Eric Forgy Discrete Martingales

Content

Introduction

The following material reviews some aspects of martingales from the perspective of discrete stochastic calculus. In particular, it is illustrated how different choices of basis vectors, i.e. basis 1-forms, for the same underlying process lead to well known Ito and Stratonovich representations in the continuum limit.

In what follows, we consider a binary tree to be a directed graph $\mathcal{G} = (\mathcal{G}_0,\mathcal{G}_1)$, where $\mathcal{G}_0$ is the set of nodes of the tree and $\mathcal{G}_1$ is the set of directed edges of the tree. Each node $(i,j)\in\mathcal{G}_0$ of the tree is indexed by two integers representing its position in space and time. For instance, consider the following diagram.

We think of time as flowing left to right and the node’s position in time is marked by the second index $j$.

In ForgySchreiber2004 and references therein (particularly those of Dimakis & Mueller-Hoissen), it is shown that a unique discrete calculus is associated to every directed graph. This discrete calculus has an associative product and a derivation satisfying the graded Leibniz rule

and

Thus, the discrete calculus is more than a mere approximation, but is a rigorously defined mathematical framework associated to any given directed graph.

The binary tree is a particularly nice directed graph with a particularly nice discrete calculus as shown in Forgy2004. On a binary tree, there are two classes of objects of interest: discrete 0-forms and discrete 1-forms. A discrete 0-form is essentially a function whose value at every node in the tree is given and is expressed as

where $\mathbf{e}^{(i,j)}$ can be thought of as a basis vector associated to the node $(i,j)$. In other words, a function on a binary tree can be thought of as a linear combination of node basis vectors.

A discrete 1-form is essentially a linear combination of directed edge basis vectors

What is particularly unique about discrete calculus as compared to the usual continuum calculus is that discrete 0-forms and discrete 1-forms do not commute. See Forgy2004 for more details, but defining coordinate 0-forms

we end up with the coordinate basis 1-forms

These satisfy the commutative relations

In the current discussion, we always set

and any continuum limit is taken while maintaining this relation. Therefore, the commutative relations we are concerned with here are

and in the continuum limit all commutative relations vanish except $[d x,x] = d t$.

Left and Right Components

On a binary tree $\mathcal{G} = (\mathcal{G}_0,\mathcal{G}_1)$, there are two basic representations of coordinate basis elements to consider: left and right bases. The two are closely related and both sum to the same coordinate basis 1-forms

Given a 0-form

we have left multiplication

and right multiplication

The left component form is to be used when multiplying functions on the left and vice versa. Multiplying on the left and right are not equivalent due to the noncommutativity of discrete 0-forms and discrete 1-forms.

Left Bases

The left basis 1-forms

correspond to the tree element

and can be inverted resulting in

Any discrete 1-form

may be expressed in terms of left bases via

In particular, the discrete 1-form

may be expressed in terms of left bases via

It was shown in Section 5.1 of Forgy2004, that the continuum limit of the above expression

corresponds to the Ito formula of stochastic calculus.

Right Bases

The right basis 1-forms

correspond to the tree element

and can be inverted resulting in

Any discrete 1-form

may be expressed in terms of left bases via

In particular, the discrete 1-form

may be expressed in terms of right bases via

The continuum limit of the above expression is given by

as introduced in Section 4 of Forgy2002.

Mixed Components

Given the left and right bases

we can define a mixed basis

so that an arbitrary discrete 1-form may be expressed as

where

and

These mixed basis 1-forms correspond to the tree elements

Using this mixed basis, the differential becomes

The continuum limit of the above expression is given by

which corresponds to the Stratonovich representation.

Discrete Martingales

On a binary tree, a discrete martingale is a function $f$ that has no drift, i.e. the time component of the differential $d f$ vanishes. The value of the time component obviously depends on the definition of the time basis function. Different time bases will correspond to different time components for the same underlying process. Above, we have considered three distinct coordinate basis elements: 1. Left Basis (Ito) - $\stackrel{\leftarrow}{d t}(i,j)$ 1. Right Basis (???) - $\stackrel{\rightarrow}{d t}(i,j)$ 1. Mixed Basis (Stratonovich) - $\stackrel{\leftrightarrow}{d t}(i,j)$

For a process to be a discrete martingale under the left basis (Ito) representation, i.e. a left martingale, the process must satisfy

As mentioned above, this has a nice continuum limit given by

In other words, for a process to be a left martingale, it must satisfy the heat/diffusion equation.

Similarly, for a process to be a discrete martingale under the right basis representation, i.e. a right martingale, the process must satisfy

This also has a nice continuum limit given by

In other words, for a process to be a right martingale, it must satisfy the time-reversed heat/diffusion equation.

When it comes to the mixed basis, i.e. the Stratonovich representation, things are little more tricky. In this case, the condition for the process to be a mixed martingale is

One solution to this would be

i.e. a constant. This corresponds to the continuum limit

However, there is another solution that can be thought of as a kind of “noise”. For instance, consider the process

that either toggles between some constant value $c$ and $-c$ or remains fixed at zero depending on its position in space. Although its behavior in the continuum limit would be quite violent (and non-smooth), this process would nonetheless be a Stratonovich (mixed) martingale.

References

• ForgySchreiber2004 - Discrete differential geometry on causal graphs, Eric Forgy, Urs Schreiber, 2004
• Forgy2004 - Financial Modeling Using Discrete Stochastic Calculus, Eric Forgy, October 2004.
• Forgy2002 - Noncommutative Geometry and Stochastic Calculus: Applications in Mathematical Finance, Eric Forgy, May 2002.

Appendix

The following material comes from a post on the Azimuth forum. I hope to spell out some additional technical details here.

Note: My RSS reader is telling me a few comments have flown by as I type this slowly, so this may seem a bit out of place, but I’ve been meaning to expand on some things I’ve said and this is a fleeting moment.

I tend to be cavalier with preciseness, but for any purposes I’ve ever coming across, saying “a martingale is a stochastic process with no drift” is usually good enough.

I like to “think geometrically” in terms of differential forms and a stochastic process (for many purposes) is an exact form

$d f.$

In my paper, we have the left component (Ito) representation (Equation 54):

$d f = \left(\partial_\mu f\right) d x^\mu + \left(\partial_t f + \frac{1}{2} g^{\mu,\nu} \partial_\mu\partial_\nu f\right) d t$

and the right component (nameless?) representation (Equation 55):

$d f = d x^\mu \left(\partial_\mu f\right) + d t \left(\partial_t f - \frac{1}{2} g^{\mu,\nu} \partial_\mu\partial_\nu f\right).$

I’ve always found it miraculous that these are both merely different representations of the same underlying stochastic process, i.e. exact 1-form, $d f$ and both are derived from the simple commutative relation

$[d x, x] = d t.$

The Stratonovich representation is the average of the two and the second order terms cancel out:

$d f = \frac{1}{2} \left[\left(\partial_\mu f\right) d x^{\mu} + d x^{\mu} \left(\partial_\mu f\right)\right] + \frac{1}{2} \left[\left(\partial_t f\right) d t + d t \left(\partial_t f\right)\right].$

You can artificially make this look like the usual differential by defining a symmetrizing product

$d f = \left(\partial_\mu f\right)\bullet d x^\mu + \left(\partial_t f\right)\bullet d t$

where

$\left(\partial_\mu f\right)\bullet d x^\mu = \frac{1}{2} \left[\left(\partial_\mu f\right) d x^{\mu} + d x^{\mu} \left(\partial_\mu f\right)\right].$

When doing things in left component (Ito) representation, the drift term vanishes, i.e. you have a “left” martingale, when

$\partial_t f + \frac{1}{2} g^{\mu,\nu} \partial_\mu\partial_\nu f = 0.$

When doing things in the right component (nameless?) rrepresentation, the drift term vanishes, i.e. you have a “right” martingale, when

$\partial_t f - \frac{1}{2} g^{\mu,\nu} \partial_\mu\partial_\nu f = 0.$

I said above (cavalierly) that there are no Stratonovich martiginales because they’d have to satisfy both the heat equation and time-reversed heat equation simultaneously. That probably isn’t quite right. The actual condition is that you need

$\left(\partial_t f\right)\bullet d t = \frac{1}{2} \left[\left(\partial_t f\right) d t + d t \left(\partial_t f\right)\right] = 0.$

Based on what has been said above, I would not be at all surprised if the above expression defines a form of “noise”.

A cool thing about this is that there is a finite/discrete version of everything above and we can check what that relation for Stratonovich martingales means in a finite world (which is easier for me to interpret). Here is the paper:

• Financial Modeling Using Discrete Stochastic Calculus