In this paper, we consider the perturbed renewal risk process. Systems of integro-differential equations for the Gerber-Shiu functions at ruin caused by a claim and oscillation are established, respectively. The explicit Laplase transforms of Gerber-Shiu functions are obtained, while the closed form expressions for the Gerber-Shiu functions are derived when the claim amount distribution is from the rational family. Finally, we present numerical examples intended to illustrate the main results.
In the present paper the change of measures technique for compound mixed renewal processes, developed in Tzaninis & Macheras [24], is applied to the ruin problem in order to compute the ruin probability and to find upper and lower bounds for it.
A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function g on a neighborhood of 0. This expansion relies on complex analysis and is expressed in terms of the residues of the function 1/(1 -- g). Under the assumption that g can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability f…
In this manuscript we consider the dual risk model with financial application, where the random gains occur under a renewal process. We particularly work the Erlang(n) case for common distribution of the inter-arrival times, from there it is easy to understand that our method or procedures can be generalised to other cases under the matrix-exponential family case. We work several and different problems involving future dividends and ruin. We also show that our results are valid even if the usual income condition is not satisfied. In most known works under the dual model, the main target under …
In this short note, we derive explicit formulas for the joint densities of the time to ruin and the number of claims until ruin in perturbed classical risk models, by constructing several auxiliary random processes.
Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time τ=τ(b)=inf{n:S_n>b}. When a drift μof the random walk S_n is 0, we derive a one-term Ed…
We study the asymptotic behavior of ruin probabilities, as the initial reserve goes to infinity, for a reserve process model where claims arrive according to a renewal process, while between the claim times the process has the dynamics of geometric Brownian motion-type Itô processes with time-dependent random coefficients. These coefficients are ``reset" after each claim time, switching to new values independent of the past history of the process. We use the implicit renewal theory to obtain power-function bounds for the eventual ruin probability. In the special case when the random drift and …
We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.
In this paper, we obtain the finite-horizon and infinite-horizon ruin probability asymptotics for risk processes with claims of subexponential tails for non-stationary arrival processes that satisfy a large deviation principle. As a result, the arrival process can be dependent, non-stationary and non-renewal. We give three examples of non-stationary and non-renewal point processes: Hawkes process, Cox process with shot noise intensity and self-correcting point process. We also show some aggregate claims results for these three examples.
We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest. We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.
We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival time distributions depending on the claims that arrived within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the Bonus-Malus feature. We discuss first asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process. Via an appropriate change of measure, ruin probabilities could be computed to a closed form formulae. A…
Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions don't admit a explicit expression. In this work we consider to approximate them by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give optimal order of uniform convergence in the approximation. We apply our results to ruin probability in the classical risk model, paying special attention to mixtures of gamma claim amounts.
Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\cdots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $j\in\mathbb{N}$ and $t\…
We study a simple random walk on Z^2 with constraints on the axis. Motivation comes from physics when particles (a gas for example, see [Dal88]) are submitted to a local field. In our case we assume that the particle evolves freely in the cones but when touching the axis a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axis.
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 $u$, then for $a>0$ the capital jumps up to the level $(1+a)u+C_i$. The ruin probability and the distribution of the time to ruin are determined. We furthermore identify the value of discounted cumulative dividend payments, for the case of a Poisson arriva…
In this paper, we investigate the ruin probabilities of non-homogeneous risk models. By employing martingale method, the Lundberg-type inequalities of ruin probabilities of non-homogeneous renewal risk models are obtained under weak assumptions. In addition, for the periodic and quasi-periodic risk models the adjustment coefficients of the Lundberg-type inequalities are obtained. Finally, examples are presented to show that estimations obtained in this paper are more accurate and the ruin probability in non-homogeneous risk models may be fast decreasing which is impossible for the case of homo…
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.
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, re…
We derive formulas for the moments of the ruin time in a Lévy risk model and use these to determine the asymptotic behavior of the moments of the ruin time as the initial capital tends to infinity. In the special case of the perturbed Cramér-Lundberg model with phase-type or exponentially distributed claims, we explicitly compute the first two moments of the ruin time. All our considerations distinguish between the profitable and the unprofitable setting.
We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to -\infty a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103-125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207-226] for ruin probabilities and the overshoot in random walk and compou…
We prove large deviation principles for two versions of fractional Poisson processes. Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d. light tail claim sizes, and a fractional Poisson claim number process. We conclude with the alternative version where all the random variables are weighted Poisson distributed. Keywords: Mittag Leffler function; renewal process; random time cha
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 large probability $1-ε$ and heavy-tailed with small probability $ε$. We analyze the seminal Gerber-Shiu function coding the joint distribution of the time to ruin, the surplus immediately before ruin, and the deficit at ruin. We derive its value as …
The ruin probability in the classical Brownian risk model can be explicitly calculated for both finite and infinite-time horizon. This is not the case for the simultaneous ruin probability in two-dimensional Brownian risk model. Resorting on asymptotic theory, we derive in this contribution approximations of both simultaneous ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity. Given the interest in proportional reinsurance, we consider in some details the case where the correlation is 1. This model is tractable…
Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. Consider a general branching process generated by $T$ and denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. We treat early generations, that is, fixed generations $j$ which do not depend on $t$. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell …
This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to construct fractional integro-differential equations for the ruin probabilities in collective renewal risk models, with inter-arrival time distributions from the aforementioned family. Gamma-time risk models and fractional Poisson risk models are two specific cases among them, whose ruin probabilities have explicit solutions, when claim sizes distributions e…
Let $\left\{\sum_{i=1}^n λ_i X_i(t), t\in [0,T]\right\}$ be an aggregate Gaussian risk process with $X_i, i\leq n$ independent Gaussian processes satisfying Piterbarg conditions and $λ_i$'s given positive weights. In this paper we derive exact asymptotics of the finite-time ruin probability given by $$\mathbb{P}\left(\sup_{t\in[0,T]}\left(\sum_{i=1}^n λ_i X_i(t)- g(t) \right)>u\right)$$ as $u\to\infty$ for some general trend function $g$. Further, we derive asymptotic results for the finite-time ruin probabilities of risk processes perturbed by an aggregate Gaussian process.
In this chapter we propose an extended concept of the expected discounted penalty function (EDPF) that takes into account new ruin-related random variables. We add to the EDPF, which was introduced in classical papers [Gerber and Shiu (1997), (1998) and Gerber and Landry (1998)], a sequence of expected discounted functions of new record minima reached by a jump of the risk process after ruin. Inspired by results of Huzak et al. (2004) and developpements in fluctuation theory for spectrally negative Lévy processes, we provide a characterization for this extended EDPF in a setting involving a cu…
Consider two insurance companies (or two branches of the same company) that divide between them both claims and premia in some specified proportions. We model the occurrence of claims according to a renewal process. One ruin problem considered is that of the corresponding two-dimensional risk process first leaving the positive quadrant; another is that of entering the negative quadrant. When the claims arrive according to a Poisson process, we obtain a closed form expression for the ultimate ruin probability. In the general case, we analyze the asymptotics of the ruin probability when the init…
In this paper we study the asymptotic decay of finite time ruin probabilities for an insurance company that faces heavy-tailed claims, uses predictable investment strategies and makes investments in risky assets whose prices evolve according to quite general semimartingales. We show that the ruin problem corresponds to determining hitting probabilities for the solution to a randomly perturbed stochastic integral equation. We derive a large deviation result for the hitting probabilities that holds uniformly over a family of semimartingales and show that this result gives the asymptotic decay of…
We consider random walks perturbed at zero which behave like (possibly different) random walks with i.i.d. increments on each half lines and restarts at $0$ whenever they cross that point. We show that the perturbed random walk, after being rescaled in a proper way, converges to a skew Brownian motion whose parameter is defined by renewal functions of the simple random walks and the transition probabilities from $0$.
In this note, we study the ultimate ruin probabilities of a real-valued L{é}vy process X with light-tailed negative jumps. It is well-known that, for such L{é}vy processes, the probability of ruin decreases as an exponential function with a rate given by the root of the Laplace exponent, when the initial value goes to infinity. Under the additional assumption that X has integrable positive jumps, we show how a finer analysis of the Laplace exponent gives in fact a complete description of the bounds on the probability of ruin for this class of L{é}vy processes. This leads to the identification …
In this paper, we adapt the classic Cramér-Lundberg collective risk theory model to a perturbed model by adding a Wiener process to the compound Poisson process, which can be used to incorporate premium income uncertainty, interest rate fluctuations and changes in the number of policyholders. Our study is part of a Master dissertation, our aim is to make a short overview and present additionally some new approximation methods for the infinite time ruin probabilities for the perturbed risk model. We present four different approximation methods for the perturbed risk model. The first method is b…
In this paper we develop a symbolic technique to obtain asymptotic expressions for ruin probabilities and discounted penalty functions in renewal insurance risk models when the premium income depends on the present surplus of the insurance portfolio. The analysis is based on boundary problems for linear ordinary differential equations with variable coefficients. The algebraic structure of the Green's operators allows us to develop an intuitive way of tackling the asymptotic behavior of the solutions, leading to exponential-type expansions and Cramér-type asymptotics. Furthermore, we obtain clo…
The generic case of pensions fund that it is not sufficiently auto financed and it is thoroughly maintained with an external financing effort is considered in this chapter. To represent the unrestricted reserves value process of this kind of funds, a time homogeneous diffusion stochastic process with finite expected time to ruin is proposed. Then it is projected a financial tool that regenerates the diffusion at some level with positive value every time the diffusion hits a barrier placed at the origin. So, the financing effort can be modeled as a renewal-reward process if the regeneration lev…
