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I'm currently looking at stochastic differential equations with irregular coefficients such as $W^{1,p}_\mathrm{loc}$. If I have \begin{align} dX_t=b(X_t) \, dt+\sigma \, dW_t, \end{align} where $b\in W^{1,1}_\mathrm{loc}$ and $\sigma$ is a constant, I can write this SDE as a ODE for every Brownian path $w$ by defining $Y_t=X_t-w_t$ and a new vector field $b^w=b(Y,t)=b(Y_t+w_t)$, so the ODE is \begin{align} dY=b^w(t,Y) \, dt \end{align} with initial condition $Y_0=y$. Since $b^w$ has Sobolev regularity I can then apply DiPerna-Lions theory (1989), which guarantees the existence and uniqueness of the flow of the ODE.

My question is now what happens if $\sigma$ is not a constant,
\begin{align} dX_t=b(X_t) \, dt+\sigma(X_t) \, dW_t. \end{align} Apparently the above argument is NOT correct in general. Is there any ways that the diffusion coefficients $\sigma(X_t)$ can be absorbed into the Brownian motion? or maybe there are some conditions on $\sigma$ under which the above argument still hold?

Many thanks!!

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@ dynamic89 : Hi, I don't get your first SDE, usually drifts are of the form $b(X_t)dt$ for diffusion equations, is it a typo or a specific formalism that I have never seen before ? If this is the case do you have any reference for the definition of such SDEs ? Best regards – TheBridge Sep 7 '12 at 13:32
Ooops typo indeed..sorry! – dynamic89 Sep 7 '12 at 13:57
Shouldn't it be $Y_t = X_t - \sigma w_t$ ? What is $W^{1,p}_{loc}$ ? – vanna Sep 7 '12 at 14:07
@vanna: yeah but it doesn't matter, $\sigma$ is just a constant anyway. $W^{1,p}_{loc}$ is Sobolev space - locally weakly differentiable functions in $L^p$ and its first weak derivative is in $L^p$ too. This part is not important but what's the conditions on $\sigma$ under which we can do such change of variables.... Thanks!! – dynamic89 Sep 8 '12 at 0:31
up vote 0 down vote accepted

To answer very partially you can under some (quite stringent) conditions use Lamperti's Transform to transfer the volatility term $\sigma(X_t)$ into the drift term.

The other way around Girsanov's transform allows you to make a change of probability measure that offsets the drift terms still under some conditions and leaves the diffusion term unchanged.

Otherwise there are some other transforms that could be usefull, timechange is one of them but you have to realise that it comes with the cost of a change in the filtration. This transform could allow you to go from $\sigma(X_t)dW_t$ to $dW'_{\phi(t)}$.

I remember a paper about Lie group theory applied to SDE in the same way that it is applied to the ODE case. I'll give you the reference what was nice about it was that (but only in the 1-dim case) it covered almost all transforms that already knew about.

Two last streams of research that could interest you are "quasi sure analysis" and "Rough Path Theory"

Best regards

Edit : Here is the Title of the paper I mentionned :
Kozlov - The Group Classification of a Scalar Stochastic Differential Equation

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Great answer! Thanks! I'll look into them. – dynamic89 Sep 11 '12 at 11:33

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