Inference for Stochastic Differential Equations

28 Jun 2021 00:18

Spun off from stochastic differential equations, and/or inference for Markov models.

To be clear, I'm considering situations in which we observe a trajectory \( x(t) \) of a stochastic process \( X \) that obeys an SDE \[ dx = a(x;\theta) dt + b(x;\theta) dW \] and want to do inference on the parameter \( \theta \). (The "parameter" here might be a whole function.)

The "easy" case is discrete-time, equally-spaced data, without loss of generality \( x(0), x(h), x(2h), \ldots x(nh) \). Because \( X \) is (by hypothesis) a Markov process, there is a conditional probability kernel \( P_h(y|x;\theta) \), which one could find by integrating the generator of the SDE, and the log-likelihood is just \[ L(\theta) = \sum_{t=1}^{n}{\log{P_h(x(th)|x((t-1)h); \theta)}} \] Of course, "just" integrating the generator is not necessarily an easy issue...

See also: Equations of Motion from a Time series