Brownian Motion

Phynance ArXiv ID: ssrn-2433826 “View on arXiv” Authors: Unknown Abstract These are the lecture notes for an advanced Ph.D. level course I taught in Spring ‘02 at the C.N. Yang Institute for Theoretical Physics at Stony Brook. The cou Keywords: Stochastic Processes, Financial Mathematics, Brownian Motion, Derivatives Pricing, Derivatives Complexity vs Empirical Score Math Complexity: 9.0/10 Empirical Rigor: 2.0/10 Quadrant: Lab Rats Why: The paper is a PhD-level lecture on advanced stochastic calculus and derivative pricing, heavily featuring formal mathematical derivations and physics-inspired path integral methods, but contains no empirical data, backtests, or implementation details. flowchart TD A["Research Goal: Model Derivatives Pricing via Stochastic Processes"] --> B["Key Methodology: Applied Brownian Motion & Itô Calculus"] B --> C["Data/Inputs: Financial Market Parameters & Hypothetical Models"] C --> D["Computational Process: Solving Stochastic Differential Equations"] D --> E["Outcome: Analytical Derivatives Pricing Frameworks"]

The Shape of Markets: Machine learning modeling and Prediction Using 2-Manifold Geometries ArXiv ID: 2511.05030 “View on arXiv” Authors: Panagiotis G. Papaioannou, Athanassios N. Yannacopoulos Abstract We introduce a Geometry Informed Model for financial forecasting by embedding high dimensional market data onto constant curvature 2manifolds. Guided by the uniformization theorem, we model market dynamics as Brownian motion on spherical S2, Euclidean R2, and hyperbolic H2 geometries. We further include the torus T, a compact, flat manifold admissible as a quotient space of the Euclidean plane anticipating its relevance for capturing cyclical dynamics. Manifold learning techniques infer the latent curvature from financial data, revealing the torus as the best performing geometry. We interpret this result through a macroeconomic lens, the torus circular dimensions align with endogenous cycles in output, interest rates, and inflation described by IS LM theory. Our findings demonstrate the value of integrating differential geometry with data-driven inference for financial modeling. ...

Why do financial prices exhibit Brownian motion despite predictable order flow? ArXiv ID: 2502.17906 “View on arXiv” Authors: Unknown Abstract In financial market microstructure, there are two enigmatic empirical laws: (i) the market-order flow has predictable persistence due to metaorder splitters by institutional investors, well formulated as the Lillo-Mike-Farmer model. However, this phenomenon seems paradoxical given the diffusive and unpredictable price dynamics; (ii) the price impact $I(Q)$ of a large metaorder $Q$ follows the square-root law, $I(Q)\propto \sqrt{“Q”}$. Here we theoretically reveal why price dynamics follows Brownian motion despite predictable order flow by unifying these enigmas. We generalize the Lillo-Mike-Farmer model to nonlinear price-impact dynamics, which is mapped to an exactly solvable Lévy-walk model. Our exact solution shows that the price dynamics remains diffusive under the square-root law, even under persistent order flow. This work illustrates the crucial role of the square-root law in mitigating large price movements by large metaorders, thereby leading to the Brownian price dynamics, consistently with the efficient market hypothesis over long timescales. ...

The Martingale Sinkhorn Algorithm ArXiv ID: 2310.13797 “View on arXiv” Authors: Unknown Abstract We develop a numerical method for the martingale analogue of the Benamou-Brenier optimal transport problem, which seeks a martingale interpolating two prescribed marginals which is closest to the Brownian motion. Recent contributions have established existence and uniqueness for the optimal martingale under finite second moment assumptions on the marginals, but numerical methods exist only in the one-dimensional setting. We introduce an iterative scheme, a martingale analogue of the celebrated Sinkhorn algorithm, and prove its convergence in arbitrary dimension under minimal assumptions. In particular, we show that convergence holds when the marginals have finite moments of order $p > 1$, thereby extending the known theory beyond the finite-second-moment regime. The proof relies on a strict descent property for the dual value of the martingale Benamou–Brenier problem. While the descent property admits a direct verification in the case of compactly supported marginals, obtaining uniform control on the iterates without assuming compact support is substantially more delicate and constitutes the main technical challenge. ...