# What is a cubature scheme?

Ideally an intuitive explanation with an example, please.

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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

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The word cubature is just a replacement for quadrature in the infinite dimensional setting, such as the Wiener space as in the answer from @TheBridge. The term is used in the context of integrating functionals of stochastic processes $$E[F(X)]$$ where X is random variable valued in a functional space such as a the solution of a SDE or simply the Brownian motion. The idea is to approximate this quantity by a quantity of the form

$$\sum_{i=1}^N p_i F(\gamma_i)$$ where $(\gamma_i)_{1 \leq i \leq N}$ are a set of deterministic functions.

So far, there are two main approaches to obtain cubature formulas in the infinite dimensional setting.

1) Functional quantization

Functional quantization for numerics with an application to option pricing By Gilles Pagès and Jacques Printems

2) Polynomial cubature

Cubature on Wiener spaces By Terry Lyons

Method (1) is an infinite-dimensional counterpart of the midpoint rule over an interval, and like with the midpoint rule, you can use Richardson Romberg extrapolation with the number of discretisation points to improve the speed of convergence.

Method (2) is a infinite-dimensional counterpart of the Gauss integration. It is exact on polynomials of the Wiener Process in the sense of iterated integrals of the Brownian motion.

Thee quantization approach is applicable to Brownian diffusions but also fractional Brownian motion and other cases.

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