Bayesian Inference for Discretely Sampled Diffusions – solving the nonlinear inverse problem

Many of the most important and fundamental models for dynamic stochastic phenomena in continuous time are based on the concept of a diffusion, whose evolution is modelled mathematically by $dX_t = b(X_t)dt + sigma(X_t) dW_t, t\ge 0$, where $W_t$ is a standard Brownian motion. Diffusions occur naturally in the physical and biological sciences, in economics and elsewhere, and their deep relationship to stochastic and partial differential equations makes them a central object of study in modern mathematics. Various specifications of the drift function b and the diffusion coefficient sigma lead to a flexible class of random continuous motions. In many scientific applications, a key challenge is to recover the parameters b, sigma from some form of observations of the diffusion. A case of key importance is the `low frequency regime’ where one samples $X_0, X_\Delta, X_{n\Delta}$ at fixed sampling distance $\Delta>0$. Computationally efficient Bayesian techniques exist to calculate certain posterior distributions in this case, but a key challenge remains in proving rigorously that such methods actually work in the sense that they successfully solve the nonlinear inverse problem that arises from having incomplete information about the diffusion. Some first results in this direction ( indicate that this is possible at least in simple scalar diffusion models, but a large number of challenges remain, particular for multi-dimensional problems which are most relevant in applications.

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