Ruin probability in a two-dimensional model with correlated Brownian motions
Peter Grandits, Maike Klein
备注：
17 pages, 17 figures, Both authors gratefully acknowledge the support by the Austrian Science Fund (Fonds zur Förderung der wissenschaftlichen Forschung) under grant P30864-N35
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 coefficients.
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 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 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.
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…
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 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…
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.
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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…
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This paper derives an exact asymptotic expression for \[ \mathbb{P}_{\mathbf{x}_u}\{\exists_{t\ge0} \mathbf{X}(t)- \boldsymbolμt\in \mathcal{U} \}, \ \ {\rm as}\ \ u\to\infty, \] where $\mathbf{X}(t)=(X_1(t),\ldots,X_d(t))^\top,t\ge0$ is a correlated $d$-dimensional Brownian motion starting at the point $\mathbf{x}_u=-\boldsymbolαu$ with $\boldsymbolα\in \mathbb{R}^d$, $\boldsymbolμ \in \mathbb{R}^d$ and $\mathcal{U}=\prod_{i=1}^d [0,\infty)$. The derived asymptotics depends on the solution of an underlying multidimensional quadratic optimization problem with constraints, which leads in some c…
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Anomalous scaling limit of a Brownian particle in a log-correlated potential
CNRS & Laboratoire Jacques-Louis Lions, Sorbonne Université（法国国家科学研究中心与雅克-路易·利翁斯实验室，索邦大学）；Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Department of Mathematics, University of Pennsylvania（宾夕法尼亚大学数学系）；Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
The purpose of this paper is to study the convergence in distribution of two subsequences of the signed cubic variation of the fractional Brownian motion with Hurst parameter $H=1/6$. We prove that, under some conditions on both subsequences, the limit is a two-dimensional Brownian motion whose components may be correlated and we find explicit formulae for its covariance function.
In this paper, we study the martingale property for a Scott correlated stochastic volatility model, when the correlation coefficient between the Brownian motion driving the volatility and the one driving the asset price process is arbitrary. For this study we verify the martingale property by using the necessary and sufficient conditions given by Bernard \emph{et al.} \cite{Bernard}. Our main results are to prove that the price process is a true and uniformly integrable martingale if and only if $ρ\in [-1,0]$ for two transformations of Brownian motion describing the dynamics of the underling a…
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is the fractional Brownian motion. We have to use two radically different models for both cases ${1\over2}\leq H<1$ and $0<H<{1\over2}$. This result provides an algorithm for the simulation of the fractional Brownian motion, which appears to be quite efficient.
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general nonlinear boundaries. The jump process can be any integer-valued process and jump sizes can have general distributions. Moreover, the jump sizes can be even correlated and/or non-identically distributed. The numerical algorithm is straightforward and easy to implement. Some numerical examples are presented.
