We analyze the classical Brownian risk models discussing the approximation of ruin probabilities (classical, γ-reflected, Parisian and cumulative Parisian) for the case that ruin can occur only on specific discrete grids. A practical and natural grid of points is for instance G(1) = {0,1,2,...}, which allows us to study the probability of the ruin on the first day, second day, and so one. For such a discrete setting, there are no explicit formulas for the ruin probabilities mentioned above. In this contribution we derive accurate approximations of ruin probabilities for uniform grids by letting the initial capital to grow to infinity.
Parisian ruin probability in the classical Brownian risk model, unlike the standard ruin probability can not be explicitly calculated even in one-dimensional setup. Resorting on asymptotic theory, we derive in this contribution an asymptotic approximations of both Parisian and cumulative Parisian ruin probability and simultaneous ruin time for the two-dimensional Brownian risk model when the initial capital increases to infinity.
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 $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $ρ\in (-1,1).$ Parisian ruin is defined as a classical ruin that happens over an extended period of time, the so-called time-in-red. We derive exact asymptotics for the non-simultaneous Parisian ruin of the company conditioned on the event of non-simultaneous ruin happening. We are interested in finding asymptotics of such problem as $u \to \infty$ and with the length of time-in-red being of order $\frac{1}{u^2},$ where $u$ represents initial capital for the companie…
Consider a multi-dimensional Brownian motion which models the surplus processes of multiple lines of business of an insurance company. Our main result gives exact asymptotics for the cumulative Parisian ruin probability as the initial capital tends to infinity. An asymptotic distribution for the conditional cumulative Parisian ruin time is also derived. The obtained results on the cumulative Parisian ruin can be seen as generalizations of some of the results derived in Debicki et al (2018, Stochastic Processes and Their Applications). As a particular interesting case, the two-dimensional Brown…
In this paper, we are concerned with the numerical solution of one type integro-differential equation by a probability method based on the fundamental martingale of mixed Gaussian processes. As an application, we will try to simulate the estimation of ruin probability with an unknown parameter driven not by the classical Lévy process but by the mixed fractional Brownian motion.
We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function $P(u)$ for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where $u$ is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of $-\ln P(u)/u$ as $u$ tends to infinity, which depends essentially on the correlation $ρ$ of the two surplus processes. In order to work out the adjustment…
Let $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $ρ\in (-1,1).$ In this contribution we derive precise approximations for cumulative Parisian ruin conditioned on the occurrence of the ruin of the aforementioned two-dimensional Brownian motion, i.e. $$\mathbb{P}\left(\begin{array}{ccc}\int_{[0,1]} \mathbf{1}(W_1^*(s)>u)ds>H_1(u) \\ \int_{[0,1]} \mathbf{1}(W_2^*(t)>au)dt>H_2(u)\end{array}\Bigg{|}\exists_{v,w \in [0,1]}\begin{array}{ccc} W_1(v)-c_1v>u \\ W_2(w)-c_2w>au \end{array}\right).$$ We study the asymptotics fo…
Let $B(t), t\in \mathbb{R}$ be a standard Brownian motion. In this paper, we derive the exact asymptotics of the probability of Parisian ruin on infinite time horizon for the following risk process \begin{align}\label{Rudef} R_u^δ(t)=e^{δt}\left(u+c\int^{t}_{0}e^{-δv}d v-σ\int_{0}^{t}e^{-δv}d B(v)\right),\quad t\geq0, \end{align} where $u\geq 0$ is the initial reserve, $δ\geq0$ is the force of interest, $c>0$ is the rate of premium and $σ>0$ is a volatility factor. Further, we show the asymptotics of the Parisian ruin time of this risk process.
This paper derives the asymptotic behavior of the following ruin probability $$P\{\exists t \in G(δ):B_H(t)-c_1t>q_1u,B_H(t)-c_2t>q_2u\}, \ \ \ u \rightarrow \infty,$$ where $B_H$ is a standard fractional Brownian motion, $c_1,q_1,c_2,q_2>0$ and $G(δ)$ denotes a regular grid $\{0,δ, 2δ,...\}$ for some $δ>0$. The approximation depends on $H$, $δ$ (only when $H\leq 1/2$) and the relations between parameters $c_1,q_1,c_2,q_2$.
This paper derives the asymptotic behavior of $$\mathbb{P} \{ \int\limits_0^\infty \mathbb{I}\Big(B_H(s)-c_1s>q_1u, B_H(s)-c_2s>q_2u\Big)ds>T_u\},\quad u \to \infty,$$ where $B_H$ is a fractional Brownian motion, $c_1,c_2,q_1,q_2>0,\ H \in (0,1), \ T_u \ge 0$ is a measurable function and $\mathbb{I}(\cdot)$ is the indicator function.
Let $B(t), t\in \mathbb{R}$ be a standard Brownian motion. Define a risk process \label{Rudef} R_u^δ(t)=e^{δt}\left(u+c\int^{t}_{0}e^{-δs}d s-σ\int_{0}^{t}e^{-δs}d B(s)\right), t\geq0, where $u\geq 0$ is the initial reserve, $δ\geq0$ is the force of interest, $c>0$ is the rate of premium and $σ>0$ is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability \mathcal{K}_S^δ(u,T_u):=\mathbb{P}\left\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u^δ(s)<0\right\}, S\ge 0, as $u\rightarrow\infty$ where $T_u$ is a bounded function. Further, we show that the Pari…
Let $\{B(t), t\ge 0\}$ be a Brownian motion. Consider the Brownian motion risk model with interest rate collection and tax payment defined by \begin{align}\label{Rudef} \widetilde{U}_γ^δ(t)=\widetilde{X}^δ(t)-γ\sup_{s\in[0,t]} \left(\widetilde{X}^δ(s)e^{δ(t-s)}-ue^{δ(t-s)}\right),t\ge0, \end{align} with $$\widetilde{X}^δ(t)=ue^{δt}+c \int_0^t e^{δ(t-v)}dv-σ\int_0^t e^{δ(t-v)}dB(v),$$ where $c>0,γ\in [0,1)$ and $δ\in \mathbb{R}$ are three given constants. When $δ=0$ and $γ\in (0,1)$ this is the risk model introduced from Albrecher and Hipp in \cite{AH2007} where the ruin probability in the infi…
In this paper we propose new iterative algorithm of calculating the joint distribution of the Parisian ruin time and the number of claims until Parisian ruin for the classical risk model. Examples are provided when the generic claim size is exponentially distributed.
In this paper we derive the exact asymptotics of the probability of Parisian ruin for self-similar Gaussian risk processes. Additionally, we obtain the normal approximation of the Parisian ruin time and derive an asymptotic relation between the Parisian and the classical ruin times.
Lundberg-type inequalities for ruin probabilities of non-homogeneous risk models are presented in this paper. By employing martingale method, the upper bounds of ruin probabilities are obtained for the general risk models under weak assumptions. In addition, several risk models, including the newly defined united risk model and quasi-periodic risk model with interest rate, are studied.
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 $(W_1(s), W_2(t)), s,t\ge 0$ be a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $ρ\in (-1,1)$ and define the joint survival probability of both supremum functionals $π_ρ(c_1,c_2; u, v)$ by $$π_ρ(c_1,c_2; u, v)=\mathbb{P}\left(\sup_{s \in [0,1]} \left(W_1(s)-c_1s\right)>u,\sup_{t \in [0,1]} \left(W_2(t)-c_2t\right)>v\right) ,$$ where $c_1,c_2 \in \mathbb{R}$ and $u,v$ are given positive constants. Approximation of $π_ρ(c_1,c_2; u, v) $ is of interest for the analysis of ruin probability in bivariate Brownian risk model as well as in the study of …
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.
In this paper we investigate the Parisian ruin probability for an integrated Gaussian process. Under certain assumptions, we find the Parisian ruin probability and the classical ruin probability are on the log-scale asymptotically the same. Moreover, for any small interval required by the risk process staying below level zero, the Parisian ruin probability and the classical one are the same also in the premise asymptotic behavior. Furthermore, we derive an approximation of the conditional ruin time.
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 expression for the corresponding infnite-time Parisian ruin probability as a limiting case. In order to obtain more analytic results, we employ a conditioning argument and derive a new expression for the classic infinite-time ruin probability in the dual risk…
In this paper we consider some generalizations of the classical d-dimensional Brownian risk model. This contribution derives some non-asymptotic bounds for simultaneous ruin probabilities of interest. In addition, we obtain non-asymptotic bounds also for the case of general trend functions and convolutions of our original risk model.
We investigate the asymptotic of ruin probabilities when the company combines the life- and non-life insurance businesses and invests its reserve into a risky asset with stochastic volatility and drift driven by a two-state Markov process. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities.
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.
This paper studies risk balancing features in an insurance market by evaluating ruin probabilities for single and multiple components of a multivariate compound Poisson risk process. The dependence of the components of the process is induced by a random bipartite network. In analogy with the non-network scenario, a network ruin parameter is introduced. This random parameter, which depends on the bipartite network, is crucial for the ruin probabilities. Under certain conditions on the network and for light-tailed claim size distributions we obtain Lundberg bounds and, for exponential claim size…
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.
In this note we consider the two-dimensional risk model introduced in Avram et al. \cite{APP08} with constant interest rate. We derive the integral-differential equations of the Laplace transforms, and asymptotic expressions for the finite time ruin probabilities with respect to the joint ruin times $T_{\rm max}(u_1,u_2)$ and $T_{\rm min}(u_1,u_2)$ respectively.
This paper investigates ruin probabilities for a two-dimensional fractional Brownian risk model with a proportional reinsurance scheme. We focus on joint and simultaneous ruin probabilities in a finite-time horizon. The risk processes of both insurance and reinsurance companies are composed of a large number of i.i.d. sub-risk processes, representing independent businesses. We derive the asymptotics as the initial capital tends to infinity.
We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals.
In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold $r$. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level $b$), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the $q$-potential measure of the process killed at Parisian ruin. T…
We explicitly find the rate of exponential long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows to reduce the pro blem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.
We used the random walk to model the problem of reserves. The classic case of a stochastic process is the example of random walks, which are used to study a set of phenomena and, particularly, as in this article, models of reserves evolution. Random walks also allow the construction of significant complex systems and are also used as an instrument of analysis, being used in the sense of giving a theoretical characteristic to other types of systems. Our goal is primarily to study reserves to see how to ensure that pension funds are sustainable. This classic approach to the study of pension fund…
We apply the theory of linear recurrence sequences to find an expression for the ultimate ruin probability in a discrete-time risk process. We assume the claims follow an arbitrary distribution with support $\{0,1,\ldots,m\}$, for some integer $m\ge2$. The method requires to find the zeroes of an $m$ degree polynomial and to solve a system of $m$ linear equations. An approximation is derived from the exact ruin formula and several numerical results and plots are provided as examples.
We study the probability of ruin before time $t$ for the family of tempered stable Lévy insurance risk processes, which includes the spectrally positive inverse Gaussian processes. Numerical approximations of the ruin time distribution are derived via the Laplace transform of the asymptotic ruin time distribution, for which we have an explicit expression. These are benchmarked against simulations based on importance sampling using stable processes. Theoretical consequences of the asymptotic formulae are found to indicate some potential drawbacks to the use of the inverse Gaussian process as a …
We investigate the asymptotic of ruin probabilities when the company invests its reserve in a risky asset with a switching regime price. We assume that the asset price is a conditional geometric Brownian motion with parameters modulated by a Markov process with a finite number of states. Using the technique of the implicit renewal theory we obtain the rate of convergence to zero of the ruin probabilities as the initial capital tends to infinity.
We consider two insurance companies with endowment processes given by Brownian motions with drift. The firms can collaborate by transfer payments in order to maximize the probability that none of them goes bankrupt. We show that pushing maximally the company with less endowment is the optimal strategy for the collaboration if the Brownian motions are correlated and the transfer rate can exceed the drift rates. Moreover, we obtain an explicit formula for the minimal ruin probability in case of perfectly positively correlated Brownian motions where we also allow for different diffusion coefficie…
The discrete time risk model with two seasons and dependent claims is considered. An algorithm is created for computing the values of the ultimate ruin probability. Theoretical results are illustrated with numerical examples.
Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.
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.
We study the discrete time risk process modelled by the skip-free random walk and we derive the results connected to the ruin probability, such as crossing the fixed level, for this kind of process. We use the method relying on the classical ballot theorems to derive these results and compare them to the results obtained for the continuous time version of the risk process. We further generalize this model by adding the perturbation and, still relying on the skip-free structure of that process, we generalize the previous results on crossing the fixed level for the generalized discrete time risk…
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…
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.
We study the rough asymptotic behaviour of a general economic risk model in a discrete setting. Both financial and insurance risks are taken into account. Loss during the first $n$ years is modelled as a random variable $B_1+A_1B_2+\ldots+A_1\ldots A_{n-1}B_n$, where $A_i$ corresponds to the financial risk of the year $i$ and $B_i$ represents the insurance risk respectively. Risks of the same year $i$ are not assumed to be independent. The main result shows that ruin probabilities exhibit power law decay under general assumptions. Our objective is to give a complete characterisation of the rel…
We investigate the Levy insurance risk model with tax under Cramér's condition. A direct analogue of Cramér's estimate for the probability of ruin in this model is obtained, together with the asymptotic distribution, conditional on ruin occurring, of several variables of interest related to ruin including the surplus immediately prior to ruin (undershoot) and shortfall at ruin (overshoot). We also compute the present value of all tax paid conditional on ruin occurring. The proof involves first transferring results from the model with no tax to the reflected process, and from there to the model…
We introduce the concept of cumulative Parisian ruin, which is based on the time spent in the red by the underlying surplus process. Our main result is an explicit representation for the distribution of the occupation time, over a finite-time horizon, for a compound Poisson process with drift and exponential claims. The Brownian ruin model is also studied in details. Finally, we analyze for a general framework the relationships between cumulative Parisian ruin and classical ruin, as well as with Parisian ruin based on exponential implementation delays.
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…
We study the asymptotic of the ruin probability for a process which is the solution of linear SDE defined by a pair of independent Lévy processes. Our main interest is the model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let $β>0$ be the root of the cumulant-generating function $H$ of the increment of the log price process $V$. We show that the ruin probability admits the exact asymptotic $Cu^{-β}$ as the initial capital $u\to\infty$ assuming only that the law of $V_T$ is non-arithmetic without any further assumptio…
In Gambler's Ruin when both players start with the same amount of money, we show the playing time stochastically increases when the games are made more fair. We give two different arguments for this fact that extend results from \cite{Pek2021}. We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases as the drift moves closer to zero; this result is not easily obtainable from available explicit formulas for the density.
This paper develops asymptotics and approximations for ruin probabilities in a multivariate risk setting. We consider a model in which the individual reserve processes are driven by a common Markovian environmental process. We subsequently consider a regime in which the claim arrival intensity and transition rates of the environmental process are jointly sped up, and one in which there is (with overwhelming probability) maximally one transition of the environmental process in the time interval considered. The approximations are extensively tested in a series of numerical experiments.
This paper obtains an asymptotic formula for the finite-time ruin probability of the compound nonhomogeneous Poisson risk model with a constant interest force, in which the claims are conditionally independent random variables with a common subexponential distribution. The paper also obtains some asymptotic relations of randomly weighted sums $\sum_{i=1}^n θ_iX_i$, in which the weights $θ_i$ $i=1,2,\cdots, n$ are positive random variables which are bounded above and the primary random variables $X_i$, $i=1,2,\cdots,n$ are conditionally independent and follow subexponential distributions.
We deal with a generalization of the classical risk model when an insurance company gets additional funds whenever a claim arrives and consider some practical approaches to the estimation of the ruin probability. In particular, we get an upper exponential bound and construct an analogue to the De Vylder approximation for the ruin probability. We compare results of these approaches with statistical estimates obtained by the Monte Carlo method for selected distributions of claim sizes and additional funds.
Inspired by works of Landriault et al. \cite{LRZ-0, LRZ}, we study discounted penalties at ruin for surplus dynamics driven by a spectrally negative Lévy process with Parisian implementation delays. To be specific, we study the so-called Gerber-Shiu functional for a ruin model where at each time the surplus process goes negative, an independent exponential clock with rate $q>0$ is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of the…
In this paper we study the draw-down related Parisian ruin problem for spectrally negative Lévy risk processes. We introduce the draw-down Parisian ruin time and solve the corresponding two-sided exit time via excursion theory. We also obtain an expression of the potential measure for the process killed at the draw-down Parisian time. As applications, new results are obtained for spectrally negative Lévy risk process with dividend barrier and Parisian ruin.
