• Corpus ID: 119591665

The intrinsic "sense" of stochastic differential equations

@article{Ryter2016TheI,
  title={The intrinsic "sense" of stochastic differential equations},
  author={Dietrich Ryter},
  journal={arXiv: Mathematical Physics},
  year={2016}
}
  • D. Ryter
  • Published 10 May 2016
  • Mathematics
  • arXiv: Mathematical Physics
A free choice of the integration sense would lead to the paradox that the number of possible equations (thus of solutions for a given model) can vary under a mere change of the variables. This is shown by a specific change which neutralizes the sense (by establishing a constant coupling with the noise). Its inverse singles out the Stratonovich sense, by means of the Ito formula. 

New forward equation for stochastic differential equations with multiplicative noise

The existing forward equation defines a probability current that disagrees with the random paths. A new one is adequate. With respect to that, the existing Ito paths consist of most probable

Exact Fokker-Planck equations for stochastic differential equations.

The validity for each noise level imposes some restrictions. Multiplicative noise must be weak, but the optimum prediction is exact and given by the noiseless motion. "Detailed balance" of steady

Stochastic differential equations with multiplicative noise: the Markov solution

Fokker-Planck equations are decisive for the Markov property. With multiplicative noise they have non-Gaussian solutions in short times. Corresponding new path increments agree with a FPE by both

Revised numerical solutions of stochastic differential equations with multiplicative noise, and the evolution of density peaks

The Fokker-Planck equation involves a drift (given by derivatives of the diffusion), which is absent in the SDE. Increments of the random paths must include that drift. This corrects the existing

References

SHOWING 1-3 OF 3 REFERENCES

Stochastic differential equations: loss of the Markov property by multiplicative noise

The solutions of SDEs with multiplicative noise are not Markovian. On a coarse-grained time scale they still are, but only in the "anti-Ito" case. This allows a simple computation of the most likely

Stochastische Differentialgleichungen (Oldenbourg

  • München, 1973) , and Stochastic Differential Equations: Theory and Applications
  • 1974

Skorochod, Stochastische Differentialgleichungen (Akademie- Verlag, Berlin

  • 1971