Model with random time horizon

64 T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 Table 1 Parameters for the numerical experiments Parameter c λ γ α x max K Value 1110 1 110 910 10 1 Due to Eq. 17 and as R i,j → 0 for j → ∞, the maximum norm of the remaining defect goes to 0 as j → ∞. Finally, we want to fit the solution to the boundary value and get the restriction U x max = 1. 18 We have three arbitrary coefficients A 0,0 , A 1,0 and A 2,0 and three equations 18, 17 and 8. Thus, our solution is the only solution of the form 3. What remains is to show that the solution is unique. The theory of differential equations with deviating arguments gives us uniqueness of the solution under the condition that the corresponding integral operator is contracting. We have chosen to prove existence and uniqueness in this way as the integral operator for Eq. 2 is of interest itself for numerical simulation. The operator is defined by the integral equation U = IU, where we define I as the expectation over time over all events at time t, claims of size y and devaluations of the expected probability of survival in each case. I U x = Z T e −λ+γ t λ Z x +ct U x + ct − y dF y + γ Uαx + ct + K dt + e −λ+γ T , 19 where T = x − x max c is the remaining time until the barrier is hit and the additive term corresponds to the case where the barrier is reached before an event occurs. We prove that the operator is strictly contracting by using the maximum norm of the difference of two solutions U and V kIU − IV k ∞ = max ≤xx max |IUx − IV x|. We write U x − V x as U − V x and obtain: kIU − IV k ∞ = max ≤xx max Z T e −λ+γ t λ Z x +ct U − V x + ct − y dF y + γ U − V αx + ct + K dt ≤ max 0x max Z T e −λ+γ t λ Z x +ct dF y + γ kU − V k ∞ dt ≤ max ≤xx max Z T e −λ+γ t λ + γ dtkU − V k ∞ ≤ 1kU − V k ∞ , where 1 = 1 − exp−λ + γ x max c 1. Therefore, we have proved contraction with an explicit contraction constant 1. Depending on the given cumulative distribution function F the contraction constant can in general be improved. In our analytic solution, we have the exponential distribution with parameter 1. This leads to a maximal contraction constant of ≈ 0.991 for the parameters in Table 1. Furthermore, due to the additive term exp −λ + γ T , we cannot have the trivial solution U ≡ 0 if x max is finite. By using Banach’s fixed point theorem on L ∞ R + we have proved that the solution of Eq. 3 is the only measurable bounded solution of Eq. 2 with Eq. 1.

4. Model with random time horizon

So far we have only discussed the down-side risk in the form of credit risk or devaluation. If we consider the case of foreign exchange risk where the role of the countries is reversed, the insurer is confronted with the positive side T. Siegl, R.F. Tichy Insurance: Mathematics and Economics 26 2000 59–73 65 of the situation. His reserve will appreciate in value by a factor α 1. Another interpretation for such a case would be that the insurer has invested the reserve in securities which pay dividends or interest non-deterministically. We are interested in a model where this possibility can be incorporated without sacrificing an analytical solution. To allow a solution also for the case α 1 we drop the upper barrier from our model also because of the mathematical intractability mentioned before, and replace the boundary condition by lim x →∞ U x = 1, 20 but we can keep the additive term K ≥ 0 that is added upon an appreciationdevaluation event. So far, the model equation has not changed safe for the support of U , which is now [0, ∞ and the boundary condition 20. However, for the case without upper bound we do not have strict contraction of the associated integral operator on the interval x ∈ [0, ∞ and thus the approach used before for proving existence and uniqueness does not hold anymore. Moreover, for numerical simulation it is very convenient that the process stops in finite time with probability 1. Thus, we introduce a modification to the previous situation. We consider a modification of the time horizon to a random variable with distribution GT . This can be interpreted as the case where the insurer stops the business after a random time. The company may then change its investment or securitization methodology. Furthermore, we assume that the company has survived in this case. This yields the model equation after similar steps as before U x = Z T P inf ≤t≤T R s ≥ 0 dGT . In the following we will consider the special case where GT is the exponential distribution with parameter η 0: cU x x − λ + γ + ηUx + λ Z x U x − y dF y + γ Uαx + K + η = 0. 21 Note that the remaining requirements for the solution λ 0, γ 0, c 0, positive claims, µ ∞ still hold, and that we do not need further restrictions to obtain Eq. 20 as the process stops in finite time with probability 1. Thus we can relax α to α 0.

5. Analytic solution