Stochastic Processes

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"]

Deep Neural Operator Learning for Probabilistic Models ArXiv ID: 2511.07235 “View on arXiv” Authors: Erhan Bayraktar, Qi Feng, Zecheng Zhang, Zhaoyu Zhang Abstract We propose a deep neural-operator framework for a general class of probability models. Under global Lipschitz conditions on the operator over the entire Euclidean space-and for a broad class of probabilistic models-we establish a universal approximation theorem with explicit network-size bounds for the proposed architecture. The underlying stochastic processes are required only to satisfy integrability and general tail-probability conditions. We verify these assumptions for both European and American option-pricing problems within the forward-backward SDE (FBSDE) framework, which in turn covers a broad class of operators arising from parabolic PDEs, with or without free boundaries. Finally, we present a numerical example for a basket of American options, demonstrating that the learned model produces optimal stopping boundaries for new strike prices without retraining. ...

Target search optimization by threshold resetting ArXiv ID: 2504.13501 “View on arXiv” Authors: Unknown Abstract We introduce a new class of first passage time optimization driven by threshold resetting, inspired by many natural processes where crossing a critical limit triggers failure, degradation or transition. In here, search agents are collectively reset when a threshold is reached, creating event-driven, system-coupled simultaneous resets that induce long-range interactions. We develop a unified framework to compute search times for these correlated stochastic processes, with ballistic- and diffusive- searchers as key examples uncovering diverse optimization behaviors. A cost function, akin to breakdown penalties, reveals that optimal resetting can forestall larger losses. This formalism generalizes to broader stochastic systems with multiple degrees of freedom. ...

How to verify that a given process is a Lévy-Driven Ornstein-Uhlenbeck Process ArXiv ID: 2501.03434 “View on arXiv” Authors: Unknown Abstract Assuming that a Lévy-Driven Ornstein-Uhlenbeck (or CAR(1)) processes is observed at discrete times $0$, $h$, $2h$,$\cdots$ $[“T/h”]h$. We introduce a step-by-step methodological approach on how a person would verify the model assumptions. The methodology involves estimating the model parameters and approximating the driving process. We demonstrate how to use the increments of the approximated driving process, along with the estimated parameters, to test the assumptions that the CAR(1) process is Lévy-driven. We then show how to test the hypothesis that the CAR(1) process belongs to a specified class of Lévy processes. The performance of the tests is illustrated through multiple simulations. Finally, we demonstrate how to apply the methodology step-by-step to a variety of economic and financial data examples. ...

Neural Operators Can Play Dynamic Stackelberg Games ArXiv ID: 2411.09644 “View on arXiv” Authors: Unknown Abstract Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader’s strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower’s best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{“follower’s best-response operator”} can be approximately implemented by an \textit{“attention-based neural operator”}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games. ...

An Algebraic Framework for the Modeling of Limit Order Books ArXiv ID: 2406.04969 “View on arXiv” Authors: Unknown Abstract Introducing an algebraic framework for modeling limit order books (LOBs) with tools from physics and stochastic processes, our proposed framework captures the creation and annihilation of orders, order matching, and the time evolution of the LOB state. It also enables compositional settings, accommodating the interaction of heterogeneous traders and different market structures. We employ Dirac notation and generalized generating functions to describe the state space and dynamics of LOBs. The utility of this framework is shown through simulations of simplified market scenarios, illustrating how variations in trader behavior impact key market observables such as spread, return volatility, and liquidity. The algebraic representation allows for exact simulations using the Gillespie algorithm, providing a robust tool for exploring the implications of market design and policy changes on LOB dynamics. Future research can expand this framework to incorporate more complex order types, adaptive event rates, and multi-asset trading environments, offering deeper insights into market microstructure and trader behavior and estimation of key drivers for market microstructure dynamics. ...

Entropy corrected geometric Brownian motion ArXiv ID: 2403.06253 “View on arXiv” Authors: Unknown Abstract The geometric Brownian motion (GBM) is widely employed for modeling stochastic processes, yet its solutions are characterized by the log-normal distribution. This comprises predictive capabilities of GBM mainly in terms of forecasting applications. Here, entropy corrections to GBM are proposed to go beyond log-normality restrictions and better account for intricacies of real systems. It is shown that GBM solutions can be effectively refined by arguing that entropy is reduced when deterministic content of considered data increases. Notable improvements over conventional GBM are observed for several cases of non-log-normal distributions, ranging from a dice roll experiment to real world data. ...

Dynamic Models and Structural Estimation in Corporate Finance ArXiv ID: ssrn-2268569 “View on arXiv” Authors: Unknown Abstract We review the last two decades of research in dynamic corporate finance, focusing on capital structure and the financing of investment. We first cover continuou Keywords: Dynamic Corporate Finance, Capital Structure, Investment Financing, Continuous Time Models, Stochastic Processes, Corporate Finance Complexity vs Empirical Score Math Complexity: 8.0/10 Empirical Rigor: 3.0/10 Quadrant: Lab Rats Why: The paper is a literature review focused on structural estimation and dynamic models, which inherently involves advanced mathematics and continuous-time frameworks, but it is a theoretical overview rather than a backtest-ready empirical study. flowchart TD A["Research Goal"] -->|Investigate dynamic models<br>in corporate finance| B["Methodology: Continuous-Time<br>Stochastic Processes"] B --> C["Data: Capital Structure<br>& Investment Data"] C --> D["Computational Process:<br>Structural Estimation"] D --> E{"Key Findings"} E --> F["Optimal Dynamic<br>Capital Structure"] E --> G["Financing Constraints<br>& Investment"]

Dynamic Models and Structural Estimation in CorporateFinance ArXiv ID: ssrn-2091854 “View on arXiv” Authors: Unknown Abstract We review the last two decades of research in dynamic corporate finance, focusing on capital structure and the financing of investment. We first cover continuo Keywords: Dynamic Corporate Finance, Capital Structure, Investment Financing, Continuous Time Models, Stochastic Processes, Corporate Finance Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 3.0/10 Quadrant: Lab Rats Why: The paper is a review of advanced theoretical models (continuous-time contingent claims, dynamic optimization) requiring heavy mathematical formalism, but it focuses on model exposition and intuition rather than presenting new data, backtests, or implementation details. flowchart TD A["Research Goal: Review Dynamic Corporate Finance Models"] --> B["Methodology: Continuous-Time Stochastic Processes"] B --> C["Data/Inputs: Firm-level financial data"] B --> D["Computational Processes: Structural Estimation"] C --> D D --> E["Outcome 1: Optimal Capital Structure"] D --> F["Outcome 2: Investment Financing Dynamics"] D --> G["Outcome 3: Macro-Financial Linkages"] E --> H["Key Findings: Models Explain Debt Heterogeneity & Investment Sensitivity"] F --> H G --> H

Review of Discrete and Continuous Processes inFinance: Theory and Applications ArXiv ID: ssrn-1373102 “View on arXiv” Authors: Unknown Abstract We review the main processes used to model financial variables. We emphasize the parallel between discrete-time processes, mainly used by econometricians for ri Keywords: Financial Modeling, Stochastic Processes, Time Series Econometrics, Discrete-time Processes, Econometrics Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 3.0/10 Quadrant: Lab Rats Why: The paper is dense with advanced mathematics like stochastic calculus, PDEs, and detailed derivations of processes (e.g., Ornstein-Uhlenbeck, fractional Brownian motion). However, it lacks backtesting, code examples beyond mention, or empirical datasets, focusing instead on theoretical review and intuition. flowchart TD A["Research Goal:\nReview & Compare Discrete vs. Continuous\nFinancial Processes"] --> B{"Methodology"} B --> C["Literature Review"] B --> D["Theoretical Analysis"] C --> E["Data/Inputs:\nEconometric Theory\nFinancial Models\nStochastic Processes"] D --> E E --> F["Computational Process:\nParallel Comparison of\nDiscrete-time vs. Continuous-time\nModeling Frameworks"] F --> G["Key Findings:\n1. Discrete-time: Preferred for Econometrics\n2. Continuous-time: Preferred for Derivatives\n3. Bridging the gap improves forecasting"]