and possibly on present or previous values of other processes too. In that case the solution process, X, is not a Markov process, and it is called an Itô process and not a diffusion process. When the coefficients depends only on present and past values of X, the defining equation is called a stochastic delay differential equation. SDEs on manifolds More generally one can extend the theory of stochastic calculus onto differential manifolds and for this purpose one uses the Fisk-Stratonovich integral. Consider a manifold , some finite-dimensional vector space , a filtered probability space with satisfying the usual conditions and let be the one-point compactification and be -measurable. A stochastic differential equation on written is a pair , such that is a continuous -valued semimartingale, is a homomorphism of vector bundles over . For each the map is linear and for each . A solution to the SDE on with initial condition is a continuous -adapted -valued process up to life time , s.t. for each test function the process is a real-valued semimartingale and for each stopping time with the equation holds -almost surely, where is the differential at . It's a maximal solution if the life time is maximal, i.e. -almost surely. It follows from the fact that for each test function is a semimartingale, that is a semimartingale on . Given a maximal solution we can extend the time of onto full and after a continuation of on we get up to indistinguishable processes. Existence and uniqueness of solutions As with deterministic ordinary and partial differential equations, it is important to know whether a given SDE has a solution, and whether or not it is unique. The following is a typical existence and uniqueness theorem for Itô SDEs taking values in n-dimensional Euclidean space Rn and driven by an m-dimensional Brownian motion B; the proof may be found in Øksendal (2003, §5.2). Let T > 0, and let be measurable functions for which there exist constants C and D such that for all t ∈ [0, T] and all x and y ∈ Rn, where Let Z be a random variable that is independent of the σ-algebra generated by Bs, s ≥ 0, and with finite second moment: Then the stochastic differential equation/initial value problem has a P-almost surely unique t-continuous solution (t, ω) ↦ Xt(ω) such that X is adapted to the filtration FtZ generated