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Estimates of the exit probability for two correlated Brownian motions. (English) Zbl 1262.60082

Summary: Given two correlated Brownian motions \((X_{t})_{t\geq 0}\) and \((Y_{t})_{t\geq 0}\) with constant correlation coefficient, we give upper and lower estimates of the probability \(\operatorname{P}(\max_{0 \leq s\leq t} X_{s}\geq a\), max \(_{ 0 \leq s\leq t} Y_{s}\geq b)\) for any \(a,b,t >0\) through explicit formulae. Our strategy is to establish a new reflection principle for two correlated Brownian motions, which can be viewed as an extension of the reflection principle for one-dimensional Brownian motion. Moreover, we also consider the nonexit probability for linear boundaries, i.e., \(\operatorname{P} (X_{t} \leq at+c,Y_{t} \leq bt+d, 0\leq t\leq T)\) for any constants \(a, b\geq 0\) and \(c,d, T >0\).

MSC:

60J65 Brownian motion
60G40 Stopping times; optimal stopping problems; gambling theory
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References:

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