Re: Girsanov theorem and singular noise

From: Hans Engler (h.engler_at_verizon.net)
Date: 03/24/05 

Date: Thu, 24 Mar 2005 16:00:05 +0000 (UTC)

> If I have understood correctly the Girsanov theorem can be used for
> computing the likelihood ratio dQ/dP of the probability laws of the
> stochastic differential equations
>
> law Q: dX_t = f(X_t,t) dt + dB_t (1)
> law P: dY_t = dB_t
>
> where X_t \in R^n and B_t \in R^n is a *standard* brownian motion. But in
> many applications the system dynamics are better modeled with SDE's of
> the form
>
> law Q': dX_t = f(X_t,t) dt + L dB_t
>
> where L \in R^{n*n} is a *singular* matrix. That is, the noise process has
> a singular covariance matrix.
>
> Now the question: Is there a way to compute the likelihood ratio dQ'/dP?

Not as a density. Take L = 0, i.e. the case of a deterministic ode. Then Q'
is supported in a single trajectory.

> Or is there a way to compute something related, for example, dQ'/dP' where
> P' is the law of
>
> law P': dY_t = L dB_t

Again the answer is no. Take L = [1 0; 0 0] in R^2, i.e. P' corresponds to
1-D Brownian motion embedded in R^2, and
f(X_t,t)= [0 1]^T, i.e. drift in an orthogonal direction. The two laws have
disjoint support in this case.

It appears that some form of compatibility is needed to get a formula for
dQ'/dP', e.g. f should leave the column space of L invariant.

Hope this helps

Hans