# How to solve the expected value of such integral

$$V_t = E^\mathbb{Q} \left[\int_t^{+\infty} e^{-r(u-t)}X_u \, du|X_t\right]$$

with a given process $X_t$ satisfied: $dX_t = (\mu-\sigma^2 \gamma) X_t \, dt + \sigma X_t \, dW_t^\mathbb{Q}$

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Please do not use homework as the only tag in your question. Add a tag that reflects the subject matter of your question. (I have done it for you this time.) –  Guess who it is. Jul 27 '12 at 0:35

• Show that $V_t = \displaystyle\int_t^{+\infty} \mathrm e^{-r(u-t)}Y_{t,u} \, \mathrm du$ where, for every $t\leqslant u$, $Y_{t,u}=E(X_u\mid X_t)$.
• Find $b$ such that $(Z_t)_{t\geqslant0}$ is a martingale, where $Z_t=\mathrm e^{bt}X_t$.
• Deduce that $Y_{t,u}=\mathrm e^{-b(u-t)}X_t$.
• Conclude that $V_t=(r+b)^{-1}X_t$, for every $r\gt-b$.