Right I am sure there must be a name for this in the literature which I very much like to know.Consider this example.Suppose you have a measurable space and a filtration (assume the standard assumptions). You have a probability measure P1 with a Brownian motion W1. You define a martingale process s(t) say with constant vol sigma and initial value s(0).Now I give you a different probability measure P2 over the same space and filtration with a Brownian motion W2. You define a martingale process with the same constant vol sigma and initial value. So this new process is "sort of a copy" of s(t) under the measure P2. What is the correct name for such relationship between these processes?Thanks

isonomy?

Thanks it looks like the correct term. (isonomy = equality in law)

Depending on your filtration, it might be called the Girsanov transform .

The law of s1(t) under P1 is identical to the law of s2(t) under P2.

and ,more broadly speaking, how to characterize the class of random processes (or might be better to say its distribution) for which there exists a measure under which any process of this class have identical distribution.

That is what I had in mind: glad you liked it, and nice googling

It seems the last could correspond to the concept "isonomy class". So far "isonomy" appears to be a bit of an esoteric term as it not in any of the textbooks I have at hand. I've found a definition of isonomy and isonomy class in the following paperCYLINDRICAL MEASURES ON TENSOR PRODUCTS OFBANACH SPACES AND RANDOM LINEAR OPERATORSNeven Elezovichttp://www.springerlink.com/content/w030661kv0p0u212/There are also some excerpts on line from the book Stochastic Differential Equations on Manifolds (London Mathematical Society Lecture Note Series) by K. D. Elworthy (1982) that define the concept. If you do a search on "isonomy stochastic" it will come up. Thanks again for your replies.

croot: I am writing a paper and would like to thank you in the acknowledgments. If this is OK with you please send your real name to manilla@email.com. Thanks.

QuoteOriginally posted by: manillaThanks it looks like the correct term. (isonomy = equality in law)if isonomy = equality in law then it looks sound too general. If we talk about SDE solutions with non degenerative diffusion then the class of SDEs that satisfies yours property is arbitrary drift coefficient and fixed diffusion coefficient. All solutions will have absolute continuous measures. Some restriction on coefficients is Novikov's condition. In general, if you do not have processes and only measures are available. Then if a class jointly absolute continuous measures (rho and rho^-1 are density) then one can construct random processes ( theorem Kolmogorov) on the space of all functions and processes probably would satisfy your property.

list: I agree with what you say, if you want to define the concept accurately it will become very involved. In my modest setting I just needed a generic term to refer to processes with the same law and a simple SDE (with constant or deterministic drift and diffusion coefficients) without having to say each time "s1(t) has the same law as s2(t)..."

If you're concerned with the finite dimensional distributions being the same, a term used is a "version" -- see Definition 1.1. here: http://math.arizona.edu/~jwatkins/notesc.pdf// or Definition 3.1.8 here: http://www-stat.stanford.edu/~adembo/ma ... s.pdf"Note that a modification has to be defined on the same probability space as the original S.P. while this is not required of versions."

QuoteOriginally posted by: manillalist: I agree with what you say, if you want to define the concept accurately it will become very involved. In my modest setting I just needed a generic term to refer to processes with the same law and a simple SDE (with constant or deterministic drift and diffusion coefficients) without having to say each time "s1(t) has the same law as s2(t)..." If question is asked in terms of stochastic processes then the answer is expected to receive in the same terms. That is the answer "s1(t) has the same law as s2(t)" for any t from [ 0 , T ] is quite correct. If you wish to describe the class we can say that there exists the prob measure that the solution of sde will have the same distribution as a solution with 0 drift and the same sigma. We can also talk about equality of the finite distributions but it is actually as to talk about measures as finite distributions generate a unique measure under some quite broad conditions.

list: OK now you've made me wonder about the generic setting. Suppose I have two stochastic processes over the same state space which are solutions for the same SDE (no restrictions here). If you look at Definition 5.45 in Watkins notes (kindly supplied by Polter above) this does not ensure they would be versions of each other (hence the definition of SDE with uniqueness in law). On the other hand being versions of each other does not seem to ensure that they would satisfy the same SDE either.