by Z and Bs, s ≤ t, and General case: local Lipschitz condition and maximal solutions The stochastic differential equation above is only a special case of a more general form where is a continous semimartingale in and is a continous semimartingal in is a map from some open nonempty set , where is the space of all linear maps from to . More generally one can also look at stochastic differential equations on manifolds. Whether the solution of this equation explodes depends on the choice of . Suppose satifies some local Lipschitz condition, i.e. for and some compact set and some constant the condition where is the Euclidean norm. This condition guarantees the existence and uniqueness of a so-called maximal solution. Suppose is continous and satisfies the above local Lipschitz condition and let be some initial condition, meaning it's a measurable function with respect to the initial σ-algebra. Let be a predictable stopping time with almost surely. A -valued semimartingale is called a maximal solution of with life time if for one (and hence all) annoucing the stopped process is a solution to the stopped stochastic differential equation on the set we have almost surely that with . is also a so-called explosion time. Some explicitly solvable SDEs Linear SDE: general case where Reducible SDEs: Case 1 for a given differentiable function is equivalent to the Stratonovich SDE which has a general solution where Reducible SDEs: Case 2 for a given differentiable function is equivalent to the Stratonovich SDE which is reducible to where where is defined as before. Its general solution is SDEs and supersymmetry In supersymmetric theory of SDEs, stochastic dynamics is defined via stochastic evolution operator acting on the differential forms on the phase space of the model. In this exact formulation of stochastic dynamics, all SDEs possess topological supersymmetry which represents the preservation of the continuity of the phase space by continuous time flow. The spontaneous breakdown of this supersymmetry is the mathematical essence of the ubiquitous dynamical phenomenon known across disciplines as chaos, turbulence, self-organized criticality etc. and the Goldstone theorem explains the associated long-range dynamical behavior, i.e., the butterfly effect, 1/f and crackling noises, and scale-free statistics of earthquakes, neuroavalanches, solar flares etc. See also Langevin dynamics Local volatility Stochastic process Stochastic volatility Stochastic partial differential equations Diffusion process Stochastic difference equation References Further reading Desmond Higham and Peter Kloeden: "An Introduction to