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Is this equation a stochastic differential equation, or a PDE?

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- Thread starter TheGhostInside
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- #1

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Is this equation a stochastic differential equation, or a PDE?

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Mute

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Basically, if you have a stochastic differential equation with brownian noise,

e.g.,

$$dX_t = a(X_t,t)dt + b(X_t,t)dB_t,$$

then one can show that this is equivalent to a PDE called the Fokker-Planck equation:

$$\frac{\partial f(x,t)}{\partial t} = -\frac{\partial}{\partial x}\left[a(x,t)f(x,t)\right] + \frac{1}{2}\frac{\partial^2}{\partial x^2}\left[ b(x,t)^2 f(x,t)\right],$$

where f(x,t) is the probability density of finding the system to have a value of x between x and x+dx at a time t.

This generalizes to more variables (see the Fokker-Planck wikipedia page for a brief intro and some further references).

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