Notebooks

Stochastic Differential Equations

30 May 2014 06:47

Non-stochastic differential equations are models of dynamical systems where the state evolves continuously in time. If they are autonomous, then the state's future values depend only on the present state; if they are non-autonomous, it is allowed to depend on an exogeneous "driving" term as well. (This may not be the standard way of putting it, but I think it's both correct and more illuminating than the more analytical viewpoints, and anyway is the line taken by V. I. Arnol'd in his book on differential equations.) Stochastic differential equations (SDEs) are, conceptually, ones where the the exogeneous driving term is a stochatic process. --- While "differential equation", unmodified, covers both ordinary differential equations, containing only time derivatives, and partial differential equations, containing both time and space derivatives, "stochastic differential equation", unmodified, refers only to the ordinary case. Stochastic partial differential equations are just what you'd think.

The solution of an SDE is, itself, a stochastic process. Heuristically, the easiest way to think of how this works is via Euler's method for solving differential equations, which is itself about the simplest possible numerical approximation scheme for an ODE. (This line of thought was apparently introduced by Bernstein in the 1920s.) To solve dx/dt = f(x), with initial condition x(0) = y, Euler's method instructs us to pick a small increment of time h, and then say that x(t+h) = x(t) + hf(x), using straight-line interpolation between the points 0, h, 2h, 3h,... Under suitable conditions on vector field f, as h shrinks, the function we obtain in this way actually converges on the correct solution. To accomodate a stochastic term on the right-hand side, say dx/dt = f(x) + E(t), where E(t) is random noise, we approximate x(t+h) - x(t) by hf(x) + E(t+h) - E(t). Then, once again, we let the time-increment shrink to zero. Doing this with full generality requires a theory of the integrals of stochastic processes, which is made especially difficult by the fact that many of the stochastic forces one would most naturally want to use, such as white noise, are ones which don't fit very naturally into differential equations! The necessary theory of stochastic integrals was developed in the 1940s by M. Loeve, K. Ito, and R. Stratonovich (all building on earlier work by, among others, N. Wiener); the theory of SDEs more strictly by Ito and Stratonovich, in slightly different forms.

Most of what one encounters, in applications, as the theory of SDEs assumes that the driving noise is in fact white, i.e., Gaussian and uncorrelated over time. On the one hand, this is less of a restriction than it might seem, because many other natural sorts of noise process can be represented as stochastic integrals of white noise. On the other hand, the same mathematical structure can be used directly to define stochastic integrals and stochastic DEs driven by a far broader class of stochastic processes; on this topic Kallenberg is a very good introduction.


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