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Name: Krzysztof Kepczynski
Talk Title: Running supremum of Brownian motion in dimension 2: exact and asymptotic results
Abstract: This paper investigates $\pi_T(a_1,a_2) = \mathbb{P}\left(\sup\limits_{t\in[0,T]} (\sigma_1B(t)-c_1t)a_1, \sup\limits_{t\in[0,T]}( \sigma_2 B(t)-c_2t)a_2\right),$ where $\{B(t) : t \geq 0\}$ is a standard Brownian motion, with $T 0, \sigma_1,\sigma_20, c_1, c_2\in\mathbb{R}.$ We derive explicit formula for the probability $\pi_T\left(a_1,a_2\right)$ and find its asymptotic behavior both in the so called many-source and high-threshold regimes.
This talk is a contributed talk at EVA 2021. View the programme here.