QuoteOriginally posted by: manillalist: 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." two stochastic processes over the same state space which are solutions for the same SDE (no restrictions here" ///usually we consider SDEs with coefficients that provide existence and uniqueness of the solution. The underling sense of this assumption does not clear. If we consider two processes S_1 (*) and S_2(*) on the same measurable space {O,F} and we state that these two are Wiener processes with respect to two measures P_1 and P_2 correspondingly it means that measures corresponding S_1 on {O,F, P_1} and S_2 on {O,F. P_2} constructed on the space continuous functions on [ 0 , T ] are the same Wiener measure. Generally speaking we do not have equality of two two processes as far as they are defined on different probability spaces.

I think we agree. If the SDE satisfies the usual Lipschitz and boundedness conditions => SDE has uniqueness in Law => process satisfying the SDE with their respective BM's will be versions of each other (I am looking at Lemma 5.3.1 in Oksendal's book). If these conditions are not assumed then equality in Law can not be guaranteed. On the other direction I think there must exist examples of processes that are modifications (same measurable space) of each other hence equal in law, but one satisfies an SDE and the other one not. And in such case the coefficients of the SDE should not be well behaved.

QuoteOriginally posted by: manillaI think we agree. If the SDE satisfies the usual Lipschitz and boundedness conditions => SDE has uniqueness in Law => process satisfying the SDE with their respective BM's will be versions of each other (I am looking at Lemma 5.3.1 in Oksendal's book). If these conditions are not assumed then equality in Law can not be guaranteed. On the other direction I think there must exist examples of processes that are modifications (same measurable space) of each other hence equal in law, but one satisfies an SDE and the other one not. And in such case the coefficients of the SDE should not be well behaved. "SDE has uniqueness in Law" in your case here it means path-wise uniqueness. we can chose a modification of the solution such that Any other solution will coincide with given for all t from [ 0 , T ] with prob. 1 at once.For example if coefficients are bounded and continuous then a solution and a Wiener process can be constructed on other space, it might be [ 0 , 1 ] in Skorohod construction or C[0, T ] in Strook&Varadhan construction. This existence is called week existence. Two different solutions with non degenerative diffusion have the same , unique measure. It is called week existence.

QuoteOriginally posted by: manillaOn the other direction I think there must exist examples of processes that are modifications (same measurable space) of each other hence equal in law, but one satisfies an SDE and the other one not. And in such case the coefficients of the SDE should not be well behaved.Take W to be the Wiener process.Let A = W.Let B = -W.Then, A and B have the same distribution at each point in time and dA = dW, while dB = -dW.Is this what you have in mind?

QuoteOriginally posted by: PolterQuoteOriginally posted by: manillaOn the other direction I think there must exist examples of processes that are modifications (same measurable space) of each other hence equal in law, but one satisfies an SDE and the other one not. And in such case the coefficients of the SDE should not be well behaved.Take W to be the Wiener process.Let A = W.Let B = -W.Then, A and B have the same distribution at each point in time and dA = dW, while dB = -dW.Is this what you have in mind?This example is a little bit different issue. Here sigma is different 1 and -1 sigma square in either case is 1. Solutions have equal distributions as far as measures corresponding solutions are completely defined by sigma square and different paths.

QuoteOriginally posted by: PolterTake W to be the Wiener process.Let A = W.Let B = -W.Then, A and B have the same distribution at each point in time and dA = dW, while dB = -dW.Is this what you have in mind?I think this example settles the question indeed (I was obviously wrong about the not well behaved coefficients ). Going back to my original question I think neither "isonomic" nor "versions of each other" are what I need. I have to say something in the lines of the processes (S1, S1(0), W1, P1) and (S2,S1(0),W2,P2) are both solutions to the SDE so and so (and because the SDE is well behaved they will be versions of each other).

QuoteOriginally posted by: manillaQuoteOriginally posted by: PolterTake W to be the Wiener process.Let A = W.Let B = -W.Then, A and B have the same distribution at each point in time and dA = dW, while dB = -dW.Is this what you have in mind?I think this example settles the question indeed (I was obviously wrong about the not well behaved coefficients ). Going back to my original question I think neither "isonomic" nor "versions of each other" are what I need. I have to say something in the lines of the processes (S1, S1(0), W1, P1) and (S2,S1(0),W2,P2) are both solutions to the SDE so and so (and because the SDE is well behaved they will be versions of each other)."versions of each other" does not the convenient term because they are defined on different prob. spaces, prob measures P1 and P2 are different. version is used for stoch processes defined on the same prob space with measure P and which coincides for each t with prob P equal to 1.it will be instructive if you try to state what you wish by considering an example processes S) (t) = W ( t ) , S1(t) = at + W ( t ) , S2(t) = bt + W(t)

QuoteOriginally posted by: list"versions of each other" does not the convenient term because they are defined on different prob. spaces, prob measures P1 and P2 are different. version is used for stoch processes defined on the same prob space with measure P and which coincides for each t with prob P equal to 1list, this is not true. Perhaps you are thinking of a modification, not a version. Recall:"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." // emphasis minemanilla, perhaps you're thinking of something along uniqueness in law, e.g. here (also note the distinction between a strong solution and a weak solution).

QuoteOriginally posted by: list"versions of each other" does not the convenient term because they are defined on different prob. spaces, prob measures P1 and P2 are different. version is used for stoch processes defined on the same prob space with measure P and which coincides for each t with prob P equal to 1.There is a definition of versions in which the probability spaces can be different is only the state space which must be the same. See Definition (1.6) in Revuz & Yor book (Continuous martingales and Brownian motion). When the probability spaces coincide they use the term modifications. Maybe this terminology is not the standard one (apologies for quoting that many textbooks !)

QuoteOriginally posted by: Poltermanilla, perhaps you're thinking of something along uniqueness in law, e.g. here (also note the distinction between a strong solution and a weak solution).Yes that is exactly what I am thinking off: Lipschitz and boundedness conditions => Path uniqueness => SDE has uniqueness in Law => versions. But I have to state this starting from the SDE, because uniqueness in Law is a property of the SDEFrom the example you supplied it is clear that modifications does not imply they are solutions for the same SDE. So what I meant originally with "sort of copy" after all this discussion is clear to me that is really "solve the same SDE (which has uniqueness in Law)"

Polter&manilla you are right in russian modification and version with respect to random processes are synonyms . Though I could forget something and if you can recall a distinction it would be good.

QuoteOriginally posted by: list Polter&manilla you are right in russian modification and version with respect to random processes are synonyms . Though I could forget something and if you can recall a distinction it would be good.I was hoping I could avoid copy-pasting ;-)In short: modification is stronger and implies version.From the notes @ http://www-stat.stanford.edu/~adembo/ma ... Definition 3.1.8. Two S.P. X and Y are called versions of one another if they have the same finite-dimensional distributions.Definition 3.1.9. A S.P. Y is called a modification of another S.P. X if P(Y(t) = X(t)) = 1 for all t.We consider next the relation between the concepts of modification and version, starting with:Exercise 3.1.10. Show that if Y is a modification of X, then Y is also a version of X.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.To be fair, I've seen some authors use "modification" and "version" interchangeably before and wondered why, perhaps linguistic reason that you bring up is an explanation.

Thank you Polter now the difference of notions is clear and it of course makes sense.

Also if you guys send me your real names (manilla@email.com) I will acknowledge you in my paper for sparing your time in this discussion (if I ever manage to get it published).

versions are called- "weakly equivalent"in Lipcer & Shiryaev p.22 of http://books.google.fr/books?id=gKtK0Cj ... &q&f=false - "stochastically equivalent in the wide sense" in Gikhman & Skorokhod (Class. of Math.) Corr Print 1st ed, Transl. S. Kotz , T.1 Ch1 Par4 Def2 p.43Since this discussion is kind of spiralling away from the W1,W2 example, I'd add that "usually in the def of weak uniqueness it is the joint of law of the solution X and the driving semimart (say Z) that is considered, HOWEVER in the case that Z is Brownian motion, weak uniqueness in this sense is a consequence of weak uniqueness considering just the law of the solution X (eg http://mech.math.msu.su/~cherny/uniq.pdf )