Some Comments on the Differential Equations Governing Markets

Some Comments on the Differential Equations Governing Markets

The current article may be of interest to mathematically inclined professionals on Wall Street and other commercial centers. The “AI Revolution” has inaugurated rapid, permanent, unpredictable changes in trading strategy. Those who read and remarked upon my recent article in New York Weekly were genuinely intrigued by the mathematical foundations of set theory in AI, even apart from any tangible profit that such study might have furnished. My 2022 submission regarding probabilistic scattering in market prediction for derivatives and currencies was received with similar grace and goodwill.

Despite the lovely attribution that directly influenced my decision to pursue the series on the “Great Equations” in Toronto, I lacked the leisure to submit a more complete deconstruction of the necessarily vague mathematical treatment rendered in New York Weekly. The tyranny of the moment impelled me to forget further enterprises in New York for the time being, as I had written some articles of mathematical and economic interest to traders in Tokyo and London. More generally, I had studied the effects of taxation on the theoretical denomination of crude for readers of Business Sun.

The essence of the current article is to explore the probabilistic results underlying a small slice of AI, and to broach subsidiary questions of more than academic importance. One of such questions introduced in my quantum scattering article for New York Weekly bears further exploration. How can the prices of derivatives be modeled by scattering processes? In the first instance, it is necessary to review a (very approximate) definition of probabilistic scattering. If one throws a handful of sand into the air on a still day, a small part of the sand will land at some appreciable distance from where it was scattered. However, most of the sand will land in a much smaller area.

If one tries to compute this area from naïve physical manipulation – following the application of Newton’s Laws to the force of the throw – one will be compelled to compute the landing-points of many tiny, irregular particles of sand individually. This task is quite close to impossible. The differential equations that govern the motion of particles scattered in a gas can only give us approximations of the area covered. These so-called “Stochastic Differential Equations” (SDE) operate in a mathematical middle-ground between the particle theory of Newton’s Laws and the pure probability (Bell curves, Z-scores, and other statistical notions). Even though we will not be able to determine the exact positions of the particles of sand, we can – with a few sensible initial conditions – determine the probability that any sufficiently large number of grains will be inside a fixed area from where we made the throw.

SDE may be studied in the setting of mathematical analysis: From this vantage, we may inquire whether a unique solution to any given problem may be found – and, if so, whether this solution “exists globally” or fails to exist after finite time. Such questions plague every mathematician, were quite prominent in my very early career in differential equations, and indirectly inspired the Toronto Telegraph pieces that discuss applications in Partial Differential Equations. Please note that my sister, also a mathematician, has produced results of high significance in atmospheric science using SDE techniques. These are of a very technical nature and appear in a search for “Jonathan Kenigson.”

From the standpoint of commercial applications, these questions have an astoundingly high significance for the prevention of unstable algorithmic trading dynamics – especially for futures and options, whose pricing frequently fails to exhibit complete fidelity to Black-Scholes (SDE) assumptions. When the existence of a single solution to an SDE cannot be found, different approximations may exist. Questions of which approximation (if any) exhibits mathematical stability and fidelity to historical data become statistical questions of estimation, interpolation, and probabilistic modeling.

The numerical modeling of approximate solutions to SDE constitutes a subfield of numerical analysis and mathematical statistics. Random walks permit the sampling of large quantities of data using crawling algorithms that progress through datasets in random increments (steps) according to an underlying (typically Normal, or Bell-Shaped distribution); Google PageRank is perhaps the most prominent algorithm of this type. Markov Processes (MP) generalize random walks to questions of multidimensional complex analysis: Instead of Normal increments, one has Markov step functions whose primary condition is “memory-free” motion. By “memory-free,” we connote that the MP doesn’t predicate future outcomes’ anticipated probabilities on past states of the process. By truncating the memory of a process, computational approximation becomes easier: It perhaps becomes possible to guarantee a desired level of accuracy in the short timescales demanded by algorithmic trading.  

Stochastic Processes (SP) are more general than MP because they can accord the past greater weight in offering future predictions. The increment function partially replaces the Markov step functions. For instance, weather models are Stochastic, according both past and present conditions some specified weight in the predictive process. When they are sufficiently smooth, SP can be characterized by Multivariable Calculus and Complex Analysis. One recovers the SDE from a “probability first” perspective. All the traditional rules of probability apply to SDE, as do all the traditional tools of Calculus.

In practice, both commanding vantages are exploited simultaneously in proofs. Their underlying similarities are found in modern mathematical physics in the classical and quantum motions of particles via Weiner Processes, Brownian Motion, Electrodynamics, Probabilistic Wave Mechanics, and Field Theory. Field Theory is the study of the conservation of energy/momentum in classical systems, its probabilistic analogues appear throughout physics, engineering, weather forecasting, missile guidance, orbital mechanics, and the motions of markets. The article I furnished for New York Statesman provides a birds-eye overview of Fields, as well as some applications in physics and mechanics.

What do all of these disciplines have in common? SDE forms a central component of all of them. Within the framework of SDE, one finds the deepest analytic results. Under generous conditions, all random processes behave similarly to “Normal” ones. Their means (averages) tend towards a Normal Distribution (Bell Curve). This remarkable property was codified by Lévy, Kolmogorov, and other collaborators in the 1930s’; in its original form, it is formally entitled the Lindeberg–Lévy Central Limit Theorem. Variations permit the relaxation of the conditions of the Lindeberg–Lévy Theorem in multiple dimensions, yielding the Berry-Esseen class of theorems. For SDE, Donsker’s Functional Central Limit Theorem guarantees that Lindeberg–Lévy conditions exist for sequences of SP.

The result – also exploited and generalized by Kolmogorov in the 1930s – applies to Markov processes as well. It furnishes a unification of the seemingly unrelated similarities among market fluctuations, quantum physics, orbital mechanics, and engineering. The proofs of Donsker-type results ultimately rely upon Sir Isaac Newton’s approximation methods for exponential functions via characteristic functions. Just as Newton revolutionized the study of mechanics with his Laws of Motion, he also furnished a theory of approximation that unifies all the Central Limit Theorems within the broad framework of Differential Calculus.

Because Differential Calculus relies upon Newtonian Approximation via numerical experiments, the same class of approximation theories applies throughout Special and General Relativity via post-Newtonian Approximations. Approximations of entire systems via their means is very nearly ubiquitous in the natural, physical, and economic sciences. My more technical treatment of these topics appears in Daily Silicon Valley for computer scientists and scientifically inclined software engineers. My intention was to produce a single article, but the topic was too weighty. At present, the intention is to continue the current discussion in the Toronto series regarding “The Great Equations.”


Dr. Jonathan Kenigson, FRSA

Springfield, TN USA