Minimizing expected time to reach a given capital level before ruin

In this paper, we consider the optimal investment and reinsurance problem for an insurance company where the claim process follows a Brownian motion with drift. The insurer can purchase proportional reinsurance and invest its surplus in one risky asset and one risk-free asset. The goal of the insurance company is to minimize the expected time to reach a given capital level before ruin. By using the Hamilton-Jacobi-Bellman equation approach, we obtain explicit expressions for the value function and the optimal strategy. We also provide some numerical examples to illustrate the results obtained in this paper, and analyze the sensitivity of the parameters.


1.
Introduction. In the past decades, optimal investment or reinsurance problems for risk models have gained a great deal of attention in actuarial literature. This is due to the fact that the insurance company can reduce its risk exposure by purchasing reinsurance and increase its profit by investing its surplus into the risky and risk-free assets. In fact, in a diffusion model blending with either of these two controls (investment and reinsurance) or both, much research has been done concerning various optimal criteria.
In this paper, we will consider another important risk-measure criterion, minimizing the expected time to reach a given capital level before ruin. This criterion is first suggested in [19]. In their work, the optimal investment problem without any constraint for an ordinary investor (no external risk process) was solved by maximizing the square of the drift divided by the square of the volatility (see (4.3) in [19]). Another related work is [16], where the authors explored the minimal expected time to reach a goal before ruin within the framework of proportional reinsurance. The optimal proportional reinsurance strategy was obtained by solving the corresponding HJB equation.
In this work, we model the claim process as a Brownian motion with drift. Furthermore, the company is allowed to purchase cheap proportional reinsurance and invest its surplus in a financial market consisting of a risky asset and a risk-free asset. The aim of the insurance company is to minimize the expected time to reach a goal before ruin by choosing control variables. In the end, we provide complete solution. In particular, the results indicate that the optimal investment strategy is not always proportional, which is different from the results in [19].
Compared to [19] and [16], the innovations of our work are as follows.
(1) In our setting, short-selling is prohibited, the proportion retained for the insurer is constrained to the interval [0, 1] and the claim process is correlated to the price process of risky asset. These lead to the failure of the approach in [19]. (2) To tackle the difficulties mentioned in (1), we apply the HJB equation approach which is also used in [16]. However, the optimization problem becomes much more complicated by involving investment, which can be seen in later sections. Most importantly, taking risk-free asset into consideration brings great difficulties in solving HJB equation. In order to solve the HJB equation, we employ tools such as the Legendre transform. Hence, our work is not a simple extension of [19] and [16]. There are many future research problems to work on.
The remainder of the paper is organized as follows. In Section 2 we give a rigorous mathematical formulation of the model. The detailed discussion of the optimization problem and the strategy are presented in Section 3. Moreover, it turns out that the minimal expected time to reach a goal before ruin and the corresponding optimal strategy have explicit expressions. In Section 4, we give some numerical examples to illustrate the results we obtained.
2. Model formulation. To motivate this model, we first introduce the classical Cramér-Lundberg model is a Poisson process with intensity β and the individual claim sizes X 1 , X 2 , · · · , independent of {N t }, are i.i.d. positive random variables with common continuous distribution F and finite first and second moments µ ∞ and σ 2 ∞ , respectively. The premium rate p is calculated via the expected value principle, that is, where η > 0 is the relative safety loading of the insurance company. In order to reduce risk due to the insurance claims, insurer is allowed to transfer part of its risk to a reinsurer. Then the reinsurer is required to pay a fraction of the claims. In return, the insurer diverts a certain amount of premiums to the reinsurer. Let q be a (fixed) retention level and let X (q) i denote the part of the claims held by the insurer. Assume that the reinsurer uses safety loading k(q)η, where the proportional factor k(q) depends only on q, then the insurer's reserve at time t is described by where the premium rate p (q,η) is According to [9], one can show that and ω t is a standard Brownian motion. Throughout this paper, we mainly focus on the case of cheap proportional reinsurance in which k(q) = 1 and X Thus, for a fixed q, the diffusion process {R t } t≥0 approximating the Cramér-Lundberg model is governed by the SDE dR (q) To give a rigorous mathematical formulation, we start with a complete probability space (Ω, F, P ) endowed with filtration {F t } t≥0 . The filtration F t represents the information available at time t and all decisions are made based on this information. The uncontrolled reserve process is given by where B t is a standard Brownian motion adapted to F t .
Apart from the reserve process R t , we suppose that the insurer is allowed to invest its surplus in a risky asset (stock or mutual fund) and a risk-free asset with interest rate r > 0. Specifically, the price process of the risky asset follows a geometric Brownian motion where µ > r and σ > 0 are constants. B S t is a standard Brownian motion with respect to F t . We use ρ S (0 < ρ S < 1) to denote the correlation coefficient between B t and B S t , that is, E(B t B S t ) = ρ S t. Let α be a strategy described by a two-dimensional stochastic process (π t , q t ), where π t represents the amount invested in the risky asset at time t and q t represents the proportion retained at time t. Then the controlled reserve process R α t evolves according to the dynamics (2.1) (iii) π t ≥ 0 (short-selling is prohibited) and t 0 π 2 s ds < ∞ almost surely for all t ≥ 0. Denote the set of all admissible strategies by α S .
3. Minimizing expected time to reach a given capital level. Suppose that the insurer is interested in minimizing the expected time to reach a goalx before ruin. For any α ∈ α S , let τ α := inf{t : R α t ≤ 0} be the ruin time, and Tx α := inf{t : R α t ≥x} be the first hitting time tox. For the Brownian motion risk model, it is well-known that τ α = inf{t : R α t = 0} and Tx α = inf{t : R α t =x}. Define the first time to reachx before ruin. The insurer aims at minimizing the expected value of T α , that is, the insurer concentrates on the following optimization problem (R · , π · , q · ) satisfies (2.1). (3.1) To solve problem (3.1), we use dynamic programming HJB equation approach described in [8]. Before this, it is necessary to show that the value function in (3.1) is well-defined.
Then, α S is not empty, that is, there exists at least one admissible strategy α ∈ α S . Proof. Suppose the initial surplus 0 < x <x, we consider a strategy α such that q t = 0 and π t = λR t for some constant λ > 0. Based on (2.1), the corresponding surplus process R α t is then given by On the other hand, from the optional stopping theorem, we have Hence, by applying the dominated convergence theorem and the Fatou lemma, we obtain E[T α ] < ∞.
In view of Lemma 3.1, we restrict the optimization problem (3.1) to α S . In addition, for control problems with infinite time horizon, it is natural to focus on stationary control strategies as the state dynamics are time-independent. Thus, we can write the objective function as Then, our aim is to compute the value function and the associated optimal investment-reinsurance strategy α * ∈ α S such that According to the standard arguments, for example, [8], we know that if the value function V (x) ∈ C 2 , then V (x) satisfies the following HJB equation In the following, we first find a solution of (3.3)-(3.5) on the class of strictly decreasing convex functions on (0,x), then verify that it is indeed the value function. The proofs of theorems and lemmas are deferred to the Appendix. Define These constants will appear in the subsequent theorems and lemmas. The solution W (x) and the corresponding strategy (π * (x), q * (x)) are as follows: and W (x) satisfies the following differential equation: where the constants M , N , A and k 1 are given in (3.6), (3.7), (3.8), and (A.25), respectively. Now, we have found a candidate solution W (x) and a candidate optimal strategy for the optimization problem (3.1), where R * t denotes the surplus process under the strategy α * . We only need to verify that W (x) is indeed the value function and α * given in (3.15) is the associated optimal strategy. Before this, we present the following lemma which shows that α * ∈ α S .
is the associated optimal investmentreinsurance strategy.
Assume that there is no risk-free asset, that is, r = 0, then the minimal expected time for the insurer to reach the goal before ruin and the associated optimal strategy can be derived similarly as that in Theorem 3.2. Here we provide the results for this case in the following corollary without detailed proof.

Numerical examples.
In this section, we provide some numerical examples to illustrate the analytical results obtained in Section 3. The parameters are given as follows: a = 0.01, r = 0.02, µ = 0.06, ρ S = 0.6, and the goalx = 1. In Figs. 1 and 2, we fix b = 0.04 and focus on the effects of the stock volatility σ on the minimal expected time and the corresponding optimal strategies. In the left panel of each figure, the insurer is allowed to invest in both the risky asset and the risk-free asset; while in the right panel, the insurer only invests in the risky asset. Specifically, we define ζ(x) = x − π * (x), the amount invested in the risk-free asset. Note that, ζ(x) < 0 means that the insurer borrows money |ζ(x)| at rate r. In Fig. 1(a), we set σ = 0.1, the results are in coincidence with case (iv) of Theorem 3.2. The optimal proportion retained is 0.3906x and the optimal amount borrowed at rate r is 2.9062x. Comparing with Fig. 1(a), we plot the value function and the associated optimal strategies when the interest rate equals zero in Fig. 1(b). Based on Corollary 3.5, the optimal choice for the insurer is to divert its entire risk incurred by claims to the reinsurer, and totally invest money in the risky asset.
In Fig. 2, we set σ = 0.01, and we find aσ − (µ − r)bρ S < 0. In accordance with Theorem 3.2 and Corollary 3.5, the results in this example coincide with case (iii). The proportion of claims retained is zero. Moreover, in light of Figs. 1 and 2, we observe that the insurer will spend much more time to reach the goal by investing in two assets than only in one risky asset. This seems intuitively reasonable when the return of the risky stock with small variation is higher than that of a risk-free bond.
In Figs. 3 and 4, we fix σ = 0.3, and change the values of b, the volatility induced by claims. In Fig. 3, we set b = 0.03, for both of the cases r = 0 and r = 0, the optimal choice for the insurer is investing no money in the risky stock, and taking part of the claims when the surplus of the insurer is less than 0.09, and then taking total claims without buying any reinsurance when the surplus of the insurer becomes larger than 0.09. Fig. 4 shows the value function and the corresponding optimal strategies for b = 0.3. Similarly to the examples considered in Figs. 1 and 2, the insurer in this situation will not buy any reinsurance, but invests a positive amount in the riskfree asset. Meanwhile, contrast to the graphs plotted in Figs. 1 and 2, we find the minimal expected time spent by the insurer for the case r = 0 is less than that for r = 0. Because the value of volatility of the risky stock we use in these examples is bigger than that used in Figs. 1 and 2, which implies that more risks will be taken by the insurer.
In Fig. 5, we set b = 0.04, σ = 0.1 and fix the initial surplus x = 0.5. We plot the graphs of the minimal expected time for various values of goal. In this case, we find the expected time for r = 0 is smaller than that for r = 0.   and the minima with q = 0, q = 1 and π = 0 are respectively attained at where M and N are given in (3.6) and (3.7). Due to the different values of π(x) and q(x), we define the following regions: Based on the above analysis, we have the following three cases to deal with.
(1) The case of aσρ S ≥ (µ − r)b In this case, from (3.6) and (3.7), we see M ≤ 0 and N ≥ 0. Therefore, we only need to consider O 1 and O 2 . First, for x ∈ O 1 , the minimum of the left-hand side of (3.3) is attained at Substituting it into (3.3) and rearranging, we get We consider the Legendre transform W 0 (y) of W (x) defined by Then we can recover W (x) from W 0 (y) by The general solution of (A.11) is W 0 (y) = C 1 y a 2 +2b 2 r a 2 + 2b 2 a 2 + 2b 2 r ln y + C 2 , in which C 1 and C 2 are constants to be determined. Since W (0) = ∞, so there exists a point y 0 , such that W 0 (y 0 ) = ∞ and W (y 0 ) = 0. Moreover, we see W 0 (y) = C 1 a 2 + 2b 2 r a 2 y 2b 2 r a 2 + 2b 2 a 2 + 2b 2 r 1 y .

This together with (A.2) and (A.3) gives
and the corresponding strategy is (π * (x), q * (x)) = 0, ax/b 2 , (A.14) for 0 < x <x. Hence, (i) is proved. If b 2 < ax, then W (x) given in (A.13) only holds on 0 < x < b 2 /a and (π * (x), q * (x)) given in (A.14) is the associated strategy on 0 < x < b 2 /a. When b 2 /a ≤ x <x, we turn to O 2 and guess that Then, the left-hand side of (3.3) attains its minimum at (π * (x), q * (x)) = (0, 1), b 2 /a ≤ x <x. Thus, whereC 1 is a constant to be determined later. Hence, we get the expression for W on b 2 /a ≤ x ≤x as From the boundary condition (3.5), we obtain thatC 2 = − x b 2 /a h(v)dv. Moreover, the values of constantsC 1 and C 2 can be determined by the fact that W (x) is continuous and continuously differentiable at x = b 2 /a, that is, Therefore, we obtain the function W on (0,x) as follows For b 2 /a ≤ x ≤x, clearly, h(x) < 0, and we also claim that h (x) > 0 on (b 2 /a,x], otherwise one could find a point x 1 ∈ (b 2 /a,x], such that h (x 1 ) < 0. By the continuously differentiable property of W (x) at x = b 2 /a, it follows easily that h (b 2 /a) = 2a 2 b 2 (a 2 +2b 2 r) > 0. Therefore, we can find one pointx ∈ (b 2 /a,x] such that h (x) = 0. Hence, we get On the other hand, taking derivatives on both sides of equation (A.17), and replacing x byx, we have h (x−) + 2 b 2 rh(x) = 0, which contradicts with the fact that h (x−) < 0 and h(x) < 0. Hence, we conclude that h (x) > 0 on (b 2 /a,x]. Moreover, (A.4) can be verified similarly as in Proposition A.1, which ends the proof of (ii).
(2) The case of aσ ≤ (µ − r)bρ S In this case, we find that M ≥ 0 and N < 0 given in (3.6) and (3.7), respectively. Thus, only O 3 needs to be considered. In this region, the left-hand side of (3.3) attains its minimum at Substituting it into (3.3), we get the differential equation satisfied by W : Analysis similar to (A.8) shows that by the boundary condition (3.5). Therefore, we have (3.10) and O 3 = {x : 0 < x < x}.
(3) The case of (µ − r)bρ 2 S < aσρ S < (µ − r)b Recalling the constants M and N defined in (3.6) and (3.7), we see that M < 0 and N < 0. Therefore, we will only consider O 4 and O 5 . If x ∈ O 4 , then (π * (x), q * (x)) = (π(x), q(x)), and HJB equation (3.3) reduces to with A defined in (3.8). As in the arguments of (A.8), the solution of (A. 19) is where k 1 is a constant and will be determined later. Now the region O 4 becomes Then, the left-hand side of (3.3) attains its minimum at The boundary conditions ( and Proof. First, we prove that W (x) is strictly decreasing and convex on [−1/M,x). Otherwise, one could find a point Substitutingc into equation (A.24) yields which is a contradiction. Therefore, W (x) < 0 on [−1/M,x). We also claim that where the last equality is obtained by the fact that π(x) = − µ−r . To demonstrate the claim, we first prove that for arbitrary x 0 ∈ (−1/M,x], the inequality W and by the boundary condition (A.26), we see Therefore, (A.24) can be rewritten as which is equivalent to (A.32) can be rewritten as for x ∈ (−1/M,x], which is impossible. Therefore, W (x) W (x) > −x cannot hold on (−1/M, x 0 ). Now, we claim that there is no constant x ∈ (−1/M,x], such that W (x) W (x) = −x, i.e., ψ(x) = −1/x. Otherwise there exists x * = inf{−1/M < x ≤x : ψ(x) = −1/x}, then combining the above conclusion we obtain ψ(x) ≥ −1/x for x ∈ (−1/M, x * ]. Thus, (A.33) Replacing x by x * in (A.24), we get . (A.34) Taking derivatives on both sides of (A.24) at x = x * implies r + (µ − r) 2 2σ 2 (x * ) 2 ψ (x * −) + the other hand, when R * t is larger than −1/M , then By (A.36), we see that the drift of R * t is positive, thus R * t tends to infinity with probability 1. Moreover, using analogous discussion of Case (1) and applying Itô s formula to W (R * t ), where W (x) is the solution given in Theorem 3.2, we find that Similarly, we can prove E x [T α * ] < ∞ for −1/M ≤ x <x.
Proof of Theorem 3.4. Let α ∈ α S , and define T n = T α ∧ n, where n is a fixed constant, then by applying Itô s formula to W (R α Tn ), we get W (R α Tn )