Path-wise solution of stochastic differential equations, leading to a new and unique stochastic integral

Abstract: SDEs are solved in two steps: (1) for short times by successive approximation in the integral equation, which leads to non-Gaussian increments when the noise is multiplicative, (2) by summing up these increments in consecutive short time intervals. This corresponds to a modified anti-Ito integral. That procedure saves the choice of an integration sense, and it also avoids an intrinsic mismatch between the standard stochastic integrals (with Gaussian increments) and the Fokker-Planck equations (with non-Gaussian solutions). As a further new feature, the local diffusion parameters (plus a noise-independent drift) are sufficient to specify the SDE. This can simplify the modelling. For the FPE it means that the diffusion matrix alone accounts for the noise (the well-known and valid anti-Ito FPE involves a noise-induced drift part that cancels with some other term).