Let [sigma]>0,[delta]>=1,b>=0, 0<p<1. Let [lambda] be a continuous and positive function in . Using the technique of moving domains (see Russo and Trutnau (2005) [9]), and classical direct stochastic calculus, we construct for positive initial conditions a pair of continuous and positive semimartingales with and where the symmetric local times , of the respective semimartingales are related through the formula Well-known special cases are the (squared) Bessel processes (choose [sigma]=2, b=0, and [lambda]2[reverse not equivalent]0, or equivalently ), and the Cox-Ingersoll-Ross process (i.e. R, with [lambda]2[reverse not equivalent]0, or equivalently ). The case 0<[delta]<1 can also be handled, but is different. If p>1, then there is no solution.

Consider a d-dimensional Brownian motion X (Xl, ... ,Xd ) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [fk(X), Xkj involving the weak first partial...

Consider a d-dimensional Brownian motion X (Xl, ... ,Xd ) and a function F which belongs locally to the Sobolev space W 1,2. We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [fk(X), Xkj involving the weak first partial...