Optimal investment for investors with state dependent income, and for insurers
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 \textit{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.
scientific article; zbMATH DE number 2169531