# On the gain of collaboration in a two dimensional ruin problem

## Abstract

We investigate the following problem: the endowment process of two insurance companies is given by a two dimensional Brownian motion with drift. How big is the gain in the probability that both companies survive, if they collaborate optimally by transfer payments, in comparison to the non collaboration case. We provide an explicit formula for the value function of the problem.

## Keywords

Ruin probabilities Optimal control problem Collaboration Two-dimensional Brownian motion## Mathematics Subject Classification

Primary 49J20 35R35 Secondary 91B70## 1 Introduction

Whereas the literature on the two main evaluation criteria for insurance companies, namely the ruin probabilities and the maximal possible dividend payments, is vast in the one dimensional case, i.e., if one considers only one company, see e.g. for an overview [1, 3], there are not so many results in the two- or multidimensional case. We mention for the dividend problem, [2, 6−8], where it is investigated, how the total dividend payments of two companies can be optimized, if they collaborate.

We want to investigate a similar problem for the ruin probabilities in this paper. Assuming that both companies can collaborate by transfer payments, and assuming that it is their goal that both companies survive, we want to examine how big the gain of collaboration in terms of this surviving probabilities is in comparison to the non-collaboration case.

The endowment processes will be modeled by two dimensional Brownian motion with drift, and we assume that the transfer process is absolutely continuous with respect to the Lebesgue measure.

It will turn out that the problem is a stochastic control problem, and let us mention that an equivalent problem is considered in [9], where the authors viewed the problem as tax policy problem. We will show the equivalence in the next section, where we introduce our model. McKean and Shepp mention in their paper (page 6591) that it is known, what the optimal strategy is, but that the value function is unknown. We shall provide an explicit formula for this value function.

Let us finally cite the papers [4, 5], where one can also find some results on multidimensional ruin problems.

## 2 The model

## 3 Construction of a classical solution for the Hamilton–Jacobi–Bellman equation

*G*. Once we have found this solution, we mirror it on the first median and show that this gives a full \(C^2\)-solution, especially \(C^2\) at the first median.

*H*, and show then that we have \(V_x \ge V_y >0\) on

*H*, which implies that the solution of (9) is also a solution of (8). The interpretation of this is that one should give all the drift to the weaker company, apart from the reserve \(\delta \).

### Proposition 3.1

*The function*\(\tilde{V}(x,y)=\frac{\delta -\overline{\mu }}{\delta } \left( e^{-(\overline{\mu }-2\delta )x}-e^{-\overline{\mu }x}\right) e^{-\overline{\mu }y}\)*solves the system* (11).

Before we prove the proposition, let us remark that we could, of course, verify Proposition 3.1 by just plugging in the explicit solution formula. But we want to demonstrate, how we found the solution.

### *Proof*

Here we have to check that \(g(\infty )=0\) is true. But this holds , because of \(C_1=0,\theta _2=-\overline{\mu }<0\). Finally, employing all the explicit values of the constants \(C_i\) and the characteristic roots, gives the asserted form of the function \(\tilde{V}(x,y)\). \(\square \)

Because of (10) an immediate corollary is

### Corollary 3.1

*The function*

*solves*(9).

As announced earlier, we now show that the candidate solution of the previous corollary fulfills the HJB system on the set *H*.

### *Proof*

All we have to show is that \(V_x(x,y)\ge V_y(x,y)>0\) on the set *H*.

*H*. To check \(V_y(x,y)>0\) is also straightforward. \(\square \)

*G*, i.e. below the first median. We do this by just mirroring at the first median, i.e. we define

### *Proof*

## 4 Verification

In this section we verify that the function *V*(*x*, *y*) of (23) is indeed the value function of our problem. We have

### Theorem 4.1

*The function**V*(*x*, *y*) *defined in* (23) *is the value function of the problem* (5), *and the optimal strategy is given by*\(u^*(X_t,Y_t)=(\overline{\mu }-\delta )\mathbf {1}_{\{Y_t\ge X_t\}}+\delta \mathbf {1}_{\{Y_t < X_t\}}\) (*which correspond to*\(c^*(X_t,Y_t)=(\mu _2-\delta )\mathbf {1}_{\{Y_t\ge X_t\}}+(-\mu _1+\delta ) \mathbf {1}_{\{Y_t < X_t\}}\)*in the original control variable*).

### *Proof*

Let \(u_t\) be an arbitrary admissible strategy with values in \([\delta ,\overline{\mu }-\delta ]\). We consider the process \(V(X_{t\wedge \tau },Y_{t\wedge \tau })\), where \(\tau :=\tau _1\wedge \tau _2\), the first time when one of the two companies is ruined.

*V*is bounded, we have in fact a uniformly integrable supermartingale \(V(X_{t\wedge \tau },Y_{t\wedge \tau })\). Hence, the supermartingale convergence theorem implies that the limit

*V*(

*x*,

*y*) is indeed the value function of the problem, and \(u_t^*\) the optimal strategy. \(\square \)

## 5 Concluding remarks

Although not direct comparable—since in these papers singular controls, modeling the accumulated transfer resp. dividend payments, are allowed, it is instructive to take a look at the structure of the optimal strategies in [2, 8].

In [2], where compound Poisson processes are used for the endowment, the strategy is rather complicated. Depending on the behavior of the value function the positive quadrant for the current endowments is divided into several subsets. On these subsets the following dividend strategies are optimal: no action at all, lump sum payment, payment at certain rates. Moreover, it is assumed that company one has always to cover the deficit of the other one, if company two is in trouble (and vice versa).

This “bailing out” assumption is also used in [8], where a diffusion approximation is used. Here the optimal strategy is easier: one possibility (the optimal strategy is not unique here) is, to keep the surplus of company 2 at zero at any time by continuously transferring money. The dividend payments are done by company one, using a barrier strategy. This solution corresponds to merging the two lines and solving a one-dimensional problem.

Let us finally remark that, allowing singular controls in our setting, ruin would occur only, if both companies are ruined simultaneously (similarly as in [8]), and the problem is again reduced to a one dimensional one.

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). Support by the “Austrian Science Fund” (Fonds zur Förderung der wissenschaftlichen Forschung), Project nr. P30864-N35, is gratefully acknowledged. Moreover, I thank an anonymous referee for carefully reading the paper and suggestions to improve it.

## References

- 1.Asmussen S, Albrecher H (2010) Ruin probabilities, vol 14, 2nd edn. Advanced series on statistical science and applied probability. World Scientific Publishing, HackensackzbMATHGoogle Scholar
- 2.Albrecher H, Azcue P, Muler N (2017) Optimal dividend strategies for two collaborating insurance companies. Adv Appl Probab 49(2):515–548MathSciNetCrossRefGoogle Scholar
- 3.Avanzi B (2009) Strategies for dividend distribution: a review. N Am Actuar J 13(2):217–251MathSciNetCrossRefGoogle Scholar
- 4.Avram F, Palmowski Z, Pistorius M (2008) Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results. Ann Appl Probab 18(6):2421–2449MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Collamore J (1996) Hitting probabilities and large deviations. Ann Probab 24(4):2065–2078MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Gerber HU, Shiu E (2006) On the merger of two companies. N Am Actuar J 10(3):60–67MathSciNetCrossRefGoogle Scholar
- 7.Grandits P (2019) A two dimensional dividend problem for collaborating companies and an optimal stopping problem. Scand Actuar J 2019(1):80−96MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Gu J, Steffensen M, Zheng H (2018) Optimal dividend strategies of two collaborating businesses in the diffusion approximation model. Math Oper Res 43(2):377–398MathSciNetCrossRefGoogle Scholar
- 9.McKean HP, Shepp LA (2006) The advantage of capitalism vs. socialism depends on the criterion. J Math Sci 139(3):6589–6594MathSciNetCrossRefzbMATHGoogle Scholar

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