Morgan Stanley Interview Question: What is the probability of a ... | Glassdoor

## Interview Question

Quantitative Analyst Interview New York, NY

# What is the probability of a Brownian motion hit 1 before

hitting -2. The Brownian motion starts at 0.

4

2/3

James on Nov 20, 2011
1

Why ?

Anonymous on Apr 21, 2012
1

Let p be the required probability
If the first move is to the right(with prob 0.5), it hits 1
If the first move is to the left, then we have to find the prob of hitting 1 before hitting 2, starting from -1, which is equal to (1-p).
So p = 0.5 + 0.5*(1-p)
p=2/3

ah on May 19, 2012
0

what is browian motion??

me on Jul 19, 2012
8

A martingale approach is easier and more intuitive. Standard Brownian motion is martingale, and with stopping time it is still a martingale. Given a stopping time tao, when it first hits 1 or -2, E[Btao] = B0 = 0 = p*1 + (1-p)*(-2). Therefore, 2/3.

Anonymous on Dec 2, 2015
2

let desired probability be p
if the first step is 1, then p(-2 reached after 1) = 1
if the first step is -1, then if second stop is -1 , we get to -2 without getting to 1
if the first step is -1, then if second step is 1, we get back to origin

hence: p = 0.5*1 + 0.5*0.5*p + 0.5*0.5*0
p = 2/3

jam on May 22, 2016
0

a stopped martingale is still a martingale.
Ans: 2/3

Ashish on Jul 15, 2019