What is a cubature scheme?

Question

Ideally an intuitive explanation with an example, please.

Answer 1

Cubature (of a given order) is a general method that allows you to do some approximate integration by being exact on a subset of integrand. If you are given a measure $M$ over for example $\mathbb R^n$ then will approach $M$ by (typically) a discrete measure $M^d=\sum_{i=1}^m \lambda_i\delta(x_i) $ such that polynomials $P$ of degree less or equal to $\gamma$ you have :
$$\int_{\mathbb R^n}P(x)dM(x)=\sum_{i=1}^m \lambda_i.P(x_i)$$

In the context of a Stochastic Diffusion processes $X_t$ defined by an SDE (ideally in a Stratonovitch form), if you have the to calculate the expectation of a functional of the diffusion path, then you can think of this as an integration over the Wiener measure. Formally this looks like :

$$E_{\mathbb{W}}[F(X_.)]=\int_{p\in Path}F(p)d\mathbb{W}(p)$$

Of course here the problem is infinite dimensional so quite hard to address in its full generallity and in a numerically tractable form.

Anyway by using Cubature over Wiener Space you can "in a way" approximate the problem by switching to another (and simpler to use) measured space over finite variation paths (recall that Wiener measure doesn't weigh finite variation path !!!) and this approximate measure is such that it matches the values Wiener Measure moments of Iterated Wiener Integral.

This transforms then the SDE into a classical ODE (that can be eventually solved analyticaly or numercaly), and finaly your expectation of your functional becomes hopefully numerically tractable.

Regards