## Optimal investment for investors with state dependent income, and for insurers.(English)Zbl 1069.91051

Let a tradeable assert be given by a continuous price process modelled as a geometric Brownian motion $dZ(t)=aZ(t)dt+bZ(t)dW(t), \quad Z(0)=z,\;a>0,\;b>0,$ where $$W(t),t\geq0,$$ is a standard Wiener process, and let $$X(t)$$ be the wealth of an investor at time $$t$$ determined by the equation $dX(t)=c(X(t))\,dt+\theta(t)\,dZ(t), \quad X(0)=s,$ where $$s\geq0$$ is the initial wealth, $$c(X(t))$$ is a positive income per time depending on the current wealth $$X(t)$$, and $$\theta(t)$$ is the number of the assert $$Z(t)$$ at time $$t$$. For a positive bounded nondecreasing utility function $$g(x)$$ let $G(\theta)=E\left(\int_0^{\tau}g(X(t))e^{-\lambda t}\,dt\right),$ where $$\tau=\inf\{t>0\colon X(t)<0\}$$ is the time of ruin which can be considered as a subjective discount factor. The problem is to find the optimal trading strategy $$\widehat\theta=\{\widehat\theta(t)$$, $$t\geq0\}$$ which maximizes $$G(\theta)$$. The value function of the problem $$V(x)$$ is a function of the current wealth $$x$$ alone. It satisfies the Bellman equation $\sup_{A}\left\{g(x)-\lambda V(x)+(c(x)+Aa)V'(x)+\tfrac{1}{2}A^2b^2V''(x)\right\} =0,\quad x>0.$ The authors prove that this equation has a smooth solution $$V(x)$$ which is the value function of the considered problem, and the maximizer $$A(x)=-aV'(x)/b^2V''(x)$$ defines the optimal investment strategy in feedback form via $$\theta(t)=A(X(t))/Z(t)$$, $$m(t)=A(X(t))$$ is the optimal amount of money invested into the tradeable asset, $$X(t)$$ is the current wealth at time $$t$$ which is the result of the trading strategy $$\theta(u),u<t$$.
The authors also deal with the optimization problem for the case of a stochastic discount rate $$\lambda(t)$$ for which they adopt the Markov chain model of R. Norberg [Appl. Stochastic Models Data Anal. 11, No. 3, 245–256 (1995; Zbl 1067.91509)]. Based on the obtained results the problem of optimal investment for an insurer with an insurance business modelled by a compound Poisson or a compound Cox process, under the presence of constant as well as (finite state space Markov) stochastic interest rate is considered.

### MSC:

 91G10 Portfolio theory 91B30 Risk theory, insurance (MSC2010) 62P05 Applications of statistics to actuarial sciences and financial mathematics 49L20 Dynamic programming in optimal control and differential games 90C39 Dynamic programming

Zbl 1067.91509
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