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 $c_1,q_1,c_2,q_2$.
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…
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 lettin…
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 $$\mathbb{P} \{ \int\limits_0^\infty \mathbb{I}\Big(B_H(s)-c_1s>q_1u, B_H(s)-c_2s>q_2u\Big)ds>T_u\},\quad u \to \infty,$$ 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…
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 …
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 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 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…
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.
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 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.
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.
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 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.
The study deals with the ruin problem when an insurance company having two business branches, life insurance and non-life insurance, invests its reserve into a risky asset with the price dynamics given by a geometric Brownian motion. We prove a result on smoothness of the ruin probability as a function of the initial capital and obtain for it an integro-differential equation understood in the classical sense. For the case of exponentially distributed jumps we show that the survival probability is a solution of an ordinary differential equation of the 4th order. Asymptotic analysis of the latte…
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 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…
For a risk process $R_u(t)=u+ct-X(t), t\ge 0$, where $u\ge 0$ is the initial capital, $c>0$ is the premium rate and $X(t),t\ge 0$ is an aggregate claim process, we investigate the probability of the Parisian ruin \[ \mathcal{P}_S(u,T_u)=\mathbb{P}\{\inf_{t\in[0,S]} \sup_{s\in[t,t+T_u]} R_u(s)<0\}, \] with a given positive constant $S$ and a positive measurable function $T_u$. We derive asymptotic expansion of $\mathcal{P}_S(u,T_u)$, as $u\to\infty$, for the aggregate claim process $X$ modeled by Gaussian processes. As a by-product, we derive the exact tail asymptotics of the infimum of a stand…
We give asymptotic analysis for probability of absorbtion $\mathsf{P}(τ_0\le T)$ on the interval $[0,T]$, where $ τ_0=\inf\{t:X_t=0\}$ and $X_t$ is a nonnegative diffusion process relative to Brownian motion $B_t$, dX_t&=μX_tdt+σX^γ_tdB_t. X_0&=K>0 Diffusion parameter $σx^γ$, $γ\in [{1/2},1)$ is not Lipschitz continuous and assures $\mathsf{P}(τ_0>T)>0$. Our main result: $$ \lim\limits_{K\to\infty} \frac{1}{K^{2(1-γ)}}\log\mathsf{P}(τ_{0}\le T) =-\frac{1}{2\E M^2_T}, $$ where $ M_T=\int_0^Tσ(1-γ)e^{-(1-γ)μs}dB_s $. Moreover we describe the most likely path to absorbtion of the normed process $…
In this note we find a formula for the supremum distribution of spectrally positive or negative Lévy processes with a broken linear drift. This gives formulas for ruin probabilities in the case when two insurance companies (or two branches of the same company) divide between them both claims and premia in some specified proportions. As an example we consider gamma Lévy process, $α$-stable Lévy process and Brownian motion. Moreover we obtain identities for Laplace transform of the distribution for the supremum of Lévy processes with randomly broken drift and on random intervals.
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 imp…
We consider the simple random walk on the $N$-dimensional integer lattice from the perspective of evaluating asymptotically the duration of play in the multidimensional gambler\apost s ruin problem. We show that, under suitable rescalings, all $p$-moments of exit-times from balls in the $L$-infinity metric, and all $p$-moments of partial-maxima values in this metric, possess associated asymptotic limit expressions, admitting two representations each. We derive for this purpose multidimensional refinements of the corresponding two-folded extension of Erd\H os-Kac theorem, which we revisit to th…
We consider a surplus process of drifted fractional Brownian motion with the Hurst index $H>1/2$, which appears as a functional limit of drifted compound Poisson risk models with correlated claims, and this is a kind of representation of a surplus with a long memory. Our interest is to construct confidence intervals of the ruin probability of the surplus when the volatility parameter is unknown. We will obtain the derivative of the ruin probability w.r.t. the volatility parameter via Malliavin calculus, and apply the delta method to identify the asymptotic distribution of an estimated ruin pro…
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 consider the mixed ratcheting-periodic dividend strategies for spectrally negative Lévy risk model, in which dividend payments can both be made continuously without falling and discretely at the jump times of an independent Poisson process. The expected net present value(NPV) of dividends paid up to ruin and the Laplace transform of the ruin time are obtained by using Lévy fluctuation theory. All the results are expressed in terms of scale functions. Finally, numerical results for Brownian motion with drift are given.
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 study the ruin problem with investment in a general framework where the business part X is a L{é}vy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a L{é}vy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a…
This paper considers a Cramér-Lundberg risk setting, where the components of the underlying model change over time. These components could be thought of as the claim arrival rate, the claim-size distribution, and the premium rate, but we allow the more general setting of the cumulative claim process being modelled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs we resample the model components from a finite number of $d$ settings. It results in a setup that is particularly suited to describe situations…
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 consider an insurance company in the case when the premium rate is a bounded non-negative random function $c_\zs{t}$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return $a$ and volatility $σ>0$. If $β:=2a/σ^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability $Ψ(u)$ as the initial endowment $u$ tends to infinity, i.e. we show that $C_*u^{-β}\leΨ(u)\le C^*u^{-β}$ for sufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{γt}$ with $γ\le 0$ we find the exact asymptotics of the ruin p…
For a given centered Gaussian process with stationary increments $\{X(t), t\geq 0\}$ and $c>0$, let $$ W_γ(t)=X(t)-ct-γ\inf_{0\leq s\leq t}\left(X(s)-cs\right), \quad t\geq 0$$ denote the $γ$-reflected process, where $γ\in (0,1)$. This process is introduced in the context of risk theory to model surplus process that include tax payments of loss-carry forward type.In this contribution we derive asymptotic approximations of both the ruin probability and the joint distribution of first and last passage times given that ruin occurs. We apply our findings to the cases with $X$ being the multiplex f…
In this paper, we discuss the Cramér-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift $a$ and volatility $σ> 0.$ By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/σ^2 > 1$. More importantly, without this assumption, we show that the probability of ruin is certain for all initial capital $u$, if $2a/σ^2 \le 1$.
Aiming for more realistic optimal dividend policies, we consider a stochastic control problem with linearly bounded control rates using a performance function given by the expected present value of dividend payments made up to ruin. In a Brownian model, we prove the optimality of a member of a new family of control strategies called delayed linear control strategies, for which the controlled process is a refracted diffusion process. For some parameters specifications, we retrieve the strategy initially proposed by Avanzi & Wong (2012) to regularize dividend payments, which is more consistent w…
Let $\{X(t),t\ge0\}$ be a centered Gaussian process and let $γ$ be a non-negative constant. In this paper we study the asymptotics of $P\{\underset{t\in [0,\mathcal{T}/u^γ]}\sup X(t)>u\}$ as $u\to\infty$, with $\mathcal{T}$ an independent of $X$ non-negative random variable. As an application, we derive the asymptotics of finite-time ruin probability of time-changed fractional Brownian motion risk processes.
Define a gamma-reflected process W_γ(t)=Y_H(t)-γ\inf_{s\in[0,t]}Y_H(s), t\ge0 with input process {Y_H(t), t\ge 0} which is a fractional Brownian motion with Hurst index H\in (0,1) and a negative linear trend. In risk theory {u-W_γ(t), t\ge 0} is referred to as the risk process with tax payments of a loss-carry-forward type. For various risk processes numerous results are known for the approximation of the first and last passage times to 0 (ruin times) when the initial reserve $u$ goes to infinity. In this paper we show that for the gamma-reflected process the conditional (standardized) first a…
This paper considers a variant of the classical Cramér-Lundberg model that is particularly appropriate in the credit context, with the distinguishing feature that it corresponds to a finite number of obligors. The focus is on computing the ruin probability, i.e., the probability that the initial reserve, increased by the interest received from the obligors and decreased by the losses due to defaults, drops below zero. Besides an exact analysis (in terms of transforms) of this ruin probability, also an asymptotic analysis is performed, including an efficient importance-sampling based simulation…
We consider a discrete-time version of the popular optimal dividend pay-out problem in risk theory. The novel aspect of our approach is that we allow for a risk averse insurer, i.e., instead of maximising the expected discounted dividends until ruin we maximise the expected utility of discounted dividends until ruin. This task has been proposed as an open problem in H. Gerber and E. Shiu (2004). The model in a continuous-time Brownian motion setting with the exponential utility function has been analysed in P. Grandits, F. Hubalek, W. Schachermayer and M. Zigo (2007). Nevertheless, a complete …
Define a $γ$-reflected process $W_γ(t)=Y_H(t)-γ\inf_{s\in[0,t]}Y_H(s)$, $t\ge0$ with input process $\{Y_H(t), t\ge 0\}$ which is a fractional Brownian motion with Hurst index $H\in (0,1)$ and a negative linear trend. In risk theory $R_γ(t)=u-W_γ(t), t\ge0$ is referred to as the risk process with tax of a loss-carry-forward type, whereas in queueing theory $W_1$ is referred to as the queue length process. In this paper, we investigate the ruin probability and the ruin time of the risk process $R_γ, γ\in [0,1]$ over a surplus dependent time interval $[0, T_u]$.
We consider a one dimensional random-walk-like process, whose steps are centered Gaussians with variances which are determined according to the sequence of arrivals of a Poisson process on the line. This process is decorated by independent random variables which are added at each step, but do not get accumulated. We study the probability that such process, conditioned to form a bridge, stays below a curve which grows at most polynomially fast away from the boundaries, with exponent less than one half. Both bounds and asymptotics are derived. These estimates are used in the manuscript "the stru…
We supplement a very recent paper of G. Markowsky concerned with the expected exit times of Brownian motion from planar domains. Besides the use of conformal mapping, we apply results from potential theory. We treat the case of a wedge-shaped region exactly, subsuming an example of Markowsky. A number of other results are presented, including for a half disc, $n$-grams, and a variety of other regions.
