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Irmina Czarna

Publications and source records attributed to Irmina Czarna.

13 recordsLinked to original sources
arXiv2019

Optimality of impulse control problem in refracted Lévy model with Parisian ruin and transaction costs

In this paper we investigate an optimal dividend problem with transaction costs, where the surplus process is modelled by a refracted Lévy process and the ruin time is considered with Parisian delay. Presence of the transaction costs implies that one need to consider the impulse control problem as a control strategy in such model. An impulse policy $(c_1,c_2)$, which is to reduce the reserves to some fixed level $c_1$ whenever they are above another level $c_2$ is an important strategy for the impulse control problem. Therefore, we give sufficient conditions under which the above described impulse policy is optimal. Further, we give the new analytical formulas for the Parisian refracted $q$-scale functions in the case of the linear Brownian motion and the Crámer-Lundberg process with exponential claims. Using these formulas we show that for these models there exists a unique $(c_1, c_2)$ policy which is optimal for the impulse control problem. Numerical examples are also provided.

math.PR↗
arXiv2019

Fluctuation theory for level-dependent Lévy risk processes

A level-dependent Lévy process solves the stochastic differential equation $dU(t) = dX(t)-ϕ(U(t)) dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy process with $ϕ_k(x)=\sum_{j=1}^kδ_j1_{\{x\geq b_j\}}$. A general rate function $ϕ$ that is non-decreasing and continuously differentiable is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of Lévy processes. We show how fluctuation identities for $U$ can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.

math.PR↗
arXiv2018

Fluctuation identities for omega-killed Markov additive processes and dividend problem

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 $ω(\cdot,\cdot)$ 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 generalizations of classical scale matrices for the MAP. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the so-called Omega model, where bankruptcy occurs at rate $ω(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider the Markov modulated Brownian motion (MMBM) and present the results for the particular choice of piecewise intensity function $ω(\cdot,\cdot)$.

math.PR↗
arXiv2018

Optimality of multi-refraction dividend strategies in the dual model

We consider the multi-refraction strategies in two equivalent versions of the optimal dividend problem in the dual (spectrally positive Lévy) model. The first problem is a variant of the bail-out case where both dividend payments and capital injections must be absolutely continuous with respect to the Lebesgue measure. The second is an extension of Avanzi et al. [4] where a strategy is a combination of two absolutely continuous dividend payments with different upper bounds and different transaction costs. In both problems, it is shown to be optimal to refract the process at two thresholds, with the optimally controlled process being the multi-refracted Lévy process recently studied by Czarna et al. [9]. The optimal strategy and the value function are succinctly written in terms of a version of the scale function. Numerical results are also given.

math.PR↗
arXiv2017

Discrete time ruin probability with Parisian delay

In this paper we evaluate the probability of the discrete time Parisian ruin that occurs when surplus process stays below or at zero at least for some fixed duration of time $d>0$. We identify expressions for the ruin probabilities within finite and infinite-time horizon. We also find their light and heavy-tailed asymptotics when initial reserves approach infinity. Finally, we calculate these probabilities for a few explicit examples.

math.PR↗
arXiv2017

Parisian ruin for a refracted Lévy process

In this paper, we investigate Parisian ruin for a Lévy surplus process with an adaptive premium rate, namely a refracted Lévy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probability of Parisian ruin of a standard Lévy insurance risk process. Despite the more general setup considered here, our main result is as compact and has a similar structure. Examples are provided.

math.PR↗
arXiv2016

Parisian quasi-stationary distributions for asymmetric Lévy processes

In recent years there has been some focus on quasi-stationary behaviour of an one-dimensional Lévy process $X$, where we ask for the law $P(X_t\in dy | τ^-_0>t)$ for $t\to\infty$ and $τ_0^-=\inf\{t\geq 0: X_t<0\}$. In this paper we address the same question for so-called Parisian ruin time $τ^θ$, that happens when process stays below zero longer than independent exponential random variable with intensity $θ$.

math.PR↗
arXiv2016

Optimal Parisian-type dividends payments discounted by the number of claims for the perturbed classical risk process

In this paper we consider a classical risk process perturbed by a Brownian motion. We analyze the value function describing the mean of the cumulative discounted dividend payments paid up to Parisian ruin time and further discounted by the number of claims appeared up to this ruin time. We identify this value function for the barrier strategy and find the sufficient conditions for this strategy to be optimal. We also consider few particular examples.

math.PR↗
arXiv2013

Parisian ruin probability for spectrally negative Lévy processes

In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally negative Levy process and the distribution of the process at time r.

math.PR↗
arXiv2011

Dividend problem with Parisian delay for a spectrally negative Lévy risk process

In this paper we consider dividend problem for an insurance company whose risk evolves as a spectrally negative Lévy process (in the absence of dividend payments) when Parisian delay is applied. The objective function is given by the cumulative discounted dividends received until the moment of ruin when so-called barrier strategy is applied. Additionally we will consider two possibilities of delay. In the first scenario ruin happens when the surplus process stays below zero longer than fixed amount of time $ζ>0$. In the second case there is a time lag $d$ between decision of paying dividends and its implementation.

q-fin.PM↗
arXiv2011

De Finetti's dividend problem and impulse control for a two-dimensional insurance risk process

Consider two insurance companies (or two branches of the same company) that receive premiums at different rates and then split the amount they pay in fixed proportions for each claim (for simplicity we assume that they are equal). We model the occurrence of claims according to a Poisson process. The ruin is achieved when the corresponding two-dimensional risk process first leaves the positive quadrant. We will consider two scenarios of the controlled process: refraction and impulse control. In the first case the dividends are payed out when the two-dimensional risk process exits the fixed region. In the second scenario, whenever the process hits the horizontal line, it is reduced by paying dividends to some fixed point in the positive quadrant where it waits for the next claim to arrive. In both models we calculate the discounted cumulative dividend payments until the ruin. This paper is the first attempt to understand the effect of dependencies of two portfolios on the joint optimal strategy of paying dividends. For example in case of proportional reinsurance one can observe the interesting phenomenon that choice of the optimal barrier depends on the initial reserves. This is in contrast with the one-dimensional Cramér-Lundberg model where the optimal choice of the barrier is uniform for all initial reserves.

q-fin.GN↗
arXiv2010

Ruin probability with Parisian delay for a spectrally negative Lévy risk process

In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time $ζ>0$. We focus on general spectrally negative Lévy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quantities that could be possibly calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tends to infinity. Finally, we analyze few explicit examples.

math.PR↗