o m=l 0 where o denotes generalized Stratonovich integration and the equality is a.s. (cf. lemma 4). We use the criteria we derive to provide some new relations between Stratonovich and Ogawa integrals which do not go through an intermediate chaos decomposition as in [1]. The results below are an outgrowth of some extensions of the Ito lemma pointed out in [2], especially lemma (4.2) there. We ...

Recently, a novel framework to handle stochastic processes has emerged from a series of studies in biology, showing situations beyond ‘Itô versus Stratonovich’. Its internal consistency can be demonstrated via the zero mass limit of a generalized Klein–Kramers equation. Moreover, the connection to other integrations becomes evident: the obtained Fokker–Planck equation defines a new type of stoc...

We consider the Langevin equation with multiplicative noise term which depends on time and space. The corresponding Fokker-Planck equation in Stratonovich approach is investigated. Its formal solution is obtained for an arbitrary multiplicative noise term given by g(x, t) = D(x)T (t), and the behaviors of probability distributions, for some specific functions of D(x), are analyzed. In particula...

The relationship of the Ito-Stratonovich stochastic calculus to studies of weakly colored noise is explained. A functional calculus approach is used to obtain an effective Fokker-Planck equation for the weakly colored noise regime. In a smooth limit, this representation produces the Stratonovich version of the ItoStratonovich calculus for white noise. It also provides an approach to steady stat...

We formulate and prove a local stable manifold theorem for stochastic differential equations (SDEs) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and Itôtype equations are treated. Starting with the existence of a stochastic flow for a SDE, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of inva...

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process (not necessarily a semi-martingale). No adaptedness of initial point or vector fields is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Stratonovich solution. This condition is s...

We consider anticipative Stratonovich stochastic differential equations driven by some stochastic process lifted to a rough path. Neither adaptedness of initial point and vector fields nor commuting conditions between vector field is assumed. Under a simple condition on the stochastic process, we show that the unique solution of the above SDE understood in the rough path sense is actually a Str...

Given a Heath–Jarrow–Morton (HJM) interest rate model M and a parametrized family of finite dimensional forward rate curves G, this paper provides a technique for projecting the infinite dimensional forward rate curve rt given by M onto the finite dimensional manifold G. The Stratonovich dynamics of the projected finite dimensional forward curve are derived and it is shown that, under the regul...