# Standard Brownian Motion Probability Computation

While studying for my exam I came across the following question:

Let $$W(t)$$ be a standard Brownian motion. Find

$$\mathbb{P}(00)$$

Attempt:

(1) Attempt 1: Finding the distribution of W(1)+W(2) which I found to be $$\mathcal{N}(0,5)$$ and same for $$3W(1)-2W(2)$$ which is $$\mathcal{N}(0,5)$$ my attempt is to prove that the two things are independent so I could separate both terms of the probability and multiply two quantities as

$$\mathbb{P}(00)$$

but this didn't work out yet.

(2) Attempt 2: Working with the inequalities to find common bounds i.e I found $$\dfrac{2}{3} Y but this isn't so useful.

$$\mathbb{P}(00)$$

$$=\mathbb{P}(0<2W(1)+\Delta W(2)<2, W(1)-2\Delta W(2)>0)$$

where $$\Delta W(2)=W(2)-W(1)$$. Given the properties of a Wiener process $$[W(1),\Delta W(2)]$$ is jointly normal with mean (0, 0) and covariance $$diag(1, 1)$$.

Now

$$\mathbb{P}(00)$$

$$=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(0<2x+y<2, x-2y>0)f(x)f(y)dxdy.$$

Where $$f$$ is the pdf of a standard normal variable. This can be evaluated numerically.