A proof of a conjecture in the Cramer-Lundberg model with investments

In the classical Cramér-Lundberg model, if the claim sizes have finite exponential moments, then it is well-known that the ruin probability decays exponentially as the initial surplus increases; see for instance the books by Asmussen [1] and Embrechts et. al [2]. For the case of heavy-tailed claims there also exists numerous results in the literature. However, when the insurance company invests in a risky asset, for example a stock, whose price is described by a geometric Brownian motion with drift a > 0 and volatility σ > 0, then the probability of ruin either decays algebraically as the initial surplus increases or the ruin is certain, provided the claim size is exponentially distributed. This result was shown by Frolova et. al [3]. Under the assumption that the claim size distributions have moment generating functions defined on a neighborhood of the origin, Constantinescu and Thommann [4] proved that if the probability of ruin decays as the initial capital u → ∞, then ρ = 2a σ 2 > 1, and that if 1 < ρ < 2, then the probability of ruin decays algebraically as the initial capital u → ∞. Furthermore, they conjectured that if ρ ≤ 1, then the ruin probability ψ(u) = 1 for all u ≥ 0.

In this paper, our main goal is to prove that the conjecture is true. This work was motivated by a paradox of risk without the possibility of reward discussed by Steele [5]. In the setting of this paradox of risk, the price of a risky asset is modeled by a geometric Brownian motion with an expected return rate a. Steele pointed out that if ρ < 1, the price of the risky asset approaches to zero with probability one, despite the fact that the expected value goes to positive infinity at an exponential rate. We observe that if the price of our risky asset is very close to zero, then even a small jump will trigger the ruin. Similarly, if the price of the risky asset drops below a threshold with probability one and if there is a positive probability that the price of the risky asset may have jumps larger than the threshold, then the ruin occurs almost surely. If the jump is modeled by a compound Poisson process, then this leads to the conjecture that is discussed in this paper.

We first recall the Cramér-Lundberg model with investments. The risk process is given by

or

where W t is the Wiener process (standard Brownian motion), N(t) is a Poisson process with parameter λ, and the claim sizes ξ i ; i = 1, 2, 3, …, are independent, identically distributed random variables, having the density function p(x), with positive mean µ and finite variance. c is the fixed rate of premium and X 0 is the initial capital. P t = N (t) j=1 ξ j . The capital X t is continuously invested in a risky asset, with relative price increments dX t = aX t dt + σX t dW t , where a > 0 and σ > 0 are the drift and volatility of the returns of the asset. Our paper is organized as follows. By assuming there is a cap on the claim size, in Section 2, we prove two important results that (1) the probability of ruin has at least an algebraic decay rate if 2a/σ 2 > 1 and (2) the price of the risky asset will drop below a threshold with probability one for all initial capital X 0 = u, if 2a/σ 2 ≤ 1. In Section 3, we prove that the conjecture is true by coupling the stochastic processes with and without the assumption on the claim sizes.

We will assume the claim size is bounded by a constant M > 0 through the entire section. In insurance, M can be understood as the limit or cap of a policy. Let T u * = inf{t > 0; X t < u * } be the first time that X t < u * , and let

be the probability of ruin, where 0 ≤ u * < u. If u * = 0, we denote the probability of ruin by ψ(u). We will discuss the probability of ruin on the Cramér-Lundberg model with investments based on (1) ρ > 1, (2) ρ = 1 and (3) ρ < 1. We first prove the following Lemma 2.1. Let X t be a stochastic process that satisfies (1.2), if 0 ≤ v ≤ u. then

Proof. We first derive a strong solution for (1.2). Let Y t = exp{( σ 2 2 -a)t -σW t }. By Itô’s formula [6], dX t Y t = X t dY t + Y t dX t + dX t dY t , and simple calculation yields

Note that Z t also satisfies (1.2) with initial condition Z 0 = v. Hence

Our main tool is Itô’s formula for semimartingales with a jump part. Let t 1 < t 2 < t 3 < … be the times where the Poisson process N(t) has a jump discontinuity. Then the jump discontinuities for P t are also at t i with jump size ξ i . Following the notations on P. 43 [6], for t > 0, and a Borel set U in R, we let

where δ t i is the Dirac δ-function centered at t i (probability measure concentrated at one point t i ). It follows that

and therefore

It is well-known, see e.g. P. 60 and P. 65 [6], that there exists a continuous process Np ((

] defines a measure, n p (dtdx), called the mean (intensity) measure of N p (dtdx) and it is given by n p (dtdx) = λp(x)dtdx.

The equation (1.1) can be rewritten as

(2.8) By (2.3), the equation (2.8) has a strong solution for each fixed initial condition and it is a

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