Optimal investment strategies in a CIR framework (Q2725291)

The authors consider the model of a financial market in which the stochastic short-term interest rate process \(r_{t}\) follows the CIR dynamics NEWLINE\[NEWLINEdr_{t}=(a-br_{t}) dt-\sigma_{r}\sqrt{r_{t}} dz_{r}(t), \quad t\geq 0,NEWLINE\]NEWLINE where the coefficients \(\sigma_{r}, a, b\) and \(r_0\) are strictly positive constants. The complete financial market contains three assets. The first is the riskless asset: its price, denoted by \(S_0(t), t\geq 0\) evolves according to equation \(dS_0(t)=S_0(t)r_{t} dt, S_0(0)=1\). The second asset is a zero-coupon bond with maturity \(T\) and price \(B(t,T)\), and the third asset is a stock with price \(S(t)\) evolves according to NEWLINE\[NEWLINEdS(t)/S(t)=r_{t} dt +\sigma_1(dz(t)+\lambda_1 dt)+ \sigma_2\sqrt{r_{t}} (dz_{r}(t)+\lambda_{r}\sqrt{r_{t}} dt),NEWLINE\]NEWLINE \(S(0)=1\), where parameters \(\sigma_1>0\), \(\sigma_2>0\), \(\lambda_1\), \(\lambda_2\) are assumed to be constant. Here the Brownian motions \(z_{r}(t)\) and \(z(t)\) are independent. NEWLINENEWLINENEWLINEThe authors solve explicitly the optimization problem \(\max_{\pi_{t}\in A}EU(Y(T))\), where \(U(y)=y^{\gamma}/\gamma\), \(\gamma\in (-\infty,1)\setminus\{0\}\); the wealth process \((Y(t))_{t\in[0,T]}\) is defined by following dynamics \(dY(t)=Y(t)\pi_{t} \text{diag}[{\mathbb S}(t)]^{-1} d{\mathbb S}(t)\), \(Y(0)=Y_0>0\), with \(\pi_{t}=[(1-\pi_{t}^{B}-\pi_{t}^{S}),\pi_{t}^{B},\pi_{t}^{S}]\), \({\mathbb S}(t)=[S_0(t),B(t,T),S(t)]\); \(A\) is the set of admissible controls. The behaviour of the solution is analyzed.