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

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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Hints:

  • 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 $r\gt-b$.
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