Apply Ito formula with integral functions

Question

I have an issue trying to apply Ito's Formula on a specific dynamics. Basically I have the following dynamics:

$df(t,T) = \frac{1}{2}\partial_T(\sigma^2(t,T))dt + \partial_T(\sigma(t,T))dW(t)$

Where $W(t)$ is a Winer process.

What I'm looking for is the dynamics of:

$B(t,T) = e^{-\int_t^T f(t,s)ds }$

I tryed to apply Ito's formula but I cannot write $B(t,T)$ as $h(t,T,f(t,T))$ because of the integral function at the exponent.

Is there any way to find such an $h$ so that I can apply Ito's formula?

In case of negative answer, how can I derive the the dynamics of $B(t,T)$ w.r.t. $t$? (i.e. the Ito differential $dB(t;T)$ )

Answer 1

Let us first compute the dynamic of $\int_t^Tf(t,s)ds$: \begin{align} d\left(\int_t^Tf(t,s)ds\right) &= -f(t,t)dt + \int_t^Tdf(t,s)ds \\ &= -f(t,t)dt + \int_t^T \left(\frac{1}{2}\partial_T(\sigma^2(t,s))dt + \partial_T(\sigma(t,s))dW(t) \right)ds \\ &= -f(t,t)dt + \alpha(t,T)dt + \Gamma(t,T)dW(t) \end{align} Now applying Ito to the function $\phi(x) = \exp(-x)$ with the process $\int_t^Tf(t,s)ds$, we have : \begin{align} dB(t,T) = B(t,T)\left[\left(f(t,t)-\alpha(t,T) + \frac12\Gamma(t,T)^2\right)dt - \Gamma(t,T)dW(t)\right] \end{align}

Answer 2

You have the form:

$f(X) = e^{-X}$

\begin{align} df(X) & = \frac{\partial f}{\partial X}dX + \frac{1}{2} \frac{\partial^2 f}{\partial X^2} (dX)^2 \\ & = -fdX + \frac{1}{2}f(dX)^2 \\ \end{align}

Substitute $dX = df(t,T)$ as given at begin then you obtain the result