Why On Earth Use Statistical Physics to Model Financial Assets?

statistical physics

Fokker–Planck

neural SDE

methodology

The equation that governs wealth distributions is not a metaphor borrowed from physics — it is the same equation, derived from the same mathematics. And modern machine learning knows how to estimate it.

Author

Anders G Frøseth

Published

March 18, 2026

The Fifth Solvay Conference, 1927. Front row includes Max Planck (second from left) and Paul Langevin (far right) — two of the physicists whose equations now govern models of wealth dynamics. Photograph by Benjamin Couprie, public domain.

The short answer

Because geometric Brownian motion is a Langevin equation. Not by analogy. Not by loose metaphor. The stochastic differential equation that underlies the Black–Scholes model, that every finance textbook writes as

\[ \frac{\mathrm{d}W}{W} = \mu \, \mathrm{d}t + \sigma \, \mathrm{d}B_t, \]

is, in the language of physics, a Langevin equation with multiplicative noise. Take the logarithm — apply Itô’s lemma — and you get

\[ \mathrm{d}x = v \, \mathrm{d}t + \sigma \, \mathrm{d}B_t, \qquad v \equiv \mu - \frac{\sigma^2}{2}, \]

which is a Langevin equation with additive noise and constant drift $v$. The probability density of log-wealth across a population then obeys the Fokker–Planck equation

\[ \frac{\partial \pi}{\partial t} = -v \frac{\partial \pi}{\partial x} + D \frac{\partial^2 \pi}{\partial x^2}, \qquad D \equiv \frac{\sigma^2}{2}. \]

This is not a modelling choice. It is a mathematical identity. If you accept that wealth grows multiplicatively with random returns — and this is standard in finance — then the distribution of wealth satisfies a Fokker–Planck equation. The only question is whether this observation is useful.

A brief history

The connection between random walks and financial markets is older than modern finance. Louis Bachelier’s 1900 thesis modelled bond prices as Brownian motion — five years before Einstein’s paper on the diffusion of pollen grains. The mathematics was the same; the applications were separated by disciplinary walls that would persist for most of the twentieth century.

Those walls started to crack in the 1990s. Physicists — trained in the statistical mechanics of many-body systems — noticed that financial markets are, in a precise sense, many-body problems: large numbers of interacting agents, stochastic dynamics, emergent macroscopic regularities (power-law tails, volatility clustering, correlation structures). The resulting field, sometimes called econophysics, produced a body of empirical work on wealth and income distributions.

Bouchaud and Mézard (2000) showed that a random-growth model with weak redistribution — essentially a system of coupled Langevin equations — generates the Pareto power law observed in the upper tail of wealth distributions. Yakovenko and collaborators mapped the full wealth distribution, finding an exponential bulk (a Boltzmann–Gibbs distribution, in physics language) with a Pareto tail. Gabaix (2009) surveyed the broader family of random-growth mechanisms that produce power laws in economics.

But most of this work remained descriptive. It characterised the shape of distributions without connecting to the decision problems that financial economics cares about — portfolio choice, asset pricing, tax incidence. And mainstream financial economics, for its part, continued to work with individual-level optimisation problems (CAPM, consumption-based asset pricing) without much interest in the distributional dynamics that the physics tools were designed for.

What the Fokker–Planck framework adds

The statistical physics paper in this series bridges the two traditions by using the Fokker–Planck equation not as a descriptive tool for distributions but as an analytical framework for policy. The central observation is simple: a proportional wealth tax at rate $\tau_w$ modifies the Langevin equation to

\[ \mathrm{d}x = (v - \tau_w) \, \mathrm{d}t + \sigma \, \mathrm{d}B_t. \]

The tax shifts the drift by a constant. The diffusion coefficient $D = \sigma^2/2$ is unchanged. In Fokker–Planck language, this is a drift-shift symmetry — the operator that governs the evolution of the wealth distribution is modified only in its first-order (advection) term. The second-order (diffusion) term, which controls the stochastic structure of the dynamics, is untouched.

This turns out to be the physical content of tax neutrality. Every departure from neutrality — book-value assessment, liquidity frictions, non-uniform taxation across assets — corresponds to a specific breaking of this symmetry: a state-dependent modification of the drift, an asset-dependent modification of the diffusion, or a coupling between agents that introduces mean-field interactions. The Fokker–Planck framework provides a taxonomy: each channel of distortion maps to a distinct modification of the equation, and the modifications are additive, so you can analyse them one at a time.

The reason this matters is that it connects the micro question (does the tax change my optimal portfolio?) to the macro question (how does the wealth distribution evolve?) through a single equation. Traditional financial economics answers the first question; econophysics answers the second. The Fokker–Planck framework answers both simultaneously, because the same equation governs individual dynamics and population-level distributions.

The empirical payoff

The real power of reformulating wealth dynamics as a Fokker–Planck problem becomes apparent when you ask: how do we estimate the drift and diffusion from data?

In the classical setting with constant coefficients, this is straightforward — you estimate a mean and a variance. But the interesting economic questions involve state-dependent dynamics: does the drift increase with wealth (returns to scale)? Does a tax shift the drift uniformly or introduce curvature? Is there a confining force at the top of the distribution (progressive taxation)?

These are questions about functions — the drift $v(x)$ and diffusion $D(x)$ as functions of the state variable — and they require nonparametric estimation. This is where the formulation pays off. The recent explosion of machine learning methods for stochastic differential equations — neural SDEs, score-based estimation, diffusion models — provides a ready-made toolkit for exactly this class of problem. These methods were developed in other contexts, but they solve the same inverse problem: given observed trajectories, recover the drift and diffusion of the underlying process. The Fokker–Planck formulation makes the connection direct, because the object being estimated is the Fokker–Planck equation.

What this means for the research programme

The wealth tax neutrality framework generates precise, testable predictions about how taxation modifies the drift and diffusion of wealth dynamics. The statistical physics formulation translates these predictions into the language of differential equations. Modern machine learning provides estimation tools for exactly this class of equation. The three layers — theory, formalism, and computation — fit together because they describe the same object: the stochastic process that governs how wealth evolves over time.

The question that opened this post — why use statistical physics? — has, I think, a pragmatic answer. When the mathematics of your problem is a stochastic differential equation, it makes sense to use the tools that were designed for stochastic differential equations — regardless of which department (economics or physics) developed them.

--- title: "Why On Earth Use Statistical Physics to Model Financial Assets?" description: "The equation that governs wealth distributions is not a metaphor borrowed from physics — it is the same equation, derived from the same mathematics. And modern machine learning knows how to estimate it." author: "Anders G Frøseth" date: "2026-03-18" categories: [statistical physics, Fokker–Planck, neural SDE, methodology] image: solvay_1927.jpg --- ![The Fifth Solvay Conference, 1927. Front row includes Max Planck (second from left) and Paul Langevin (far right) — two of the physicists whose equations now govern models of wealth dynamics. Photograph by Benjamin Couprie, [public domain](https://commons.wikimedia.org/wiki/File:Solvay_conference_1927.jpg).](solvay_1927.jpg) ## The short answer Because geometric Brownian motion *is* a Langevin equation. Not by analogy. Not by loose metaphor. The stochastic differential equation that underlies the Black–Scholes model, that every finance textbook writes as $$ \frac{\mathrm{d}W}{W} = \mu \, \mathrm{d}t + \sigma \, \mathrm{d}B_t, $$ is, in the language of physics, a Langevin equation with multiplicative noise. Take the logarithm — apply Itô's lemma — and you get $$ \mathrm{d}x = v \, \mathrm{d}t + \sigma \, \mathrm{d}B_t, \qquad v \equiv \mu - \frac{\sigma^2}{2}, $$ which is a Langevin equation with additive noise and constant drift $v$. The probability density of log-wealth across a population then obeys the Fokker–Planck equation $$ \frac{\partial \pi}{\partial t} = -v \frac{\partial \pi}{\partial x} + D \frac{\partial^2 \pi}{\partial x^2}, \qquad D \equiv \frac{\sigma^2}{2}. $$ This is not a modelling choice. It is a mathematical identity. If you accept that wealth grows multiplicatively with random returns — and this is standard in finance — then the distribution of wealth satisfies a Fokker–Planck equation. The only question is whether this observation is *useful*. ## A brief history The connection between random walks and financial markets is older than modern finance. Louis Bachelier's 1900 thesis modelled bond prices as Brownian motion — five years before Einstein's paper on the diffusion of pollen grains. The mathematics was the same; the applications were separated by disciplinary walls that would persist for most of the twentieth century. Those walls started to crack in the 1990s. Physicists — trained in the statistical mechanics of many-body systems — noticed that financial markets are, in a precise sense, many-body problems: large numbers of interacting agents, stochastic dynamics, emergent macroscopic regularities (power-law tails, volatility clustering, correlation structures). The resulting field, sometimes called econophysics, produced a body of empirical work on wealth and income distributions. Bouchaud and Mézard (2000) showed that a random-growth model with weak redistribution — essentially a system of coupled Langevin equations — generates the Pareto power law observed in the upper tail of wealth distributions. Yakovenko and collaborators mapped the full wealth distribution, finding an exponential bulk (a Boltzmann–Gibbs distribution, in physics language) with a Pareto tail. Gabaix (2009) surveyed the broader family of random-growth mechanisms that produce power laws in economics. But most of this work remained descriptive. It characterised the *shape* of distributions without connecting to the *decision problems* that financial economics cares about — portfolio choice, asset pricing, tax incidence. And mainstream financial economics, for its part, continued to work with individual-level optimisation problems (CAPM, consumption-based asset pricing) without much interest in the distributional dynamics that the physics tools were designed for. ## What the Fokker–Planck framework adds The [statistical physics paper](/research/wealth-tax/index.qmd) in this series bridges the two traditions by using the Fokker–Planck equation not as a descriptive tool for distributions but as an *analytical framework for policy*. The central observation is simple: a proportional wealth tax at rate $\tau_w$ modifies the Langevin equation to $$ \mathrm{d}x = (v - \tau_w) \, \mathrm{d}t + \sigma \, \mathrm{d}B_t. $$ The tax shifts the drift by a constant. The diffusion coefficient $D = \sigma^2/2$ is unchanged. In Fokker–Planck language, this is a *drift-shift symmetry* — the operator that governs the evolution of the wealth distribution is modified only in its first-order (advection) term. The second-order (diffusion) term, which controls the stochastic structure of the dynamics, is untouched. This turns out to be the physical content of [tax neutrality](/research/wealth-tax/index.qmd). Every departure from neutrality — book-value assessment, liquidity frictions, non-uniform taxation across assets — corresponds to a specific *breaking* of this symmetry: a state-dependent modification of the drift, an asset-dependent modification of the diffusion, or a coupling between agents that introduces mean-field interactions. The Fokker–Planck framework provides a taxonomy: each channel of distortion maps to a distinct modification of the equation, and the modifications are additive, so you can analyse them one at a time. The reason this matters is that it connects the *micro* question (does the tax change my optimal portfolio?) to the *macro* question (how does the wealth distribution evolve?) through a single equation. Traditional financial economics answers the first question; econophysics answers the second. The Fokker–Planck framework answers both simultaneously, because the same equation governs individual dynamics and population-level distributions. ## The empirical payoff The real power of reformulating wealth dynamics as a Fokker–Planck problem becomes apparent when you ask: how do we *estimate* the drift and diffusion from data? In the classical setting with constant coefficients, this is straightforward — you estimate a mean and a variance. But the interesting economic questions involve state-dependent dynamics: does the drift increase with wealth (returns to scale)? Does a tax shift the drift uniformly or introduce curvature? Is there a confining force at the top of the distribution (progressive taxation)? These are questions about *functions* — the drift $v(x)$ and diffusion $D(x)$ as functions of the state variable — and they require nonparametric estimation. This is where the formulation pays off. The recent explosion of machine learning methods for stochastic differential equations — neural SDEs, score-based estimation, diffusion models — provides a ready-made toolkit for exactly this class of problem. These methods were developed in other contexts, but they solve the same inverse problem: given observed trajectories, recover the drift and diffusion of the underlying process. The Fokker–Planck formulation makes the connection direct, because the object being estimated *is* the Fokker–Planck equation. ## What this means for the research programme The [wealth tax neutrality framework](/research/wealth-tax/index.qmd) generates precise, testable predictions about how taxation modifies the drift and diffusion of wealth dynamics. The statistical physics formulation translates these predictions into the language of differential equations. Modern machine learning provides estimation tools for exactly this class of equation. The three layers — theory, formalism, and computation — fit together because they describe the same object: the stochastic process that governs how wealth evolves over time. The question that opened this post — why use statistical physics? — has, I think, a pragmatic answer. When the mathematics of your problem is a stochastic differential equation, it makes sense to use the tools that were designed for stochastic differential equations — regardless of which department (economics or physics) developed them.