## My Papers

Here are some of my papers that relate to phynance:

**Noncommutative Geometry and Stochastic Calculus:
Applications in Mathematical Finance**

May 20, 2002

**Abstract**

The present report contains an introduction to some elementary concepts in noncommutative differential geometry. The material extends upon ideas first presented by Dimakis and Mueller-Hoissen. In particular, stochastic calculus and the Ito formula are shown to arise naturally from introducing noncommutativity of functions (0-forms) and differentials (1-forms). The abstract construction allows for the straightforward generalization to lattice theories for the direct implementation of numerical models. As an elementary demonstration of the formalism, the standard Black-Scholes model for option pricing is reformulated.

**Financial Modelling Using Discrete Stochastic Calculus**

2004

**Abstract**

In the present report, a review of discrete calculus on directed graphs is presented. It is found that the binary tree is a special directed graph that contains both the exterior calculus and stochastic calculus as different continuum limits are taken. In the latter case, we arrive at something that may be referred to as “discrete stochastic calculus.” The resulting discrete stochastic calculus may be applied to any stochastic financial model and is guaranteed to produce results that converge in the continuum limit. Discrete stochastic calculus is applied to the Black-Scholes model for an illustration. The resulting algorithm generated by discrete stochastic calculus agrees with that of the Cox-Ross-Rubinstein model, as it should. The results presented here are preliminary and are intended to encourage others to learn discrete stochastic calculus and apply it to more complicated financial models.

**Geometric Issue and Sector Selection for Performance Attribution**

May 17, 2002

**Abstract**

This paper represents an overview of the fully geometric approach to performance attribution developed by Menchero of Vestek. In addition to providing further insights into the subtleties involved with geometric attribution, two new fully geometric approaches to attributions are provided. The first represents a slight extension of that provided by Menchero. The second constitutes a slight extension of the exponential approach developed by Carino of the Frank Russell Company. The advantages of the extended exponential approach include absolute transparency in interpretation of the results as well as a direct parallel to the algebraic aproach.

Here are some of my non-phynance related papers:

**Differential Geometry in Computational Electromagnetics**

Dissertation, University of Illinois at Urbana-Champaign, 2002

**Abstract**

This dissertation consists of two parts. Part I constitutes an introduction to some of the elementary concepts of the classical theory of differential geometry that are of relevance to computational electromagnetics. The motivation here is to aid researchers in overcoming the significant investment in mathematics that is typically required in order to learn and apply the theory to problems of practical interest in engineering. Another motivation for introducing the mathematical background in such detail is to provide a springboard to the algebraic model that takes up the second part of the dissertation.

The algebraic model of Part II represents a somewhat radical approach to computation. In this approach, rather than take the classical theory based on the continuum as a given and constructing approximate numerical techniques from there, work is done toward constructing an alternative to the continuum theory that is built up from scratch in a discrete setting. The goal is to take each mathematical objects that is necessary in order to write down the classical electromagnetic theory and develop a corresponding discrete analog. The theory builds upon standard concepts in algebraic topology and is motivated by recent progress in noncommutative differential geometry.

**Discrete Differential Geometry on Causal Graphs**

Eric Forgy and Urs Schreiber, July 2004

**Abstract**

Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric is encoded in a deformation of the inner product on that space, which is the crucial technique of this paper. We study the general case but find that drastic and possibly vital simplifications occur when the underlying lattice is topologically hypercubic, in which case we explicitly construct mimetic analogues of the volume form, the Hodge star operator, and the Hodge inner product for arbitrary discrete geometries. It turns out that the formalism singles out a pseudo-Riemannian metric on topologically hypercubic graphs with respect to which all edges are lightlike. We study such causal graph complexes in detail and consider some of their possible physical applications, such as lattice Yang-Mills theory and lattice fermions.

**A Time-Domain Method With Isotropic Dispersion and Increased Stability on an Overlapped Lattice**

Eric Forgy and Weng Chew

IEEE Transactions on Antennas and Propagation, July 2002

*(Submitted in December 1998!)*

*Note: This research won 1st Prize in the highly-competitive Student Paper Competition at the 1999 IEEE Antennas and Propagation Society International Symposium. It was essentially my MS thesis.*

**Abstract**

A time-domain method on an overlapped lattice is presented for the accurate and efficient simulation of electromagnetic wave propagation through inhomogeneous media. The method comprises a superposition of complementary approximations to electromagnetic theory on a lattice. The discrete space–time (DST) method, is set on a pair of dual lattices whose field components are collocated on their respective lattice sites. The other, the time-domain element (TDE) method, is set on overlapping dual lattices whose field components are noncollocated. The TDE method is shown to be a generalization and reinterpretation of the Yee algorithm. The benefits of the combined algorithm over comparable methods include: 1) increased accuracy over larger bandwidths; 2) increased stability allowing larger time steps; 3) local stencil-satisfying boundary conditions on interfaces; 4) self-contained mathematical framework; and 5) it is physically intuitive.

**A Discrete Exterior Calculus and Electromagnetic Theory on a Lattice**

Eric Forgy and Weng Chew, 2000

**Abstract**

A highly accurate FDTD formulation was developed on an overlapped cubic grid that greatly reduces numerical dispersion errors. However, errors in in the FDTD method arise not only from numerical dispersion, but from geometrical modelling as well. Although representing a significant progress in addressing the numerical dispersion problem, it is still confined to a cubic grid with the subsequent “stair-casing” geometric approximations that it entails. The material presented represents a fundamentally new paradigm for finite-difference methods which hopes to address both issues of numerical dispersion and geometrical modelling. It involves a rigorous mathematical framework based on concepts from topology and differential geometry. Particularly, it involves a construction of a discrete analog to the calculus of differential forms. It should be noted that the use of differential forms, and their lattice counterparts, is well known within the field of algebraic topology. However, the original contribution here lies in the introduction of a metric onto the lattice. It is with the metric that the adjoint exterior derivative may be defined, which is required for most physical systems not the least of importance being Maxwell’s equations.

Eric,

Your PhD thesis was on computational E&M and differential geometry, right? Since I am interested in both fields, I would love to take a look at the thesis. I googled for it and found nothing. Would you mind sending it via email? If you prefer not to, I understand.

Thanks for the attention,

-R

rod.June 8, 2008 at 2:57 am

Hi Rod,

No problem. Yes, my dissertation was on differential geometry in computational electromagnetics. It was also a kind of “Introduction to Differential Geometry for Engineers”. Since Maxwell’s equations are naturally written in the language of differential forms, i.e.

and ,

it seemed natural to try to find discrete analogues to differential forms. This turned into a bigger search for a general theory of discrete differential geometry.

I was actually meaning to get in touch with you again. I hope all is well!

I recently read your article

discrete, Amorphous Physical Models

and had some things to say since the topic is near and dear to my heart.

The soft copies of both my MS thesis and PhD dissertation are not readily available so I’ll have to dig around a bit. I’ll send them (and post them hear) once I’ve got them. In the meantime, if you haven’t seen it you might find a paper I wrote with Urs Schreiber interesting:

Discrete Differential Geometry on Causal Graphs

Urs and I (and a few others) discussed the topic over on the String Coffee Table (which is now inactive and the members have moved to the n-Category Cafe) in the thread

The Search for Discrete Differential Geometry.

More recently, we discussed the topic a bit in relation to Jenny Harrison’a work here

Harrison’s Geometric Calculus

but the discussion ultimately deteriorated unfortunately.

It is a great topic that I don’t think has been explored as deeply as it could be.

phorgyphynanceJune 8, 2008 at 6:14 am

Eric,

Thanks for the reply 🙂

I found that paper of yours,

Financial Modelling Using Discrete Stochastic Calculus, long ago and I have been highly intrigued by it ever since. This whole “continuous Vs discrete” thing has been a constant source of fascination for years, and it all started while studying Physics, eh eh. Probably sounds like adejá vuto you.Over the last year or so, I have been following the work some people at Caltech have been doing on Discrete Differential Geometry. I have gathered a bunch of papers (including the ones you wrote with Urs Schreiber, which I found on arXiv.org), but I am afraid my math skills aren’t yet enough to attack this area. I studied EE, not Math / Phys, and as such, Differential Geometry is not something I am terribly familiar with.

In any case, if you ever find a soft copy of your thesis and don’t mind to email it, I would definitely be most grateful.

Last but not least, you’re in SoCal, right? I will be moving to SoCal next summer, and it would be cool to meet up for a beer if the opportunity ever arises.

Cheers!

-R

rod.June 12, 2008 at 5:45 pm

There you go! I managed to find both my PhD dissertation and a paper summarizing my MS thesis (along with some others).

The Caltech guys were certainly aware of my work, but I’m not sure if the grad student ever saw my thesis although they could have and should have since it was published in 2002 (before they even started). My first paper on discrete geometry was written in 2000.

When you get to SoCal, let me know. It’d be great to have a few :beer:

By the way, if you have any questions regarding any of my papers, please don’t hesitate to ask. You know where to find me 🙂

Cheers!

phorgyphynanceJune 12, 2008 at 9:33 pm