I'm working through a textbook on SP and I'm trying to do some of the exercises.

Here's the question followed by my attempt at a solution.

For a standard brownian motion started at x=-3, find P(X1 + X2 > -1)

Let Y = X1+X2

We want P(Y > -1)

Finding the distribution of Y:

E(Y) = E(X1 - 3 + X2 - 3)

= E(X1 - 3) + E(X2 - 3)

= 0 - 3 + 0 - 3

= -6

Var(Y) = Var(X1 - 3) + Var(X2 - 3) + 2*Cov(X1-3, X2-3) (cov is given by the minimum of {1,2})

= 1 + 2 + 2*(1)

=5

So Y ~ N(-6, 5)

In R, I find

Code:

```
> 1- pnorm(-1,-6,sqrt(5))
[1] 0.01267366
```