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I'm on the lookout for a numerical method that can solve both a deterministic and stochastic equation. In the deterministic case, I know that a fourth order RK method is a valuable one, very effective. Unfortunately, there has not been applied to stochastic equations successfully (at least as far as I know).

Now what I want to know is if a numerical method that can solve both equations (roughly I mean, in comparison to the analytic solutions) exists and, in that case, what would be. A stochastic equation analytically solvable would be the Black-Scholes one, for instance.

  • Would it be possible to work on fitted / sliding averaged data instead of working with the equation itself? – Guimoute Oct 30 '18 at 16:42
  • You know already about Euler-Maruyama and Milstein's method? – Lutz Lehmann Oct 30 '18 at 16:55
  • Hi to both of you and thanks for commenting! Yes, I know both methods and I know that both are pretty useful when solving the Black Scholes eq in particular, but when I apply them to a deterministic equation the results leave much to be desired in comparison to RK. W.r.t the comment of Guimoute, I don't get the point. What do you mean? –  Oct 30 '18 at 17:12
  • Euler method is always good enough using the right timestep. In computational neuroscience is common to integrate both deterministic and stochastic equation using just the Euler method. – heracho Oct 30 '18 at 18:34

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There are methods for solving these kinds of equations in DifferentialEquations.jl. Stochastic differential equations are a form of mixed deterministic and stochastic equation and solving them is shown in the SDE tutorial. Mixing discrete stochasticity with deterministic equations is shown in the jump equation tutorial. While written naively in Julia, it is accessible in Python via the package diffeqpy. Notice that this has some example stochastic differential equations in the README.

Chris Rackauckas
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