Cumulative Integration with regard to Vasicek Model’s Bond Price and its Forward Price

stochastic-analysisstochastic-calculusstochastic-integralsstochastic-pdestochastic-processes

(My Question)

Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M.

$$E\left[ \exp \left( – \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \right) \right]$$

(Thank you for your help in advance.)

(Cross-link)

I have posted the same question on https://quant.stackexchange.com/questions/47522/cumulative-integration-with-regard-to-vasicek-models-bond-price-and-its-forward/47524#47524

(Original Question)

Solve $P(t, T)$ with the following model

$$dr_t=-br_t dt + \sigma dB_t$$

(My consideration)

  • Fist,

$$r_u=e^{-bu} r_0 + \int^u_0 \sigma e^{-b(u-s)} dB_s$$

  • Second,

\begin{eqnarray}
P(t, T) &=& E \left[ \exp \left( – \int^T_t r_u du \right) \middle| \mathcal{F}_t \right] \\
&=& E\left[ \exp \left( – \int^T_t \left(e^{-bu} r_0 + \int^u_0 \sigma e^{-b(u-s)} dB_s \right) du \right) \middle| \mathcal{F}_t \right] \\
&=& E\left[ \exp \left( – \int^T_t e^{-bu} r_0 du – \int^T_t \int^u_0 \sigma e^{-b(u-s)} dB_s du \right) \middle| \mathcal{F}_t \right] \\
&=& \frac{r_0}{b} (e^{-bT}-e^{-bt}) E\left[ \exp \left(- \int^T_t \int^u_0 \sigma e^{-b(u-s)} dB_s du \right) \middle| \mathcal{F}_t \right]
\end{eqnarray}

  • Third, I assume to use the following formula, but I cannot have any idea to replace the integration order.

$$E\left[ \exp \left( \int^T_t f(s) dB_s \right) \middle| \mathcal{F}_t \right] = \exp \left( \frac{1}{2} \int^T_t f(s)^2 ds \right) $$

Thank you for your help in advance.

Best Answer

(Cumulative Integration Formula Replacing $du$ and $dB_s$)

  • I have developed formulas to solve this by myself!

\begin{eqnarray} \int^t_0 \int^u_0 dB_s \ du &=& \int^t_0 \int^u_s du \ dB_s \\ \int^T_t \int^u_0 dB_s \ du &=& \int^T_0 \int^u_s du \ dB_s - \int^t_0 \int^u_s du \ dB_s \end{eqnarray}

  • Therefore, we can use the following formula as I assume.

$$E\left[ \exp \left( \int^T_t f(s) dB_s \right) \middle| \mathcal{F}_t \right] = \exp \left( \frac{1}{2} \int^T_t f(s)^2 ds \right) $$

  • I solved the Vasicek Bond Price computation using the formulas.

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