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Suppose we have an SDE, which is the Wiener process with drift

$dr_t=c dt+\sigma dB_t$, where $B_t$ is brownian

I want to find $\mathbb{E}[e^{-\int_0^t r_s ds} |r_t=r]$

so my approach is this : write the SDE as : $r_t-r_0=ct+\int\sigma dB_t$

Then I know $r_t$ is distributed as a normal. but then how can i get the distribution of $\int_0^t r_s ds$ and hence the expectation?

thanks

  • 0
    Does "OU process" stand for the [Ornstein-Uhlenbeck process](http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) ? The SDE you wrote is that of the Wiener process with drift.2011-10-27
  • 0
    i mean Wiener process with drift2011-10-27

2 Answers 2