Constraint on Girsanov Transformation

[quotes]

If the drift of a Brownian motion is constant, Radon-Nikodym derivative could be applied to Girsanov transform the drift to zero, or from zero to a constant.

This is no doubt.

However, I doubt if it can also be applied to a stochastic drift, say, a mean-reverting stochastic drift.

Just like Sum(b*dW)=bW but Sum(WdW)\=W*W

and Sum( VdW)\=V*W where V is another Brownian Motion indepedent of W.

I doubt if Girsanov transformation can be applied to stochastic drift or interest rate.

 

[amir]

this is a good question.

If the drift of a Brownian motion is constant, Radon-Nikodym derivative could be applied to Girsanov transform the drift to zero, or from zero to a constant..

but what are you referring to by the drift! both the diffusion's drift and the girsanov kernel can be called drift of BM

 

[quotes]

To simplify, just the drift of the diffusion.

 

[amir]

as you know the only constraint on the drift is that it has to an adapted process. I am also curios to see if there is any work answering your question.

btw, this statement seems a bit unclear because this is not what the Girsanov's transformation does with the drift

... to Girsanov transform the drift to zero, or from zero to a constant

 

[quotes]

as you know the only constraint on the drift is that it has to an adapted process. I am also curios to see if there is any work answering your question.

btw, this statement seems a bit unclear because this is not what the Girsanov's transformation does with the drift

"adapted process" might mean that for any given small time interval, we already know the drift before hand, therefore drift is held constant in that interval and not stochastic compared to the volatile part of Brownian Motion.

The Girsanov's Transformation originally involves only one Brownian Motion with completion of squares. Given a simple Brownian motion, it may or may not have drift given different probabilities. You can easily transform the drift from zero to some or from some to zero, or even some to another some (importance sampling), the key is to divide different densities of the same time.

Shreve's book deals with Radon-Nikodym derivative with two Brownian Motions, which actually complicates the problem. I wish I could find Girsanov's original work

On transforming a certain class of stochastic processes by absolutely continuous substitution of measures (1960)

and show you this concept.

 

[DStahl]

"adapted process" might mean that for any given small time interval, we already know the drift before hand, therefore drift is held constant in that interval and not stochastic compared to the volatile part of Brownian Motion.

The Girsanov's Transformation originally involves only one Brownian Motion with completion of squares. Given a simple Brownian motion, it may or may not have drift given different probabilities. You can easily transform the drift from zero to some or from some to zero, or even some to another some (importance sampling), the key is to divide different densities of the same time.

Shreve's book deals with Radon-Nikodym derivative with two Brownian Motions, which actually complicates the problem. I wish I could find Girsanov's original work

On transforming a certain class of stochastic processes by absolutely continuous substitution of measures (1960)

and show you this concept.

An adapted process is measurable with respect to the filtration of the Wiener process. This definition does not preclude the possibility of stochastic drift, at least as far as I can tell.

 

[quotes]

Sorry I misinterpret the definition of "adapted" as "previsible". What I meant in post 5 is that if drift is "previsible", then it can be treated as constant in discrete time intervals like the weights in a self-financing portfolio.

Therefore my challenge on the theorem still stands.

It is fine to integrate drift terms if it is differentiable and variation-bounded in classical sense. But Ito integration is needed if it is stochastic.
e.g.
(\int a_t \,dW_t)
(\int (A-W_t)\,dW_t)

Do you integrate the two with the same method?
Let's look at the Girsanov' Theorem more closely:

The probability density function for dW=w without drift is
(\frac{1}{\sqrt{2\pi dt} }e^{-\frac{w^2}{2dt})

and the one with drift a is
(\frac{1}{\sqrt{2\pi dt} }e^{-\frac{(w-adt)^2}{2dt})

If we divide the latter on the former we get the Radon-Nikodym derivative in differential form Adding drift, while the reverse Eliminating drift.

(e^{aw-\frac{a^2}{2} dt} \, ; \,e^{-aw-\frac{a^2}{2} dt} )

or

(e^{adW-\frac{a^2}{2} dt}\, ; \,e^{-adW-\frac{a^2}{2} dt})

Now if it is multiplied through time horizon we get the Radon-Nikodym derivative in integral form.

(exp\{aw_1-\frac{a_1^2}{2} dt\}exp\{aw_2-\frac{a_2^2}{2} dt\}...=exp\{\int a_t dW -\frac{a_t^2}{2} dt\})


However the question arises if a_t is stochastic. We can no longer add up aw's or a_tdW as that simple Ito integral, say a_t = A-W_t or the integration just needs simplification? like the

(\int Za \,dW) ?

Note I use (a) instead of (\theta) for simplification.

 

[DStahl]

Using the Shreve "proof" he simply uses the fact that Ito integrals (whether a_t is stochastic or not) are martingales if a_t is adapted. He shows that ( Z_t \tilde{W}_t ) is a martingale under P (regardless of ( \theta_t ) ) and then uses Lemma 5.2.2 to show that ( \tilde{W}_t ) is then a martingale under the risk neutral measure. This is enough (by Levy) to show that ( \tilde{W}_t ) is a brownian motion under the risk neutral measure.

 

[quotes]

Using the Shreve "proof" he simply uses the fact that Ito integrals (whether a_t is stochastic or not) are martingales if a_t is adapted. He shows that ( Z_t \tilde{W}_t ) is a martingale under P (regardless of ( \theta_t ) ) and then uses Lemma 5.2.2 to show that ( \tilde{W}_t ) is then a martingale under the risk neutral measure. This is enough (by Levy) to show that ( \tilde{W}_t ) is a brownian motion under the risk neutral measure.

On page 212, he indeed states (\Theta (u) ) needs to be an adapted process, and the convergence condition comes from 4.3.1, where Ito integral is defined. He further showed an example in 4.3.8 that the integrand in an Ito integral can be stochastic (in fact can jump) but is still a martingale. So I understand the Radon-Nikodym derivative might involve unsimplified Ito integrals like the examples of mine or Shreve's. My lack of knowledge on Ito-integral caused the doubt.

Thank you for referring to Shreve's book. My proof above sometimes appears as heuristics in lecture notes for it does not check the convergence constraint on Ito integral.

 

[quotes]

No. Shreve's proof is only a sketch. I found the drifted Brownian Motion must be at least Markov to determine the time differential of Radon-Nikodym derivative before multiplying them up.