an ... more We consider the problem of maximizing the expected utility of discounted dividend payments of an insurance company whose reserves are modeled as a Cramér risk process with Erlang claims. We focus on the exponential claims and power and logarithmic utility functions. Finally we also analyze asymptotic behaviour of the value function and identify the asymptotic optimal strategy. We also give the numerical procedure of finding considered value function. 1
In this paper we consider the problem of the quantile hedging from the point of view of a better ... more In this paper we consider the problem of the quantile hedging from the point of view of a better informed agent acting on the market. The additional knowledge of the agent is modelled by a filtration initially enlarged by some random variable. By using equivalent martingale measures introduced in Amendinger (2000) and Amendinger, Imkeller and Schweizer (1998) we solve the problem for the complete case, by extending the results obtained in Föllmer and Leukert (1999) to the insider context. Finally, we consider the examples with the explicit calculations within the standard Black-Scholes model.
Abstract. We give precise asymptotic estimates of the tail behavior of the distribution of the su... more Abstract. We give precise asymptotic estimates of the tail behavior of the distribution of the supremum of a process with regenerative increments. Our results cover four qualitatively different regimes, involving both light tails and heavy tails, and are illustrated with examples arising in queueing theory and insurance risk.
We consider an equity-linked contract whose payoff depends on the lifetime of policy holder and t... more We consider an equity-linked contract whose payoff depends on the lifetime of policy holder and the stock price. We provide best strategy
For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation iden... more For spectrally negative Markov Additive Processes (MAPs) we generalize classical fluctuation identities developed in Zolotarev (1964), Takács (1967),
We analyse an additive-increase and multiplicative-decrease (aka growth-collapse) process that gr... more 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 oneand 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↑ and L↑. Here Z↑ is described in terms of multiple explicit sums,...
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Papers by Zbigniew Palmowski