Chuancun Yin

The time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process perturbed by diffusion

Abstract The paper studies the joint distribution of the time of ruin, the surplus prior to ruin and the deficit at ruin for the classical risk process that is perturbed by diffusion. We prove that the expected discounted penalty satisfies an integro-differential equation of renewal type, the solution of which can be expressed as a convolution formula. The asymptotic behaviour of the expected discounted penalty as the initial capital tends to infinity is discussed.

Exit problems for jump processes with applications to dividend problems

This paper investigates the first passage times to flat boundaries for hyper-exponential jump (diffusion) processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot), the joint distribution of the process and running suprema (infima) are obtained. The processes recover many models appearing in the literature such as the compound Poisson risk models, the diffusion perturbed compound Poisson risk models, and their dual models. As applications, we present explicit expressions of the dividend formulae for barrier strategy and threshold strategy.

Optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes: An alternative approach

The optimal dividend problem proposed in de Finetti [1] is to find the dividend-payment strategy that maximizes the expected discounted value of dividends which are paid to the shareholders until the company is ruined. Avram et al. [9] studied the case when the risk process is modelled by a general spectrally negative Levy process and Loeffen [10] gave sufficient conditions under which the optimal strategy is of the barrier type. Recently Kyprianou et al. [11] strengthened the result of Loeffen [10] which established a larger class of Levy processes for which the barrier strategy is optimal among all admissible ones. In this paper we use an analytical argument to re-investigate the optimality of barrier dividend strategies considered in the three recent papers.

The first passage time problem for mixed-exponential jump processes with applications in insurance and finance

This paper studies the first passage times to constant boundaries for mixed-exponential jump diffusion processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot) are obtained. As applications, we present explicit expression of the Gerber-Shiu functions for surplus processes with two-sided jumps, present the analytical solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms, and give a closed-form expression on the price of the zero-coupon bond under a structural credit risk model with jumps.

On optimality of the barrier strategy for a general Lévy risk process

We consider the optimal dividend problem for the insurance risk process in a general Levy process setting. The objective is to find a strategy which maximizes the expected total discounted dividends until the time of ruin. We give sufficient conditions under which the optimal strategy is of barrier type. In particular, we show that if the Levy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy. This approach was inspired by the work of Avram et al. [F. Avram, Z. Palmowski, M.R. Pistorius, On the optimal dividend problem for a spectrally negative Levy process, The Annals of Applied Probability 17 (2007) 156-180], Loeffen [R. Loeffen, On optimality of the barrier strategy in De Finettis dividend problem for spectrally negative Levy processes, The Annals of Applied Probability 18 (2008) 1669-1680] and Kyprianou et al. [A.E. Kyprianou, V. Rivero, R. Song, Convexity and smoothness of scale functions with applications to De Finettis control problem, Journal of Theoretical Probability 23 (2010) 547-564] in which the same problem was considered under the spectrally negative Levy processes setting.

On occupation times for a risk process with reserve-dependent premium

Consider a risk reserve process under which the reserve can generate interest. For constants a and b such that a<b, we study the occupation time T a,b (t), which is the total length of the time intervals up to time t during which the reserve is between a and b. We first present a general formula for piecewise deterministic Markov processes, which will be used for the computation of the Laplace transform of T a,b (t). Explicit results are then given for the special case that claim sizes are exponentially distributed. The classical model is discussed in detail. *Research supported by RGC of Hong Kong SAR (Grant No. HKBU/2075/98P). **Research supported by NSF of China(Grant No. 19801020).

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory

We consider the spectrally negative Lévy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson’s formula is provided.

The joint distribution of the hitting time and place to a sphere or spherical shell for Brownian motion with drift

Let Xt be a standard d-dimensional Brownian motion with drift c started at a fixed X0, and let T be the hitting time for a sphere or concentric spherical shell. By using an appropriate martingale, a Laplace-Gegenbauer transform of the joint distribution of T and XT is determined.

A perturbed risk process compounded by a geometric Brownian motion with a dividend barrier strategy

Abstract In this paper, we consider a perturbed risk process (in which the inter-occurrence times are generalized Erlang(n)-distributed) compounded by a geometric Brownian motion. Integro-differential equations with certain boundary conditions for the moment-generating function and the mth moment of the present value of all dividends until ruin are derived. We also derive integro-differential equations with boundary conditions for the Gerber–Shiu function. Some special cases are considered in details.

Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process

We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.