Optimal investment and consumption models with non-linear stock dynamics
The author studies the optimal investment model of a single agent who manages his portfolio by investing in a bond and a stock account when intermediate consumption is allowed. The price of the bond \(B_{t}\) solves \(d B_{t}= rB_{t} dt,\;B_0=B\), where \(r>0\) is the interest rate. The price of the stock is modelled as solution of problem \[ d S_{t}=\mu(S_{t})S_{t} dt+ \sigma(S_{t})S_{t} dW_{t},\quad S_0=S\geq 0, \] where \(W_{t}\) is a Brownian motion. The investor rebalances his portfolio dynamically by choosing at any time \(s\in[t,T]\) the amounts \(\pi_{s}^{0}\) and \(\pi_{s}\) to be invested respectively in the bond and the stock accounts. He also consumes out of his bond holdings at a rate \(C_{s}\). His total wealth satisfies the budget constraint \(X_{s}=\pi_{s}^{0}+\pi_{s}\) and the stochastic differential equation \[ d X_{s}=rX_{s} ds+ (\mu(S_{s})-r)\pi_{s} ds- C_{s} ds+ \sigma(S_{s})\pi_{s} dW_{s}, \quad X_{t}=x\geq 0,\;0\leq t\leq s\leq T. \] The investor's objective is to maximize his expected utility \[ J(x,S,t;\pi,C)=E\left[\int_{t}^{T}U(C_{s}) ds+ \Phi(X_{T})/X_{t}=x, S_{t}=S\right], \] where the utility function \(U\) and the bequest function \(\Phi\) are of the form \(U(c)=c^{\gamma}/\gamma\), \(\Phi(x)= x^{\gamma}/\gamma\). The value function of the investor is \(v(x,S,t)= \sup_{(\pi,C)\in{\mathcal A}}J(x,S,t;\pi,C)\), where \(\mathcal A\) is some set of admissible control processes \((\pi_{s},C_{s})\). The author proves that the value function \(v\) is given by \(v(x,S,t)=x^{\gamma}w^{1-\gamma}(S,t)/ \gamma\), where \(w(S,t)\) is the solution of the problem \[ w_{t}+{1\over 2}\sigma^2(S)S^2w_{ss}+\left[\mu(S)S+ {\gamma(\mu(S)-r)S\over (1-\gamma)}\right]w_{s}+{\gamma\over 1-\gamma} \left[r+{(\mu(S)-r)^2\over 2\sigma^2(S)(1-\gamma)}\right]w+1=0 \] \[ w(S,T)=1,\quad \text{and}\quad w(0,t)=\left(1+{1-\gamma\over r\gamma}\right) e^{(r\gamma/(1-\gamma))(T-t)}-{1-\gamma\over r\gamma}. \]
scientific article; zbMATH DE number 1370335