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Ok, I'm new to stochastic calculus and I'm having some troubles with a simple exercise that I don't seem to get. Here it is:

Recalling that $\mathbb{E}[e^{W_t}]=e^{\frac{t}{2}}$ compute $\mathbb{E}[e^{W_t}|\mathcal{F}_s]$, with $s\leq t$.

Now the expectation that I'm supposed to remember is simply an expectation of a lognormal distribution with mean zero and variance $t$. I don't get the second point: how can the introduction of filtration affect the expected value? I would appreciate an explanation as general as possible, in order to avoid having problems with similar exercises even in the future... thanks!

BCLC
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james42
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  • "how can the introduction of filtration affect the expected value?" Wait, what do they explain about filtrations in your notes, already? – Did May 30 '15 at 08:10
  • Now I don't have notes at hand, but a filtration is an increasing family of sigma algebras, fancy term that stands for "information". Now, if the process above is a martigale, its expectation conditioned w.r.t. a filtration doesn't change, but since it isn't (non zero drift after applying Itö's lemma), we shoud expect that conditioning affects expected value, but how? – james42 May 30 '15 at 08:16
  • Why does this have a downvote? – BCLC May 30 '15 at 08:42
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    Maybe becouse somebody thinks that is an easy question! xD – james42 May 30 '15 at 08:52
  • ale42, re "Now, if the process above is a martigale, its expectation conditioned w.r.t. a filtration doesn't change" --> actually, conditional expectation on a random variable, sigma-algebra or filtration is a random variable. – BCLC May 30 '15 at 08:58

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$\mathbb{E}[e^{W_t}|\mathcal{F}_s]$

$=\mathbb{E}[e^{W_s}e^{W_t - W_s}|\mathcal{F}_s]$

$=e^{W_s} \mathbb{E}[e^{W_t - W_s}|\mathcal{F}_s]$ (why?)

$=e^{W_s} \mathbb{E}[e^{W_t - W_s}]$ (why?)

Now note that $W_t - W_s$ is a random variable distributed $N(0, t-s)$.

Do you remember moment generating functions?

As to how filtration can affect expectation, in general:

$E[X|Y] \neq E[X]$

where Y can be an event, random variable, $\sigma$-algebra or filtration.

Do you remember conditional expectation?

BCLC
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    Ok, now i see what you have did. You added and subtracted $W_s$ and brought $e^{W_s}$ outside the expectation since it is $\mathcal{F}_s$ measurable. But why after those manipulations you can get rid of the conditioning w.r.t. filtration? – james42 May 30 '15 at 08:51
  • Independence http://en.wikipedia.org/wiki/Brownian_motion#Mathematics If you want, you can check out some of my martingale-related questions last year:

    http://math.stackexchange.com/questions/962613/prove-a-t-w-t3-3t-w-t-a-martingale

    http://math.stackexchange.com/questions/974969/prove-x-is-a-martingale

    http://quant.stackexchange.com/questions/14955/determine-ew-p-w-q-w-r

    http://quant.stackexchange.com/questions/14956/show-that-eb-t-mathscrf-s-b-s

     – BCLC May 30 '15 at 08:55
  • Btw if you want, look up p.13-14 here http://www.math.zju.edu.cn/zlx/forphd/stochcalcuslecture.pdf There is a more general version of your question – BCLC May 30 '15 at 09:00
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    How can I do it? I don't see any "accept answer" button in my iphone interface xD – james42 May 30 '15 at 09:00
  • Ahh, Apple, my mortal enemy http://youtube.com/watch?v=jBeDOvxatKw&feature=youtu.be&t=166 – BCLC May 30 '15 at 12:59