Optimal time to invest when the price processes are geometric Brownian motions

Abstract.

Let $X_1(t)$, $\cdots$, $X_n(t)$ be $n$ geometric Brownian motions, possibly correlated. We study the optimal stopping problem: Find a stopping time $\tau^*<\infty$ such that \[ \sup_{\tau}{\Bbb E}^x\Big\{ X_1(\tau)-X_2(\tau)-\cdots -X_n(\tau)\Big\}={\Bbb E}^x \Big\{ X_1(\tau^*)-X_2(\tau^*)-\cdots -X_n(\tau^*)\Big\} , \] the $\sup$ being taken all over all finite stopping times $\tau$, and ${\Bbb E}^x$ denotes the expectation when $(X_1(0), \cdots, X_n(0))=x=(x_1,\cdots, x_n)$. For $n=2$ this problem was solved by McDonald and Siegel, but they did not state the precise conditions for their result. We give a new proof of their solution for $n=2$ using variational inequalities and we solve the $n$-dimensional case when the parameters satisfy certain (additional) conditions.