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Discrete Stochastic Calculus

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This post is part of a series

In the previous post of this series, we found that when Cartesian coordinates are placed on a binary tree, the commutative relations are given by

  • [dx,x] = \frac{(\Delta x)^2}{\Delta t} dt
  • [dt,t] = \Delta t dt
  • [dx,t] = [dt,x] = \Delta t dx.

There are two distinct classes of discrete calculus depending on the relation between \Delta x and \Delta t.

Discrete Exterior Calculus

If we set \Delta x = \Delta t, the commutative relations reduce to

  • [dx,x] = \Delta t dt
  • [dt,t] = \Delta t dt
  • [dx,t] = [dt,x] = \Delta t dx

and in the continuum limit, i.e.  \Delta t\to 0, reduce to

  • [dx,x] = 0
  • [dt,t] = 0
  • [dx,t] = [dt,x] = 0.

In other words, when \Delta x = \Delta t, the commutative relations vanish in the continuum limit and the discrete calculus converges to the exterior calculus of differential forms.

Because of this, the discrete calculus on a binary tree with \Delta x = \Delta t will be referred to as the discrete exterior calculus.

Discrete Stochastic Calculus

If instead of \Delta x = \Delta t, we set (\Delta x)^2 = \Delta t, the commutative relations reduce to

  • [dx,x] = dt
  • [dt,t] = \Delta t dt
  • [dx,t] = [dt,x] = \Delta t dx

and in the continuum limit, i.e.  \Delta t\to 0, reduce to

  • [dx,x] = dt
  • [dt,t] = 0
  • [dx,t] = [dt,x] = 0.

In this case, all commutative relations vanish in the continuum limit except [dx,x] = dt.

In the paper:

I demonstrate how the continuum limit of the commutative relations give rise to (a noncommutative version of) stochastic calculus, where dx plays the role of a Brownian motion.

Because of this, the discrete calculus on a binary tree with (\Delta x)^2 = \Delta t will be referred to as the discrete stochastic calculus.

To date, discrete stochastic calculus has found robust applications in mathematical finance and fluid dynamics. For instance, the application of discrete stochastic calculus to Black-Scholes option pricing was presented in

and the application to fluid dynamics was presented in

Both of these subjects will be addressed in more detail as part of this series of articles.

It should be noted that discrete calculus and its special cases of discrete exterior calculus and discrete stochastic calculus represent a new framework for numerical modeling. We are not taking continuum models built on continuum calculus and constructing finite approximations. Instead, we are building a robust mathematical framework that has finiteness built in from the outset. The resulting numerical models are not approximations, but exact models developed within a finite numerical framework. The framework itself converges to the continuum versions so that any numerical models built within this framework will automatically converge to the continuum versions (if such a thing is desired).

Discrete calculus provides a kind of meta algorithm. It is an algorithm for generating algorithms.

Written by Eric

August 25, 2012 at 12:38 pm