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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:

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

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

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