ISSN:
1432-1122
Keywords:
Key words: Geometric Brownian motion, optimal stopping time, continuation region, stopping set JEL classification: D81 Mathematics Subject Classifications (1991): 60G40, 93E20, 60H10, 90A09
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Economics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s007800050042
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