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Name: Konrad Krystecki

Talk Title: Two-dimensional ruin for Brownian motions with drift dependant on initial capital

Abstract: Denote by $(W_1(s), W_2(t)), s,t\ge 0$ a bivariate Brownian motion with standard Brownian motion marginals and constant correlation $\rho \in (-1,1).$ In this contribution we study the asymptotics following ruin probability model. $$p_{\alpha, \red{\beta}, \rho}(\vk c, a) = \mathbb{P}\{\sup_{t \in [0,T]}W_1(t)-c_1 u^{\red{\alpha}} t > u^{\red{\beta}}, W_2(t)-c_2 u^{\red{\alpha}} t > au^{\red{\beta}}\},$$. as $u \to \IF.$ Constants $c_1,c_2, \alpha \in \mathbf{R}, \red{\beta \geq 0}$ model the behaviour of the drift function, which is dependent on the initial capitals $u, au, a \le 1.$ Moreover, we give the connection and give asymptotics for the two-dimensional many-source ruin model. $$\mathbb{P}\{\sup_{t \in [0,T]}\sum_{i=1}^N B_i(t)-c_1 t > N, \sum_{i=1}^N B_i'(t)-c_2 t > N\},$$. $$\mathbb{P}\{\sup_{t \in [0,T]}\sum_{i=1}^N B_i(t)-c_1 t > N, \sum_{i=1}^N B_i'(t)-c_2 t > N\},$$ where $B_i, B_i'$ are standard Brownian motions, $Cov(B_i(t),B_i'(t))=\rho t,$ as $N \to \IF.$

This talk is a contributed talk at EVA 2021. View the programme here.