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 March 5th, 2017, 12:44 PM #1 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 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
 March 5th, 2017, 01:15 PM #2 Global Moderator   Joined: May 2007 Posts: 6,416 Thanks: 558 What expression do you have for $Y_t$? Last edited by mathman; March 5th, 2017 at 01:17 PM.
 March 5th, 2017, 01:21 PM #3 Senior Member   Joined: Feb 2015 From: london Posts: 121 Thanks: 0 $\displaystyle Y_t = \frac{1}{X_t}$

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