The following question is taken from Mark Joshi's Quant Job Interview.
Question: What is the joint distribution of $W_t$ and $I(t)$ where $(W_t)_{t\geq 0}$ is Brownian motion and $$I(t) = \int_0^t W_s \,ds?$$
When one wants to find joint density, one can start at joint CDF followed by partial differentiation with respect to each random variable.
In this case, however, we have $$P(W_t \leq x, I(t) \leq y).$$ I do not know how to evaluate the probability above.
If they are independent, then I know how to evaluate the product as each of the random variable follows a normal distribution.
