# Simulation of hitting time of brownian motion with drift

I want to generate the hitting time of brownian motion with drift (upper and lower depending on some binomial random variable $$\delta = 0,1$$).

$$\tau^{up} = inf(\tau : \mu \tau + \sigma W_{\tau}\ge h)$$

$$\tau^{down} = inf(\tau : \mu \tau + \sigma W_{\tau}\le -h)$$

The probability density functions are

$$p^{up}(s)= \frac{h}{\sqrt{2\pi\sigma^2s^3}}\exp{(-\frac{(h-\mu s)^2}{2\sigma^2 s})}$$

$$p^{down}(s)= \frac{h}{\sqrt{2\pi\sigma^2s^3}}\exp{(-\frac{(h+\mu s)^2}{2\sigma^2 s})}$$

The random variable with first one density can be generated using Wald distribution. But the second one is the problem. Should I use some numercial method for this like Inverse transform sampling, Acceptance-Rejection Method or Monte-Carlo, or it can be simplefied to common distribtuion.

• If the drift $\mu \gt 0$ then there may be a positive probability that you never hit $-h$ – Henry Nov 28 '18 at 0:11