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March 5th, 2017, 11:44 AM   #1
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Stochastic differential equation

Let $\displaystyle X_t$ be the price of a stock, where

$\displaystyle \frac{dX_t}{X_t} = u dt + \sigma dW_t$
$\displaystyle X_t = X_0 e^{(u-0.5\sigma^2)t +\sigma W_t}$

where u and $\displaystyle \sigma$ are strictly positive constance and $\displaystyle W_t$ is Brownian motion and has distribution ~N(0,t)

Let $\displaystyle Y_t = \frac{1}{X_t}$

The question asks to show that $\displaystyle E[X_t]E[Y_t] = e^{\sigma^2t}$

I have calculated that $\displaystyle E[X] = X_0 e^{ut}$, which seems correct from the web, but every time I try to calculate $\displaystyle E[X_t]E[Y_t]$ I just end up with 1

Any help to try and prove $\displaystyle E[X_t]E[Y_t] = e^{\sigma^2t}$ would be much appreciated
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March 5th, 2017, 12:15 PM   #2
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What expression do you have for ?

Last edited by mathman; March 5th, 2017 at 12:17 PM.
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March 5th, 2017, 12:21 PM   #3
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$\displaystyle Y_t = \frac{1}{X_t}$
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