Stochastic differential equations

One of the things I noticed at the recent Snowbird meeting was an increase in interest in stochastic differential equations (SDEs) or Langevin equations.  They arise wherever noise is involved in a dynamical process.  In some instances, an SDE comes about as the continuum approximation of a discrete stochastic process, like the price of a stock.  In other cases, they arise as a way to reintroduce stochastic effects to mean field differential equations originally obtained by averaging over a large number of stochastic molecules or neurons.  For example, the Hodgkin-Huxley equation describing action potential generation in neurons is a mean field approximation of the stochastic transport of ions (through ion channels) across the cell membrane, which can be modeled as a multi-state Markov process usually simulated with the Gillespie algorithm (for example see here).  This is computationally expensive so adding a noise term to the mean field equations is a more efficient way to account for stochasticity.

In one dimension, an SDE has the form

\frac{dx}{dt} = f(x) + g(x)\eta(t)    (1)

where \eta(t) is a noise term.  This could have any form but usually it is taken to be Gaussian white noise obeying \langle \eta(t) \rangle = 0, \langle \eta(t)\eta(t')\rangle = \delta(t-t'), where the brackets imply ensemble average over the noise distribution.  Now, one of the problems with using white noise is that two time points are uncorrelated no matter how close together so the noise can’t be continuous and the variance is infinite.  Hence, the SDE as written above doesn’t really exist.  There are two ways to deal with this.  Physicists take the attitude that the white noise is not really white noise so that for very short times it is well-behaved. This approach goes under the name Stratonovich formulation.

Mathematicians interpret the SDE as really being an integral equation and write the SDE as

dx = f(x) dt + g(x) dW       (2)

where dW is a white noise process that obeys   \langle w(\tau) w(\tau') \rangle = \tau, where w(\tau)=\int_t^{t+\tau} dW.  This is known as the Ito formulation. The problem with the Ito formulation is that the rules for differentiation and integration must be changed to something called Ito calculus. The deepness of the mathematical concepts involved in SDEs also makes treatments on the topic widely divergent.  If you read a book from the physicist’s point of view like for example Risken versus a more mathematical one like Oksendal, you may not even know they were writing about the same topic.

I started out using the Stratonovich formulation of SDEs but have since switched exclusively to Ito.  The main reason is that numerical simulations of SDEs are most natural in the Ito formulation.  If we discretize equation (2) with a single Euler step of time step \Delta, we obtain

x_{n+1}=x_n + f(x_n) \Delta + g(x_n) Z_n \sqrt{\Delta}

where Z_n is a normally distributed random variable.  If you discretize in the Stratonovich formulation then g(x) must be evaluated at the mid-point between the time steps.

However, when it comes to “solving” SDEs, I put on my physicist’s cap and use two approaches.  One is to solve the SDEs directly to obtain moments of x using path integrals (e.g. see here).  The other approach is to solve the Fokker-Planck equation for the probability density function of x.  The Fokker-Planck equation is a parabolic partial differential equation of the form

\frac{\partial}{\partial t}P(x,t) = -\frac{\partial }{\partial x}f(x) P +\frac{1}{2}\frac{\partial^2}{\partial x^2}g(x)^2 P

for Ito and

\frac{\partial}{\partial t}P(x,t) = -\frac{\partial }{\partial x}(f(x) + g(x)'g(x))P +\frac{1}{2}\frac{\partial^2}{\partial x^2}g(x)^2 P

for Stratonovich.  The Fokker-Planck equation is a “drift-diffusion”  equation.  In the Ito formulation, the drift term is given by the deterministic part of the SDE and the diffusion term is the square of the stochastic part divided by two.  In the Stratonovich formulation, there is an extra drift term that arises from the stochastic forcing.  The difference arises from the difference between Ito and regular calculus.

One crucial point about the Fokker-Planck equation is that it is linear.  In fact, what the Fokker-Planck equation does is to transform a nonlinear SDE  into a linear PDE.  In some sense you have traded nonlinearity for heterogeneity and extra-dimensions.  I have always found this to be fascinating.  It is a way to demonstrate that solving linear heterogeneous PDEs can be as hard as solving nonlinear SDEs.  However, depending on what question you have one may be more useful than the other.  For example, if you are interested in the equilibrium probability distribution of x then the Fokker-Planck equation reduces to a heterogeneous second order ODE, which can generally be solved.  Finally, I want to stress that the Fokker-Planck equation is always linear by definition.  Some papers in the literature have called single-variable continuity equations for probability densities nonlinear Fokker-Planck equations.  This is technically incorrect.   The systems these papers describe are high dimensional systems that do have multi-variable Fokker-Planck equations.   The so-called nonlinear Fokker-Planck equations that appear are usually derived heuristically.  However, they can be more formally derived by marginalization over the other variables followed by a truncation of a moment (BBGKY) hierarchy.  Again, there is a trade-off between nonlinearity and extra dimensions.

Typos fixed 2011-6-5

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3 thoughts on “Stochastic differential equations

  1. Fascinating. I had looked into SDEs but I couldn’t figure out which formulation I should learn about so I moved on to other things. What book would you recommend to learn more about Ito calculus? My undergrad was in math and physics.

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  2. Hi,

    Good question. I’m not sure what book to read to learn about Ito calculus. Maybe Gardiner’s book Handbook of Stochastic Methods would give the physicist’s point of view? If you have a good background in analysis then you can try Oksendal or Shreve’s books.

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