Non-Standard Analysis: Infinitesimals in Finance

Non-Standard Analysis (NSA), introduced by Abraham Robinson (1960), provides a rigorous foundation for Infinitesimals—numbers that are smaller than any positive real number but greater than zero. In 2026, NSA is the primary bridge between the discrete "ticks" of High-Frequency Trading (HFT) and the continuous smooth curves of Stochastic Calculus.

1. The Hyperreal Number System ( $\mathbb{R}^*$ )

NSA extends the real numbers $\mathbb{R}$ to the Hyperreals $\mathbb{R}^*$ , which include:

Standard Reals: $1, \pi, \sqrt{2}$.
Infinitesimals ( $\epsilon$ ): Numbers such that $| \epsilon | < 1/n$ for all $n \in \mathbb{N}$ .
Infinite Numbers ( $\omega$ ): Numbers such that $\omega > n$ for all $n \in \mathbb{N}$ .

2. Bridging the Discrete-Continuous Gap

In Finance, the price process is actually a sequence of discrete trades. However, the math of Black-Scholes uses continuous-time Brownian Motion ( $dW_t$ ).

A. The Hyperreal Random Walk

Using NSA, we model Brownian motion as a Hyperfinite Random Walk with infinitesimal steps of size $\Delta t = \epsilon$ .

The Benefit: Unlike standard measure theory, which is highly abstract, NSA allows quants to treat SDEs as simple algebraic equations over the hyperreals. It makes the Itô Integral as intuitive as a standard summation.

3. High-Frequency Trading (HFT) Microstructure

Modern HFT systems operate at sub-millisecond scales where market "liquidity" looks non-continuous.

Micro-Burst Modeling: Hyperreal time-scales allow researchers to model "bursts" of volatility that appear instantaneous in $\mathbb{R}$ but have a detailed, sequential structure in $\mathbb{R}^*$ .
Infill Asymptotics: Deriving the limit properties of estimators as the frequency of observation goes to infinity (infinitesimal intervals).

4. Non-Standard Errors (NSE)

A 2025 research focus is the Non-Standard Error—the uncertainty introduced by the variation in researchers' modeling choices. Even with identical data, differences in infinitesimal assumptions lead to divergent volatility estimates in HFT environments.

External Deep Dive:

Non-standard Analysis (Wikipedia) — Comprehensive overview of Robinson's framework.
Hyperreal Number (Wikipedia) — Formal construction of the $\mathbb{R}^*$ field.
Internal Set (Wikipedia) — The critical set-theoretic distinction in NSA.

See Also:

Numerical Methods — Finite-precision arithmetic.
Real Analysis — The standard real number system.
Quantitative Finance Research Hub