Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints
This paper deals with a multidimensional model for securities with random volatilities. This model includes correlation between the assets and the stochastic factors influencing the volatilities. The author considers finite time horizon investment model on \([0,T]\) with one riskless bond and \(n\) risky assets with random volatilities. The price process of the bond \(S^0\) asset is given by \(dS^0_{t}=rS_{t}^0 dt\), where \(r\in R\) is the interest rate. The price of the \(n\) risky assets \(S\) are modeled as \[ dS_{t}=\text{diag}(S_{t})(b(Y_{t})dt+\sigma(Y_{t})dW_{t}+\overline\sigma(Y_{t})d\overline W_{t}), \] where \(Y\in R^{d}\) is the stochastic factors with dynamics \(dY_{t}=\eta(Y_{t})dt+dW_{t}\). Here \(W\) is a \(d\)-dimensional standard Brownian motion and \(\overline W\) is an \(m\)-dimensional standard Brownian motion independent of \(W\). The author considers an agent with a CRRA utility function \(U(x)=x^{\delta}/\delta\), \(x\geq 0\), for \(\delta<1\), \(\delta\neq 0\). The value function of the agent has following form \[ v(t,x,y)=\sup_{\pi\in{\mathcal A}(K)}E[U(X_{t})\mid X_{t}=x,\;Y_{t}=y], \quad (t,x,y)\in[0,T]\times \mathbb{R}_{+}\times \mathbb{R}^{d}, \] where \(X_{t}\) is the wealth process; \({\mathcal A}(K)\) is the set of admissible portfolio controls; portfolio \(\pi\in K\); \(K\) is closed convex set in \(\mathbb{R}^{d}\). The corresponding Bellman equation is derived. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. The existence of a smooth solution to this semilinear equation is proved. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation.
semilinear partial differential equation
scientific article; zbMATH DE number 1882089
scientific article