Stochastic Differential Equations

Chaos, Ito-Stratonovich dilemma, and topological supersymmetry ArXiv ID: 2512.21539 “View on arXiv” Authors: Igor V. Ovchinnikov Abstract It was recently established that the formalism of the generalized transfer operator (GTO) of dynamical systems (DS) theory, applied to stochastic differential equations (SDEs) of arbitrary form, belongs to the family of cohomological topological field theories (TFT) – a class of models at the intersection of algebraic topology and high-energy physics. This interdisciplinary approach, which can be called the supersymmetric theory of stochastic dynamics (STS), can be seen as an algebraic dual to the traditional set-theoretic framework of the DS theory, with its algebraic structure enabling the extension of some DS theory concepts to stochastic dynamics. Moreover, it reveals the presence of a topological supersymmetry (TS) in the GTOs of all SDEs. It also shows that among the various definitions of chaos, positive “pressure”, defined as the logarithm of the GTO spectral radius, stands out as particularly meaningful from a physical perspective, as it corresponds to the spontaneous breakdown of TS on the TFT side. Via the Goldstone theorem, this definition has a potential to provide the long-sought explanation for the experimental signature of chaotic dynamics known as 1/f noise. Additionally, STS clarifies that among the various existing interpretations of SDEs, only the Stratonovich interpretation yields evolution operators that match the corresponding GTOs and, consequently, have a clear-cut mathematical meaning. Here, we discuss these and other aspects of STS from both the DS theory and TFT perspectives, focusing on links between these two fields and providing mathematical concepts with physical interpretations that may be useful in some contexts. ...

One model to solve them all: 2BSDE families via neural operators ArXiv ID: 2511.01125 “View on arXiv” Authors: Takashi Furuya, Anastasis Kratsios, Dylan Possamaï, Bogdan Raonić Abstract We introduce a mild generative variant of the classical neural operator model, which leverages Kolmogorov–Arnold networks to solve infinite families of second-order backward stochastic differential equations ($2$BSDEs) on regular bounded Euclidean domains with random terminal time. Our first main result shows that the solution operator associated with a broad range of $2$BSDE families is approximable by appropriate neural operator models. We then identify a structured subclass of (infinite) families of $2$BSDEs whose neural operator approximation requires only a polynomial number of parameters in the reciprocal approximation rate, as opposed to the exponential requirement in general worst-case neural operator guarantees. ...

Convergence in probability of numerical solutions of a highly non-linear delayed stochastic interest rate model ArXiv ID: 2510.04092 “View on arXiv” Authors: Emmanuel Coffie Abstract We examine a delayed stochastic interest rate model with super-linearly growing coefficients and develop several new mathematical tools to establish the properties of its true and truncated EM solutions. Moreover, we show that the true solution converges to the truncated EM solutions in probability as the step size tends to zero. Further, we support the convergence result with some illustrative numerical examples and justify the convergence result for the Monte Carlo evaluation of some financial quantities. ...

Non-Linear and Meta-Stable Dynamics in Financial Markets: Evidence from High Frequency Crypto Currency Market Makers ArXiv ID: 2509.02941 “View on arXiv” Authors: Igor Halperin Abstract This work builds upon the long-standing conjecture that linear diffusion models are inadequate for complex market dynamics. Specifically, it provides experimental validation for the author’s prior arguments that realistic market dynamics are governed by higher-order (cubic and higher) non-linearities in the drift. As the diffusion drift is given by the negative gradient of a potential function, this means that a non-linear drift translates into a non-quadratic potential. These arguments were based both on general theoretical grounds as well as a structured approach to modeling the price dynamics which incorporates money flows and their impact on market prices. Here, we find direct confirmation of this view by analyzing high-frequency crypto currency data at different time scales ranging from minutes to months. We find that markets can be characterized by either a single-well or a double-well potential, depending on the time period and sampling frequency, where a double-well potential may signal market uncertainty or stress. ...

To Trade or Not to Trade: An Agentic Approach to Estimating Market Risk Improves Trading Decisions ArXiv ID: 2507.08584 “View on arXiv” Authors: Dimitrios Emmanoulopoulos, Ollie Olby, Justin Lyon, Namid R. Stillman Abstract Large language models (LLMs) are increasingly deployed in agentic frameworks, in which prompts trigger complex tool-based analysis in pursuit of a goal. While these frameworks have shown promise across multiple domains including in finance, they typically lack a principled model-building step, relying instead on sentiment- or trend-based analysis. We address this gap by developing an agentic system that uses LLMs to iteratively discover stochastic differential equations for financial time series. These models generate risk metrics which inform daily trading decisions. We evaluate our system in both traditional backtests and using a market simulator, which introduces synthetic but causally plausible price paths and news events. We find that model-informed trading strategies outperform standard LLM-based agents, improving Sharpe ratios across multiple equities. Our results show that combining LLMs with agentic model discovery enhances market risk estimation and enables more profitable trading decisions. ...

A new architecture of high-order deep neural networks that learn martingales ArXiv ID: 2505.03789 “View on arXiv” Authors: Syoiti Ninomiya, Yuming Ma Abstract A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning models. The behaviour of deep neural networks based on this architecture, when applied to the problem of pricing financial derivatives, is also examined. The core of this new architecture lies in the high-order weak approximation algorithms of the explicit Runge–Kutta type, wherein the approximation is realised solely through iterative compositions and linear combinations of vector fields of the target SDEs. ...

Neuro-Symbolic Traders: Assessing the Wisdom of AI Crowds in Markets ArXiv ID: 2410.14587 “View on arXiv” Authors: Unknown Abstract Deep generative models are becoming increasingly used as tools for financial analysis. However, it is unclear how these models will influence financial markets, especially when they infer financial value in a semi-autonomous way. In this work, we explore the interplay between deep generative models and market dynamics. We develop a form of virtual traders that use deep generative models to make buy/sell decisions, which we term neuro-symbolic traders, and expose them to a virtual market. Under our framework, neuro-symbolic traders are agents that use vision-language models to discover a model of the fundamental value of an asset. Agents develop this model as a stochastic differential equation, calibrated to market data using gradient descent. We test our neuro-symbolic traders on both synthetic data and real financial time series, including an equity stock, commodity, and a foreign exchange pair. We then expose several groups of neuro-symbolic traders to a virtual market environment. This market environment allows for feedback between the traders belief of the underlying value to the observed price dynamics. We find that this leads to price suppression compared to the historical data, highlighting a future risk to market stability. Our work is a first step towards quantifying the effect of deep generative agents on markets dynamics and sets out some of the potential risks and benefits of this approach in the future. ...

On the Hull-White model with volatility smile for Valuation Adjustments ArXiv ID: 2403.14841 “View on arXiv” Authors: Unknown Abstract Affine Diffusion dynamics are frequently used for Valuation Adjustments (xVA) calculations due to their analytic tractability. However, these models cannot capture the market-implied skew and smile, which are relevant when computing xVA metrics. Hence, additional degrees of freedom are required to capture these market features. In this paper, we address this through an SDE with state-dependent coefficients. The SDE is consistent with the convex combination of a finite number of different AD dynamics. We combine Hull-White one-factor models where one model parameter is varied. We use the Randomized AD (RAnD) technique to parameterize the combination of dynamics. We refer to our SDE with state-dependent coefficients and the RAnD parametrization of the original models as the rHW model. The rHW model allows for efficient semi-analytic calibration to European swaptions through the analytic tractability of the Hull-White dynamics. We use a regression-based Monte-Carlo simulation to calculate exposures. In this setting, we demonstrate the significant effect of skew and smile on exposures and xVAs of linear and early-exercise interest rate derivatives. ...

A Mean Field Game between Informed Traders and a Broker ArXiv ID: 2401.05257 “View on arXiv” Authors: Unknown Abstract We find closed-form solutions to the stochastic game between a broker and a mean-field of informed traders. In the finite player game, the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a Gâteaux derivative approach, we characterise the solution to the game with a system of forward-backward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory, on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients. ...

Instabilities of Super-Time-Stepping Methods on the Heston Stochastic Volatility Model ArXiv ID: 2309.00540 “View on arXiv” Authors: Unknown Abstract This note explores in more details instabilities of explicit super-time-stepping schemes, such as the Runge-Kutta-Chebyshev or Runge-Kutta-Legendre schemes, noticed in the litterature, when applied to the Heston stochastic volatility model. The stability remarks are relevant beyond the scope of super-time-stepping schemes. Keywords: super-time-stepping schemes, Heston stochastic volatility model, Runge-Kutta-Chebyshev, numerical stability, stochastic differential equations, Equity (Derivatives Pricing) ...