# How to calculate this conditional expectation

Let $X$ be a normal distribution with mean $\mu$ and variance $\sigma^2$. Furthermore we have a constant $\rho$. Suppose our probability space is carrying a filtration. Now I want to calculate the following conditional expectation:

$$E[e^X\mathcal1_{\{X>f(\rho)\}}\mid\mathcal{F}_t]$$

where $f$ is a deterministic function. Similar I want also to calculate

$$E[\mathcal1_{\{X>f(\rho)\}}\mid\mathcal{F}_t]$$

How can I calculate this? Thanks for you help

hulik

• 'carrying a filtration' is too vague. can you be more specific ? – mike Dec 19 '12 at 12:22
• @mike we can assume that it satisfies the usual conditions, i.e. right continues and $P$-complete. – user20869 Dec 19 '12 at 12:39
• It is impossible to say anything meaningful since we don't know what ${\cal F}_t$ is. – Yury Dec 20 '12 at 2:07

As @Yury explained in the comments, it is impossible to say anything meaningful without knowing the relationship of $X$ and $\mathcal F_t$. For example, if $X$ is $\mathcal F_t$-measurable then $$\mathbb E(\mathrm e^X\mathbf 1_{X\gt f(\rho)}\mid\mathcal F_t)=\mathrm e^X\mathbf 1_{X\gt f(\rho)},\qquad\mathbb P(X\gt f(\rho)\mid\mathcal F_t)=\mathbf 1_{X\gt f(\rho)}.$$ On the other hand, if $X$ is independent of $\mathcal F_t$ then $$\mathbb E(\mathrm e^X\mathbf 1_{X\gt f(\rho)}\mid\mathcal F_t)=\mathbb E(\mathrm e^X\mathbf 1_{X\gt f(\rho)}),\qquad\mathbb P(X\gt f(\rho)\mid\mathcal F_t)=\mathbb P(X\gt f(\rho)).$$