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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:


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


How can I calculate this? Thanks for you help


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'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
up vote 0 down vote accepted

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 $$ \mathbb E(\mathrm e^X\mathbf 1_{X\gt f(\rho)}\mid\mathcal F_t)=\mathbb E(\mathrm e^X\mathbf 1_{X\gt f(\rho)}),\qquad\mathbb P(X\gt f(\rho)\mid\mathcal F_t)=\mathbb P(X\gt f(\rho)). $$

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