# Computing conditional expectation of function of Ito process

In some projects I got to a conditional expectation of the form:

$$E\left[\exp\left\{\int_s^tX(r)f(r)dr\right\}\mid\mathcal F_s\right]$$

where $$t\geq s$$, $$f$$ is nice enough and $$X$$ is an Ito process and $$\mathcal F_s$$ is the information about $$X$$ up to time $$s$$. I want to compute this. Is there anyway to do so?

I tried Taylor expanding $$\exp$$, I tried to apply Ito formula in a few clever ways and nothing works out nicely. Is there anyway of computing this?

Disclaimer. I'm using $$T$$ instead of $$t$$ for the terminal time and $$t_0$$ instead of $$s$$ for the initial time. I am ignoring issues arising due to a lack of regularity and, as you say, assuming that the drift, diffusion, and $$f$$ are nice enough.
Since $$X$$ is an Ito process and hence Markov, it is sufficient to consider $$v(t,x)\equiv\mathbb{E}\left[\exp\left\{\int_{t}^{T}X(s)f(s)ds\right\}\,\middle|\, X(t)=x\right].$$ By the Feynman-Kac formula, it follows that \begin{align*} v_{t}+Lv+xfv & =0, & \text{on }[t_{0},T)\times\mathbb{R}\phantom{.}\\ v(T,\cdot) & =1, & \text{on }\mathbb{R}. \end{align*} where $$L$$ is the infinitesimal generator of $$X$$.
You can solve this PDE numerically to get the value of $$v(t_0, X(t_0))$$. Alternatively, you can use Monte Carlo simulation on the original expectation, but this is generally slower than the solving the PDE numerically due to the problem's low dimensionality.