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Let $\{B(t), t\ge 0\}$ be a Brownian motion. Consider the Brownian motionrisk model with interest rate collection and tax payment defined by\begin{align}\label{Rudef}\widetilde{U}_\gamma^\delta(t)=\widetilde{X}^\delta(t)-\gamma\sup_{s\in[0,t]}\left(\widetilde{X}^\delta(s)e^{\delta(t-s)}-ue^{\delta(t-s)}\right),t\ge0,\end{align} with $$\widetilde{X}^\delta(t)=ue^{\delta t}+c \int_0^t e^{\delta(t-v)}dv-\sigma \int_0^t e^{\delta(t-v)}dB(v),$$ where $c>0,\gamma \in [0,1)$and $\delta \in \mathbb{R}$ are three given constants. When $\delta=0$ and$\gamma \in (0,1)$ this is the risk model introduced from Albrecher and Hipp in\cite{AH2007} where the ruin probability in the infinite time horizon has beenexplicitly calculated. In the presence of interest rate $\delta\neq0$, thecalculation of ruin probability for this risk process for both finite andinfinite time horizon seems impossible. In the paper, based on asymptotictheory we propose an approximation for ruin probability and ruin time when theinitial capital $u$ tends to infinity. Our results are of interest given thefact that this can be used as benchmark model in various calculations.

Ruin problem for Brownian motion risk model with interest rate and tax payment