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Let $X_t$ be a solution to the SDE, $dX_t=-aX_t \; dt+\sigma \; dW_t$, $a>0$, $\sigma>0$, $X_0=\text{constant}$ where $W_t$ is Brownian. What is the joint distribution of $(X_t, \int_0^t X_s \; ds)$?

I have calculated the solution to the SDE, I have $X_t=e^{-at}[X_0+\int_0^t \sigma e^{as} \; dW_s]$, I need to also get $\int_0^t e^{-as}[X_0+\int_0^s \sigma e^{au} \; dW_u] \; ds$ but I am not sure how to calculate that, and also the distribution of $(X_t, \int_0^t X_s \; ds)$

Thanks for your help

I have also shown that $X_t$ is a gaussian , but I am not sure how to find the distribution of the second element in the vector.

  • 2
    Leon: are you related to the author of [this question](http://math.stackexchange.com/q/74783/6179)?2011-10-22
  • 0
    We are working doing some practice questions together2011-10-22

2 Answers 2