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A generalization of the Cramér–Lundberg risk model perturbed by a diffusion is proposed. Aggregate claims of an insurer follow a compound Poisson process and premiums are collected at a constant rate with additional random fluctuation. The insurer is allowed to invest the surplus into a risky asset with volatility dependent on the level of the investment, which permits the incorporation of rational investment strategies as proposed by Berk and Green (2004). The return on investment is modulated by a Markov process which generalizes previously studied settings for the evolution of the interest rate in time. The Gerber–Shiu expected penalty–reward function is studied in this context, including ruin probabilities (a first-passage problem) as a special case. The second order integro-differential system of equations that characterizes the function of interest is obtained. As a closed-form solution does not exist, a numerical procedure based on the Chebyshev polynomial approximation through a collocation method is proposed. Finally, some examples illustrating the procedure are presented.

The risk process presented by Gerber (1970) extends the classical model of risk theory introducing a Brownian diffusion. The total claims follow a compound Poisson process {Xt,t≥0}{Xt,t≥0} with Lévy measure View the MathML sourceλf(x)dx, λλ being the intensity of arrivals and ff the density of jumps. The collection of premiums is driven by a Wiener process View the MathML sourceWtc independent of XtXt with drift cc and volatility σσ, thus the perturbed risk process with initial surplus uu is given by equation(1) View the MathML sourcedRt=cdt+σdWtc−dXt,R0=u. Turn MathJax on This process has been considered by Dufresne and Gerber (1991) where a defective renewal equation was derived for the probability of ruin ψ(u)=Pr(τ<∞)ψ(u)=Pr(τ<∞) where τ=inf{t≥0:Rt<0}τ=inf{t≥0:Rt<0}. A review of the research on this type of process can be found in Asmussen and Albrecher (2010), Chapter 11. Generalizations of the model are treated in Li and Garrido (2005), Sarkar and Sen (2005), and Morales (2007), whereas Ren (2005) gives explicit formulae to calculate the ruin probability and related quantities for phase-type distributed claims. Let us now allow the insurer to invest the reserves UtUt into an asset with time-dependent Markov-modulated return rate (drift) ΔtΔt and volatility κ(Ut)κ(Ut), that possibly depends on the amount invested UtUt, driven by a Wiener process View the MathML sourceWtI independent of the risk process RtRt equation(2) View the MathML sourcedUt=(Δtdt+κ(Ut)dWtI)Ut+dRt,U0=R0=u. Turn MathJax on The drift parameter ΔtΔt is governed by a finite state homogeneous Markov process with state space {δ1,…,δn}{δ1,…,δn}, intensity matrix Q=(qij)n×nQ=(qij)n×n and initial state δiδi. For example, ΔtΔt can be used to model the risk free rate announced by a central bank that evolves according to the Markov process by, for instance, 25 basis point jumps. The state space would be in this case e.g., 1.00%,1.25%,1.50%,1.75%,2.00%,…,9.00%.1.00%,1.25%,1.50%,1.75%,2.00%,…,9.00%. Turn MathJax on This environment offers considerable versatility in capturing the evolution of interest rates since any diffusion model to forecast the yield curve can be approximated arbitrarily well by continuous time Markov chains; see Kushner and Dupuis (1992). Variation of the volatility according to the size of the funds invested is justified, for example, by Berk and Green (2004) as an implication of their study of the performance of mutual funds and resulting rational capital flows. A particular shape of κκ suggested in the cited paper, View the MathML sourceκ(u)=σru, yields a surplus process in the form of an affine diffusion that was studied by Avram and Usabel (2008) in this context. Many practical ideas support a fund-dependent volatility, for instance the possibility to obtain more efficient portfolios, due to transaction costs, when more money is available. Model (2) is a generalization of the process considered most frequently in the literature where the return rate and the volatility are constant in time, Δt=δΔt=δ, κ(⋅)=σrκ(⋅)=σr, like in Paulsen (1993), Paulsen and Gjessing (1997), Wang (2001), Ma and Sun (2003), Gaier and Grandits (2004), Grandits (2005), Cai and Yang (2005) and Wang and Wu (2008). The stochastic differential equation (2) can be arranged into equation(3) View the MathML sourcedUt=(c+ΔtUt)dt+σ2+κ2(Ut)Ut2dWt−dXt Turn MathJax on with initial condition (U0,Δ0)=(u,δi)(U0,Δ0)=(u,δi). The expected penalty–reward function (see Gerber and Landry (1998)) is introduced equation(4) View the MathML sourceϕti(u)=E[π(Uτ)I(τ≤t)+P(Ut)I(τ>t)∣U0=u,Δ0=δi] Turn MathJax on where τ=inf{s≥0:Us<0}τ=inf{s≥0:Us<0}. If ruin occurs before the time horizon tt, the penalty π(Uτ)π(Uτ) applies to the overshoot UτUτ at the ruin. Otherwise, the reward function P(Ut)P(Ut) applies to the reserves at time tt. The concept of the expected penalty–reward function presented in Gerber and Shiu (1997) and Gerber and Shiu (1998) is a quite general framework comprising several quantities of interest as a special case, such as the time to ruin, the amount at and immediately prior to ruin or survival probabilities. For further analysis the smoothed version of the function View the MathML sourceϕti(u) will be considered, namely its Laplace–Carson transform in time defined as View the MathML sourceΥαi(u)=∫0∞αe−αtϕti(u)dt. Turn MathJax on Further, letting HαHα be an exponentially distributed random variable with parameter αα, the former expression may be viewed as a penalty–reward function with an exponentially killed time horizon (see expression (6) in Avram and Usabel (2008)) equation(5) View the MathML sourceΥαi(u)=∫0∞αe−αtϕti(u)dt=E(ϕHαi(u))=E(π(Uτ)I(τ≤Hα)+P(UHα)I(τ>Hα)∣U0=u,Δ0=δi) Turn MathJax on where the last equality comes from substituting the definition of View the MathML sourceϕti(u), in (4). The function View the MathML sourceΥαi(u) is analytically more tractable than the original function while, at the same time, retains a probabilistic interpretation as a penalty–reward function considering an exponential random time horizon HαHα. The results in this paper are organized as follows: in Section 2 an integro-differential system that characterizes the function of interest View the MathML sourceΥαi(u) is derived and the existence of the solution discussed. In Section 3 a numerical method to approximate the solution of the system via Chebyshev polynomials is considered and Section 4 offers some numerical illustrations.

A general model for the risk process of an insurance company is presented allowing arbitrary distributions of the claim sizes, a Wiener fluctuation in premium collection and investment in a, possibly, risky asset. The evolution of the return rate is modulated by Markov process implementing a non-constant interest rates in a risk process. In particular, we suggest the possibility of interpretation as interest rates announced by a central bank that in practice move by a quarter percentile jumps. A method is obtained to calculate the Gerber–Shiu expected penalty–reward function in this framework that comprises several interesting particular cases such as the calculation of ruin probabilities or moments of the deficit at ruin. The method is based on Chebyshev polynomial approximations and shows an outstanding convergence rate.