In this paper, we investigate the features of higher order Gubinelli derivatives controlled rough paths having an arbitrary positive Holder index. There is used a notion (α, β)-rough map on basis which sufficient conditions are given for uniqueness. Using theorem uniqueness analogue Doob – Meyer with index proved. final section prove that law local iterated logarithm fractional Brownian motion ...

Let X be a regular linear diffusion whose state space is an open interval E ⊆ R. We consider a diffusion X∗ which probability law is obtained as a Doob htransform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. This is the dual of X with respect to h(x)m(dx) where m(dx) is the speed measure of X. Examples include the case where X∗ is X conditi...

A reduction procedure for stochastic differential equations based on symmetries including Girsanov random transformations is proposed. In this setting, a new notion of reconstruction given, involving the expectation values functionals solution to SDE and theorem general proved. Moreover, notable case under closed subclass quasi Doob presented. The theoretical results are applied models relevant...