March 5th, 2017, 11:44 AM  #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 $\displaystyle \frac{dX_t}{X_t} = u dt + \sigma dW_t$ $\displaystyle X_t = X_0 e^{(u0.5\sigma^2)t +\sigma W_t}$ 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, 12:15 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,552 Thanks: 597 
What expression do you have for ?
Last edited by mathman; March 5th, 2017 at 12:17 PM. 
March 5th, 2017, 12:21 PM  #3 
Senior Member Joined: Feb 2015 From: london Posts: 121 Thanks: 0 
$\displaystyle Y_t = \frac{1}{X_t}$


Tags 
differential, equation, stochastic 
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