Hi there !I'm trying to figure out whether the solution to a SDE of the formis a Gaussian process. Clearly, it is the case if is linear. But does this hold in general ?Many thanks

Well, not only when a2(x)=k0 is cst, but clearly when a2(x)=k0*x+k1 for cst k0,k1. But not sure for non-linear a2(x) fonctions...

QuoteOriginally posted by: FredBTHi there !I'm trying to figure out whether the solution to a SDE of the formis a Gaussian process. Clearly, it is the case if is linear. But does this hold in general ?Many thankswrite Characteristic Function or functional and using CF of Gaussian process it should be not difficult to find some sufficient condition to state G.

Hi list,can you be a little bit more specific or give some references pls ? Not that familiar with this matter. Tx !

denote X ( t ) = X ( T ; t , y ) and consider J ( t , y ) = E exp i u X ( T ; t , y ). write Kolmogorov equation for J ( t , y ). It might be that solution can be written in a closed form. But of course the SDE represents nonlinear transformation of the Gaussian measure generated by w () on [ 0 , T ] and does not to be Gaussian. Just check the situation for Gaussian variables. The solution of your SDE represents rv in say functional space C[0 , T ]. Hence of we could not state something in 1-dimensional case we could not state that in infinite dimensional space. For rv we have x = a_1 *a_2 ( x ) + w where a_1 is a constant, and w is a N ( 0 , 1 ) rv and a_2 is a function. the problem is to find a class function a_2 ( ) for which x is a Gaussian. This class then can be supposed to be the solution in the general case.