Turnpike behavior of long-term investments
The uthors study the asymptotic behavior of the optimal portfolio policy in the following financial model. Let \(A_{\tau}\) denote the dollar amount invested in the risky asset when there are \(\tau\) periods to time horizon \(T\). Let the wealth dynamics \(W_{s}\) satisfy the stochastic differential equation \(dW_{s}=[rW_{s}+(\mu-r)A_{s}]ds+\sigma A_{s}dw(s)\), \(W_{t}=x\), \(x\geq 0\), \(s\in(t,T]\), where \(\mu>0\), \(\sigma>0\), \(w(t)\) is a standard Brownian motion, \(r\) is a constant riskless interest rate. Let \({\mathcal A}\) be the set of admissible policies. Given the initial level of wealth at \(t\), \(W_{t}=x\), we want to find the policy \(A\in{\mathcal A}\) that solves the portfolio problem \(\sup\limits_{A\in{\mathcal A}}E[U(W_{T})| W_{t}=x]\) a.s., where \(U\) is the utility function. The optimal policy is given by \(A_{\tau}^{*}= A(W^{*}_{\tau},\tau)\) where \(W^{*}_{\tau}\) is the optimal wealth trajectory. The main result of the paper is the following: If the utility function \(U\) is increasing, concave and twice continuously differentiable, \(U''\) is nondecreasing, \(U(0)=0,\;U(x)\leq K(1+x)^{\gamma}\), \(\lim_{x\to\infty}{U'(x)\over x^{\gamma-1}}=1\) for \(0<\gamma<1\), \(K>0\), then \(A^{*}(x)=A_{*}(x)={\mu-r\over \sigma^{2}(1-\gamma)}x,\;x\geq 0\). Here \(A^{*}(x)=\limsup_{\tau\to\infty,y\to x}A(y,\tau)\), \(A_{*}(x)=\liminf_{\tau\to\infty,y\to x}A(y,\tau)\).
scientific article; zbMATH DE number 1336673