Neural Brownian Motion

TL;DR

This paper introduces Neural-Brownian Motion, a novel stochastic process defined via neural network-driven non-linear expectations, with a representation theorem and stochastic calculus developed for modeling learned uncertainty.

Contribution

It formulates a new class of stochastic processes using neural networks within a non-linear expectation framework, including a representation theorem and a Girsanov-type theorem.

Findings

Existence and uniqueness of the Neural-Brownian Motion as a strong solution.

Development of stochastic calculus for the process.

Demonstration of measure change and drift learning through neural parameters.

Abstract

This paper introduces the Neural-Brownian Motion (NBM), a new class of stochastic processes for modeling dynamics under learned uncertainty. The NBM is defined axiomatically by replacing the classical martingale property with respect to linear expectation with one relative to a non-linear Neural Expectation Operator, $\varepsilon^\theta$, generated by a Backward Stochastic Differential Equation (BSDE) whose driver $f_\theta$ is parameterized by a neural network. Our main result is a representation theorem for a canonical NBM, which we define as a continuous $\varepsilon^\theta$-martingale with zero drift under the physical measure. We prove that, under a key structural assumption on the driver, such a canonical NBM exists and is the unique strong solution to a stochastic differential equation of the form ${\rm d} M_t = \nu_\theta(t, M_t) {\rm d} W_t$. Crucially, the volatility function $\nu_\theta$ is not postulated a priori but is implicitly defined by the algebraic constraint $g_\theta(t, M_t, \nu_\theta(t, M_t)) = 0$, where $g_\theta$ is a specialization of the BSDE driver. We develop the stochastic calculus for this process and prove a Girsanov-type theorem for the quadratic case, showing that an NBM acquires a drift under a new, learned measure. The character of this measure, whether pessimistic or optimistic, is endogenously determined by the learned parameters $\theta$, providing a rigorous foundation for models where the attitude towards uncertainty is a discoverable feature.