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We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black–Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility and the probability of ruin. Following recent research, we assume that investment strategies have to be deterministic. This leads to deterministic control problems, which are quite easy to solve. Moreover, it turns out that there are some interesting links between the optimal investment strategies of these problems. Finally, we also show that this approach works in the Lévy process framework.

Inspired by [

Mathematically, the mean-variance problem for the restricted class of strategies leads to a deterministic control problem directly, without the problem of facing the non-separability of the target function. In the classical adapted case, it is necessary to link the mean-variance problem to an auxiliary linear-quadratic problem first (see, e.g., [

In this paper, we do

The paper is organized as follows: In the next section, we introduce the insurance model and the mean-variance problem along with some standing assumptions. Then, we explain how to reduce the problem in general to a stochastic linear-quadratic problem. Next, we solve the problem within the classical framework of adapted, i.e., wealth-dependent investment strategies. In

We suppose that the risk reserve process,

The insurance company is now allowed to invest the risk reserve into the financial market. A classical trading strategy

In the next section, we explain the standard way to transform this problem into a classical stochastic control problem, which will then be solved in the subsequent sections. In order to obtain non-trivial problems, we assume that:

Problem (MV) can be solved via the well-known Lagrange multiplier technique. The discussion in this section follows [

Obviously, the value of (MV) is equal to

From Lemma 1, we see that it is sufficient to look for a saddle point

Suppose

The implication of Lemma 2 is that any optimal solution of

We will first solve problem

Finally, we want to solve problem (MV). Thus, we have to compute

In this section, we assume now that the investment strategy has to be pre-determined, i.e., that the process,

Finally, we solve problem (MVD). First note that:

Densities of the optimal terminal wealth for (MVD) and (MV).

Obviously, for this time horizon (

In this section, we will briefly discuss some other optimality criteria for the investment problem with deterministic investment strategies. Of course, when the solution of the classical stochastic control problem with adapted investment strategies yields an optimal strategy, which is itself deterministic, then this strategy is also optimal in the smaller class of deterministic strategies. A situation like this can occur when we consider the probability of ruin or the expected exponential utility as a target function. We discuss these cases below. However, we start this section with the observation that in the mean-variance framework, our optimal deterministic investment strategy is not only optimal w.r.t. to minimizing the variance.

The variance or standard deviation is, of course, just one way to measure risk. Suppose now that

First note that in both cases, because of

As a consequence, the optimal investment strategy we obtained is very robust w.r.t. the choice of risk measure. Indeed, it does not depend on the precise risk measure, as long as we agree to take a law invariant and positive homogeneous one. Indeed, the result is also valid for a function,

In this subsection, we consider the problem of maximizing

Another popular ‘risk measure’ in the actuarial sciences is the probability of ruin of a controlled risk reserve. When we consider the classical situation of

The standard model for the risk reserve process of an insurance company is the so-called Cramér–Lundberg model. It assumes that the risk reserve process follows a Lévy process given as the difference of the premium income process and the claims that have been paid out so far. More precisely, it is usually assumed that:

Finally, we solve the problem (MVD). Note that:

As a result, we see that the optimal control depends only on the drift of the risk reserve (here,

We have shown that stochastic control problems with deterministic investment strategies lead to deterministic control problems that are, in general, easier to solve. In particular, in the case of a Brownian setting, the terminal wealth has a normal distribution under any admissible deterministic investment strategy. This leads to some very favorable properties, like the insensitivity of the optimal control w.r.t. to a class of target functions. Moreover, there are some interesting links between these problems. Optimal deterministic investment strategies for mean-variance problems, for example, correspond to optimal investment strategies for an insurance company with exponential utility. Finally, we also show that the current approach works in the setting of Lévy processes.

The authors declare no conflict of interest.