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2 votes
Accepted

Variance of an Itô Integral

The variance of $$ \int_t^TZ_s-Z_t\ ds $$ is \begin{align}&\textstyle \mathbb E\left[\left(\int_t^TZ_s-Z_t\ ds\right)\left(\int_t^TZ_u-Z_t\ du\right)\right]=\displaystyle\int_t^T\int_t^T\mathbb E\...
Kurt G.'s user avatar
  • 14.8k
1 vote
Accepted

Find a PDE for $f$ satisfying $f(t,Y_t) = \exp(- \frac{\gamma^2}{2} t + \gamma W_t) E[\exp(\frac{\gamma^2}{2} T - \gamma W_T) F(Y_T) | \mathcal{F}_t]$

Note that we can rewrite the random variable in the following way: $$\begin{aligned} &e^{-\gamma^2t/2+\gamma W_t}E^P[e^{\gamma^2T/2-\gamma W_T}F(Y_T)|\mathscr{F}_t]\\ &=e^{-\gamma^2t/2+\gamma ...
Snoop's user avatar
  • 15.5k
1 vote
Accepted

System of Stratonovitch SDEs $dX = \sigma X \circ dW$ to a system of Ito SDEs

This is true. In the one-dimensional case the transformation between the two types of integrals is, as we know, $$ \underbrace{\int_0^tY_s\circ\,dW_s}_{\text{Stratonovich}}=\underbrace{\int_0^tY_s\,...
Kurt G.'s user avatar
  • 14.8k
1 vote

Solving this SDE $dX_t = aX_tdt + bdW_t$, $X_0 = x$ to find $E[X_t^2]$

This question is old but there are errors in signs and many typos in the accepted answer above. The answer is also has many upvotes (and rightfully so). But still, it posseses an issue for a reader. ...
Mr.Gandalf Sauron's user avatar

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