Dynamic-planning agent system use case: deep research on deriving differential equations from financial data
May 25, 2025
The research team at AI Search
email@aisearchllc.com
Introduction
We are currently developing a large language model (LLM) agent system that dynamically adapts it’s planning to the evidence being collected to solve problems. Once finished, we plan to write a paper describing the final agent system. This article is intended to show how our system is currently working for an interesting example: a literature search on how to derive differential equations from financial data. We show the internal workings of the system. In particular how the agents extract the relevant bits of information from from complex text and then use these extracts to adapt their planning.
Why use an agent system?
Agent systems generally divide a problem into parts that can be separately tackled by specialized agents that may be endowed with tools to reach their goals. While the agents themselves may be organized in many possible ways (loosely integrated, tightly scripted, or something in between) the common feature is the division of labor towards solving a complex task. In this sense agent systems extend the capabilities of LLMs through a piecewise approach.
In the same direction of extending LLMs, another strong argument for agent systems involves information not available to the LLM during its training. This can simply be due to the fact that recent events were not included in the LLM’s training data. Also proprietary information would not be known to the LLM as well as the information that may be obtained from arbitrary tools (e.g., simulations of proprietary systems). Perhaps the enduring advantage of agent systems lies simply on their capability of customizing LLMs to particular applications.
Use case: conduct deep research on financial data modeling
To illustrate how agent systems work let us consider the following query:
Is it possible to extract laws in the form of differential equations from financial data? If so, what are the insights or main points of the corresponding algorithms?
This is a multipart query that may, to a large extent, be answered by LLMs using their current knowledge. Answering the query above does not require, for the most part, proprietary knowledge or special tools. Thus, this usecase offers a fair way to compare agent systems with LLMs by not exploiting the obvious agent advantages of access to proprietary information and tools. Under these conditions, in order to improve over LLMs the agent system will have to reason about how to best answer the user query as well as support it’s claims with references where appropriate. Let us first look at the LLM responses.
Direct LLM responses
Google’s AI overview (Gemini)
AI Overview
Yes, it's possible to extract differential equations from financial data, and algorithms like machine learning and sparse regression are used for this purpose. These algorithms help identify underlying dynamical laws and their governing equations from empirical financial data.
Insights and Main Points of Corresponding Algorithms:
· Machine Learning:
· Machine learning models, including neural networks, can be trained to predict the behavior of financial assets based on historical data and identify complex relationships.
· They can learn from data without explicit programming of the underlying differential equations, making them suitable for modeling complex financial systems.
· Physics-informed machine learning (PIML) incorporates prior knowledge about the governing differential equations, boundary conditions, and initial conditions into the model, enhancing accuracy and interpretability.
· PIML loss functions grade predictions not only on their accuracy but also on whether they are consistent with the governing differential equations.
· Sparse Regression:
· Sparse regression techniques, such as the Basis Pursuit algorithm, can identify the relevant terms in the differential equation that best describe the system's dynamics.
· It can effectively identify the most important features in financial data, helping to uncover underlying relationships.
· Sparse regression can be combined with other techniques like denoising to smooth the signal and reduce noise.
· Data-Driven PDE Discovery:
· Machine learning algorithms can be used to extract features from financial data and identify the building blocks of PDEs that govern spatiotemporal data.
· The method focuses on robust features that are meaningful and have a physical background, allowing for transferability to new observations.
· This approach allows for the identification of governing equations directly from data, even under complex conditions, according to ScienceDirect.
· Differential Equation Discovery:
· Algorithms like those proposed in a Nature study can identify nonlinear differential equations that describe system dynamics from empirical data.
· They integrate denoising techniques, sparse regression, and bootstrap confidence intervals to quantify the uncertainty of estimates.
· These methods have the potential to impact the understanding of complex systems, especially where data is abundant but developing mathematical models is challenging.
· Black-Scholes Equation:
· The Black-Scholes equation, a well-known partial differential equation, is used to model the price of European options.
· It represents the classical heat or diffusion equation, highlighting the applicability of PDEs in financial modeling.
· The equation helps in pricing options and other derivatives by considering the underlying asset's price, volatility, and time to maturity.
Gemma 3
## Extracting Laws as Differential Equations from Financial Data: A Deep Dive
Yes, it is indeed possible to attempt to extract "laws" in the form of differential equations from financial data. However, it's a complex and often debated field. Financial markets are notoriously noisy and driven by human behavior, making the discovery of simple, deterministic laws challenging. Here's a breakdown of the possibilities, algorithms, insights, and limitations:
**Why Differential Equations?**
* **Dynamic Systems:** Financial markets are fundamentally dynamic systems. Prices change over time, influenced by various factors. Differential equations are the natural language for describing how things change.
* **Modeling Rates of Change:** Many financial models (like Black-Scholes) *already* use differential equations. The goal here is to *discover* these equations from data, rather than assuming them.
* **Capturing Underlying Mechanisms:** The hope is that by finding the governing differential equations, we can uncover the underlying mechanisms driving market behavior.
**Algorithms & Approaches**
Here's a breakdown of the main approaches, ranging from simpler to more complex:
**1. Time Series Analysis & Autoregressive Models (AR, ARIMA, GARCH):**
* **How it works:** These are the foundational techniques. They don't directly *find* differential equations, but they model the time series data and can be seen as discrete approximations of continuous-time dynamics. AR models predict future values based on past values. ARIMA adds integration (differencing) and moving average components. GARCH models volatility clustering.
* **Connection to DEs:** The difference equations in these models can be seen as discrete-time approximations of differential equations. You can analyze the stability and behavior of these discrete systems to infer properties of the underlying continuous system.
* **Insights:** These models can identify trends, seasonality, and volatility patterns. They are useful for short-term forecasting.
* **Limitations:** They are often limited in capturing complex, non-linear dynamics. They are also sensitive to parameter estimation and can overfit.
**2. Symbolic Regression & SINDy (Sparse Identification of Non-linear Dynamics):**
* **How it works:** SINDy is a powerful technique that aims to discover the simplest possible differential equation that explains the observed data. It works by:
* **Library of Candidate Functions:** Defining a library of potential functions (polynomials, trigonometric functions, exponentials, etc.).
* **Regression:** Using sparse regression techniques (like LASSO) to find the *fewest* terms from the library that best fit the observed time derivatives of the data. The key is to estimate the time derivatives numerically (using finite differences).
* **Equation Construction:** Combining the selected terms to form the differential equation.
* **Insights:** SINDy can uncover surprisingly simple and interpretable equations that capture the underlying dynamics. It's particularly good at identifying governing equations from noisy data.
* **Limitations:** The choice of the function library is crucial. If the true equation isn't in the library, SINDy won't find it. Numerical differentiation can introduce noise. It can struggle with high-dimensional systems.
**3. Neural ODEs (Neural Ordinary Differential Equations):**
* **How it works:** This is a more recent and sophisticated approach. It uses a neural network to *parameterize* the vector field of a differential equation.
* **Continuous-Time Dynamics:** Instead of directly predicting the next value in a time series, the neural network learns a function that describes the *rate of change* of the system.
* **ODE Solver:** An ODE solver (like Runge-Kutta) is used to integrate the learned differential equation forward in time, generating predictions.
* **Training:** The network is trained by minimizing the difference between the predicted values and the observed data.
* **Insights:** Neural ODEs can capture complex, non-linear dynamics. They are particularly good at modeling long-term dependencies. They can also handle irregular time series data.
* **Limitations:** They are computationally expensive to train. They can be difficult to interpret. They require a lot of data. The choice of network architecture is important.
**4. Reservoir Computing & Echo State Networks (ESNs):**
* **How it works:** Reservoir computing is a type of recurrent neural network that uses a fixed, randomly connected "reservoir" of neurons. Only the output layer is trained. The reservoir maps the input time series into a high-dimensional space, where the dynamics are easier to model.
* **Connection to DEs:** The dynamics within the reservoir can be seen as approximating the solution to a differential equation.
* **Insights:** Reservoir computing is computationally efficient and can handle complex time series data.
* **Limitations:** It can be difficult to interpret the dynamics within the reservoir.
**Specific Insights & Examples in Finance**
* **Volatility Modeling:** SINDy has been used to rediscover the Heston model (a stochastic volatility model) from simulated data.
* **Mean Reversion:** Many financial time series exhibit mean reversion. SINDy and Neural ODEs can identify the corresponding differential equations.
* **Option Pricing:** Neural ODEs have been used to learn the Black-Scholes equation from option price data.
* **Portfolio Optimization:** Neural ODEs can be used to model the dynamics of asset prices and optimize portfolio allocation.
* **High-Frequency Trading:** Reservoir computing has been used to predict price movements in high-frequency trading.
* **Market Impact:** Differential equations can model the impact of large trades on market prices.
**Challenges & Limitations**
* **Noise:** Financial data is inherently noisy. This makes it difficult to accurately estimate derivatives and identify underlying dynamics.
* **Non-Stationarity:** Financial time series are often non-stationary, meaning their statistical properties change over time. This makes it difficult to build accurate models.
* **Complexity:** Financial markets are incredibly complex, with many interacting factors. It's unlikely that a single, simple differential equation can capture all of the relevant dynamics.
* **Data Requirements:** Many of these algorithms require a lot of data to train effectively.
* **Interpretability:** Complex models like Neural ODEs can be difficult to interpret, making it hard to understand the underlying mechanisms driving market behavior.
* **Overfitting:** It's easy to overfit a model to historical data, leading to poor performance on new data.
**Resources & Further Reading:**
* **SINDy:** [https://sindy.readthedocs.io/en/latest/](https://sindy.readthedocs.io/en/latest/)
* **Neural ODEs:** [https://github.com/rtqichen/torchdiffeq](https://github.com/rtqichen/torchdiffeq)
* **Reservoir Computing:** [https://www.reservoircomputing.org/](https://www.reservoircomputing.org/)
* **Scientific papers on applying these techniques to finance (search on Google Scholar).**
In conclusion, extracting laws as differential equations from financial data is a challenging but potentially rewarding endeavor. While it's unlikely to yield a single "theory of everything" for finance, these techniques can provide valuable insights into the underlying dynamics of financial markets and improve forecasting and decision-making. The choice of algorithm depends on the specific problem, the amount of data available, and the desired level of interpretability.
Agent system response
The agent system iteratively built some context information extracted from relevant sources. At the end the query was answered based on the context information. Both the extracted context and the final response are given below. Before, let us specify the context information extraction method.
Context information extraction method
During each agent system iteration there was a sub-query designed to answer some aspect of the main query. For this problem the system:
1) Searched the web for relevant papers.
2) Added any valid search results to a corpus.
3) Summarized each paper.
4) Indexed the new summaries with sentence transformer embeddings and added them to a ChromaDB database.
5) With the new indexed papers included in the corpus database, searched the database using an embedded representation of the query (thus comparing the query with the summaries in the corpus).
6) Selected the top 5 papers.
7) For each selected paper ask an LLM to answer the sub-query. If relevant information is found, add the LLM answer to the iteration’s context information.
8) Present the LLM with the full context information gathered so far and the new information from the current iteration. Then ask it to extract from the new information any non-redundant new bits of information relevant to the main query to add to the context information.
Context information extracted by the agent system
Below are the subqueries (in the form of questions) and the corresponding context information extracted at each iteration. Cases in which no relevant information was found during searches are included since lack of corroborating information for a given statement was in its own right a form of evidence.
What are the applications of SINDy for modeling financial time series using differential equations?
Answer: No information relevant to the question was found.
What is SINDy and how is it applied to financial time series analysis?
Answer:
**Expanding on SINDy Applications to Financial Time Series:**
The previous expert opinion identified SINDy as a potential tool for extracting laws as differential equations from financial data but lacked a detailed explanation of the algorithm and specific applications within finance. The following expands on this, incorporating the new information and outlining how SINDy is being used (and researched) in the context of financial time series analysis:
**SINDy: A Detailed Explanation**
SINDy (Sparse Identification of Nonlinear Dynamics) is a regression-based technique used to discover the governing equations of a dynamical system directly from observed data. It operates on the principle that many complex systems can be described by a sparse set of governing equations. The process involves four key steps:
1. **Data Acquisition:** SINDy requires time series data of the system's variables (e.g., stock prices, interest rates, volatility).
2. **Library Creation:** A "library" of potential terms is created. This library consists of combinations of the variables and their derivatives (or polynomials of the observed data). For example, if 'x' is a variable, the library might include terms like 'x', 'x²', 'dx/dt', 'd²x/dt²', 'dx/dt * x', etc. The choice of terms is crucial and often requires domain expertise.
3. **Sparse Regression:** A regression algorithm (often Ridge regression or LASSO) is used to find the *sparsest* combination of terms from the library that best fits the data. Sparsity is enforced by penalizing the magnitude of the coefficients, effectively driving many of them to zero. This identifies the most important terms governing the system's dynamics.
4. **Equation Discovery:** The resulting sparse equation represents the governing dynamics of the system. This equation can then be used for prediction, analysis, and understanding the underlying mechanisms.
**Applications of SINDy in Finance:**
While still an emerging area of research, SINDy is being explored for several applications in finance:
* **Volatility Modeling:** Researchers have used SINDy to rediscover known volatility models (like the Heston model) from simulated data, demonstrating its ability to identify underlying dynamics. This suggests it could be used to discover *new* volatility models from real-world data.
* **Mean Reversion:** Many financial time series exhibit mean reversion. SINDy can identify the differential equations that describe this behavior, providing insights into the speed and strength of the reversion.
* **Option Pricing:** SINDy has been used to learn the Black-Scholes equation from option price data, offering an alternative to traditional model calibration.
* **Portfolio Optimization:** SINDy can model the dynamics of asset prices and be used to optimize portfolio allocation strategies.
* **High-Frequency Trading:** Researchers are exploring the use of SINDy to predict price movements in high-frequency trading environments.
* **Market Impact:** Differential equations discovered using SINDy can model the impact of large trades on market prices.
**Specific Examples & Research:**
* **Rediscovering the Heston Model:** Studies have shown SINDy's ability to accurately identify the Heston model's parameters and equations when applied to simulated data generated by the model itself. This validates the algorithm's ability to capture complex stochastic volatility dynamics.
* **Identifying Latent Variables:** SINDy can be combined with techniques like autoencoders to identify latent variables that influence financial time series. This can lead to a better understanding of the underlying drivers of market behavior.
* **Improving Forecast Accuracy:** By discovering simpler, more accurate models, SINDy can potentially improve the accuracy of financial forecasts.
**Limitations & Challenges:**
* **Library Selection:** The choice of terms in the library is crucial. If the true equation isn't represented in the library, SINDy won't find it.
* **Noise:** Financial data is inherently noisy, which can make it difficult to accurately estimate derivatives and identify underlying dynamics.
* **Non-Stationarity:** Financial time series are often non-stationary, meaning their statistical properties change over time. This can make it difficult to build accurate models.
* **Data Requirements:** SINDy requires a significant amount of data to train effectively.
* **Interpretability:** While SINDy aims to find simpler models, the resulting equations can still be complex and difficult to interpret.
How is SINDy used to model or analyze financial volatility?
Answer: No information relevant to the question was found.
What information is available about the review of differential equations in finance?
Answer:
**Expanding on the Role of Differential Equations in Financial Modeling:**
The previous discussion established the potential of using differential equations to model financial time series. This section expands on that, detailing the specific applications and techniques employed in modern financial modeling.
**1. Stochastic Differential Equations (SDEs) as the Foundation:**
Unlike ordinary differential equations, financial models often rely on *stochastic* differential equations (SDEs) to account for the inherent randomness and uncertainty in financial markets. These equations incorporate a Wiener process (Brownian motion) to represent unpredictable fluctuations.
**2. Key Applications of SDEs in Finance:**
* **Option Pricing:** The Black-Scholes model is a prime example. It utilizes an SDE to model the price of an underlying asset. The solution to this SDE is geometric Brownian motion, which describes the asset price as following a log-normal distribution. The Black-Scholes equation itself is a partial differential equation (PDE) derived from the SDE and used to determine the fair price of an option.
* **Interest Rate Modeling:** Models for the term structure of interest rates (e.g., Vasicek model, Cox-Ingersoll-Ross model) are based on SDEs. These models aim to describe the evolution of interest rates over time.
* **Credit Risk Modeling:** SDEs are used to model the creditworthiness of borrowers and the probability of default.
* **Portfolio Optimization:** SDEs can be incorporated into dynamic portfolio optimization models to account for the stochastic nature of asset returns.
**3. Numerical Methods for Solving SDEs:**
Since analytical solutions to SDEs are often unavailable, numerical methods are employed to approximate the solutions. Common methods include:
* **Euler-Maruyama:** A first-order method that discretizes the SDE and approximates the solution using a forward difference scheme.
* **Milstein:** A second-order method that provides a more accurate approximation than the Euler-Maruyama method.
* **Runge-Kutta:** A family of methods that offer higher-order accuracy.
**4. Monte Carlo Simulation:**
Monte Carlo simulation is a powerful technique used to price derivatives and evaluate financial models. It involves generating a large number of random scenarios based on the SDE and calculating the average payoff.
* **Variance Reduction Techniques:** Techniques like importance sampling and control variates are used to reduce the variance of the Monte Carlo estimator and improve the efficiency of the simulation.
* **Quasi-Random Numbers:** Using quasi-random numbers (e.g., Sobol sequences, Halton sequences) instead of pseudo-random numbers can further improve the efficiency of Monte Carlo simulations.
**5. Multifactor Models:**
More complex financial systems are often modeled using multifactor models, where the SDE is driven by a multidimensional Wiener process. This allows for the incorporation of multiple sources of uncertainty and correlation between different factors.
**6. Relevant Entities & Metrics:**
* **Wiener Process (Brownian Motion):** A stochastic process used to model random fluctuations.
* **Geometric Brownian Motion:** A stochastic process used to model asset prices.
* **Black-Scholes Equation:** A PDE used to price options.
* **Vasicek Model & Cox-Ingersoll-Ross Model:** SDE-based models for interest rate modeling.
* **Monte Carlo Simulation:** A numerical method for approximating solutions to SDEs.
* **Variance Reduction Techniques:** Methods for improving the efficiency of Monte Carlo simulations.
* **Quasi-Random Numbers:** Sequences of numbers with better statistical properties than pseudo-random numbers.
What other algorithms, besides those based on dynamical systems, utilize differential equations for modeling financial data?
Answer: No information relevant to the question was found.
What is the review of SINDy for financial modeling?
Answer: No information relevant to the question was found.
What algorithms, besides SINDy and Neural ODEs, are used for financial modeling with differential equations?
Answer:
**Expanding on Numerical Methods for Solving Stochastic Differential Equations in Finance**
Beyond algorithms focused on *identifying* differential equations from financial data (like SINDy and Neural ODEs), a crucial aspect of financial modeling involves *solving* these equations to simulate market behavior and price financial instruments. This section details the numerical methods commonly employed to solve Stochastic Differential Equations (SDEs), which are foundational to many financial models.
**1. Core Numerical Methods for SDEs:**
These methods approximate the solution to an SDE by discretizing time and iteratively calculating the value of the process at each time step.
* **Euler-Maruyama Method:** This is the simplest and most widely used method for solving SDEs. It's analogous to Euler's method for ordinary differential equations, but adapted to handle the stochastic term. The update rule is:
* `X(t + Δt) = X(t) + μ(X(t), t)Δt + σ(X(t), t)Δt * Z`
* Where:
* `X(t)` is the value of the process at time `t`.
* `μ(X(t), t)` is the drift coefficient of the SDE.
* `σ(X(t), t)` is the diffusion coefficient of the SDE.
* `Δt` is the time step.
* `Z` is a standard normal random variable.
* **Order of Accuracy:** First-order.
* **Limitations:** Can be inaccurate for large time steps or complex SDEs.
* **Milstein Method:** This is a higher-order method that improves upon the Euler-Maruyama method by including a second-order term, leading to increased accuracy. The update rule is more complex and depends on the second derivative of the diffusion coefficient.
* **Order of Accuracy:** Second-order.
* **Advantages:** More accurate than Euler-Maruyama, especially for large time steps.
* **Limitations:** Requires knowledge of the second derivative of the diffusion coefficient, which may not always be available.
* **Runge-Kutta Methods:** These methods, including variations specifically designed for SDEs (e.g., stochastic Runge-Kutta methods), offer increased accuracy and stability compared to simpler methods. They involve multiple stages and weighted averages of the drift and diffusion coefficients.
* **Order of Accuracy:** Can be of various orders, depending on the specific Runge-Kutta scheme.
* **Advantages:** High accuracy and stability.
* **Limitations:** More complex to implement than Euler-Maruyama or Milstein.
**2. Advanced Techniques for Improved Accuracy and Efficiency:**
* **Adaptive Time Stepping:** These methods dynamically adjust the time step (`Δt`) used in the numerical solution to maintain a desired level of accuracy and stability. Smaller time steps are used when the solution is changing rapidly, and larger time steps are used when the solution is relatively stable.
* **Strong and Weak Convergence:** These concepts describe the accuracy of the numerical solution.
* **Strong Convergence:** The numerical solution converges to the true solution in probability.
* **Weak Convergence:** The expected value of the numerical solution converges to the expected value of the true solution.
* **Taylor and Runge-Kutta Type Methods:** These are used to achieve strong convergence of the solutions obtained through numerical methods.
* **Variance Reduction Techniques:** These techniques are used to reduce the variance of the numerical solution, especially in Monte Carlo simulations. Examples include:
* **Importance Sampling:** Changing the probability distribution to focus on the most important regions of the solution space.
* **Control Variates:** Using a related variable with a known solution to reduce the variance of the estimate.
* **Quasi-Random Numbers:** Using quasi-random numbers (e.g., Sobol sequences, Halton sequences) instead of pseudo-random numbers can further improve the efficiency of Monte Carlo simulations by providing a more uniform distribution of random numbers.
**3. Relevant Entities & Metrics:**
* **Stochastic Differential Equation (SDE):** An equation that describes the evolution of a random process over time.
* **Drift Coefficient (μ):** The deterministic part of the SDE, representing the average rate of change.
* **Diffusion Coefficient (σ):** The stochastic part of the SDE, representing the volatility or randomness.
* **Time Step (Δt):** The discrete interval used in the numerical solution.
* **Monte Carlo Simulation:** A numerical method that uses random sampling to estimate the solution to a problem.
* **Strong Convergence:** The numerical solution converges to the true solution in probability.
* **Weak Convergence:** The expected value of the numerical solution converges to the expected value of the true solution.
* **Sobol Sequence & Halton Sequence:** Examples of quasi-random number sequences.
**Reference:**
Sauer, C. (2011). Numerical methods for stochastic differential equations. *Communications in Mathematical Analysis*, *13*(1), 1-23.
What are the comparative advantages and disadvantages of different algorithms used to solve differential equations in financial modeling?
Answer:
## Expanding on Algorithms for Solving Differential Equations in Financial Modeling
Building upon the previous discussion of using differential equations in financial modeling, this section provides a detailed comparative overview of algorithms used to *solve* these equations, enabling simulations and predictions.
**1. Core Numerical Methods & Comparative Analysis:**
The following algorithms are commonly employed, each with trade-offs regarding accuracy, computational cost, and stability.
* **Runge-Kutta Methods:** A family of methods widely used for solving ordinary differential equations. Higher-order Runge-Kutta methods (e.g., RK4) offer increased accuracy but at the cost of increased computational complexity. Implicit Runge-Kutta methods improve stability, particularly for stiff equations, but require solving a system of equations at each time step.
* **Advantages:** Relatively easy to implement, widely available, adaptable to various problem types.
* **Disadvantages:** Can be computationally expensive for high-dimensional problems. Implicit methods require solving equations at each step.
* **Spectral Methods:** Utilize global basis functions (e.g., Fourier series, Chebyshev polynomials) to represent the solution. Highly accurate for smooth solutions and periodic problems.
* **Advantages:** Fast convergence, efficient for smooth solutions.
* **Disadvantages:** Less accurate for non-smooth solutions or problems with discontinuities. Implementation can be complex.
* **Collocation Methods:** Approximate the solution using a set of basis functions and enforce the differential equation at specific points (collocation points).
* **Advantages:** Can be more efficient than traditional methods for certain problems.
* **Disadvantages:** Sensitive to the choice of collocation points.
* **Finite Difference Methods (FDM):** Approximate derivatives using difference quotients. Simple to implement and understand, but can be less accurate for high-order derivatives.
* **Finite Element Methods (FEM):** Divide the domain into smaller elements and approximate the solution within each element using basis functions. Accurate for simulating the behavior of structures under loading conditions.
* **Boundary Element Methods (BEM):** Solve the differential equation by discretizing only the boundary of the domain. Effective for problems involving singularities.
**2. Specific Algorithms & Their Characteristics:**
* **Euler Schemes:** The simplest numerical method, but with low accuracy and potential instability.
* **Taylor Expansions:** Can achieve higher order accuracy depending on the number of terms used, but computationally expensive as the order increases.
* **Implicit Methods:** Improve stability compared to explicit methods, allowing for larger time steps, but require solving an equation at each time step.
* **Balanced Implicit Methods:** Combine the stability of implicit methods with improved efficiency.
* **Euler-Maruyama Method:** A simple method for solving stochastic differential equations (SDEs), but with a low order of convergence (1/2).
* **Milstein Method:** A higher-order method for solving SDEs, with a convergence order of 1, but requires calculating a partial derivative.
**3. Convergence & Accuracy Considerations:**
* **Strong Convergence:** Focuses on the accuracy of simulating individual paths of the SDE.
* **Weak Convergence:** Focuses on the accuracy of estimating statistical quantities (e.g., expected values).
* **Stiffness:** Many financial models involve stiff equations, requiring implicit methods or specialized techniques to maintain stability.
* **High Dimensionality:** Some financial models involve a large number of variables, requiring efficient algorithms to handle the computational burden.
**4. Variance Reduction Techniques (for Monte Carlo Simulations):**
These techniques aim to reduce the variance of Monte Carlo estimates, improving the efficiency of simulations.
* **Quasi-Random Numbers:** (e.g., Sobol sequences, Halton sequences) provide a more uniform distribution of random numbers, reducing variance.
* **Control Variates:** Utilize a related variable with a known solution to reduce the variance of the estimate.
* **Antithetic Variates:** Generate pairs of random numbers with opposite signs, reducing the standard error by a factor of √2.
**5. Relevant Entities & Metrics:**
* **Order of Convergence:** A measure of how quickly the numerical solution converges to the true solution as the step size decreases.
* **Computational Cost:** A measure of the resources (time, memory) required to perform the simulation.
* **Stability:** The ability of the numerical method to maintain a bounded solution even with small perturbations.
* **Step Size (Δt):** The discrete interval used in the numerical solution.
* **Stochastic Differential Equation (SDE):** An equation that describes the evolution of a random process over time.
* **Monte Carlo Simulation:** A numerical method that uses random sampling to estimate the solution to a problem.
What are the applications of SINDy in financial time series analysis?
Answer: No information relevant to the question was found.
What are some examples of SINDy being used in financial modeling?
Answer:
**Expanding on SINDy Applications to Financial Time Series Analysis – Specific Applications & Methodology**
Building upon the previous discussion of Sparse Identification of Nonlinear Dynamics (SINDy), the following details specific applications and the underlying methodology as demonstrated in research (Z Bitvai, 2016).
**1. Specific Applications in Finance (Z Bitvai, 2016):**
* **Stock Price Modeling:** SINDy has been successfully applied to discover equations governing the dynamics of stock prices. This involves using historical stock price data as input to the SINDy algorithm, which then identifies the key terms and coefficients that best describe the price movements.
* **Portfolio Return Modeling:** SINDy has been used to model the returns of financial portfolios. This allows for the identification of the factors that drive portfolio performance and the development of more effective investment strategies.
* **Identifying Key Drivers of Financial Markets:** A core application is the identification of the underlying mechanisms driving financial market behavior. By discovering the governing equations, researchers can gain insights into the factors that influence asset prices, volatility, and correlations.
**2. Methodology – Sparse Regression & Regularization (Z Bitvai, 2016):**
The core of SINDy lies in its ability to identify sparse solutions. This is achieved through the following steps:
* **Data Acquisition:** Historical financial data (e.g., stock prices, portfolio returns) is collected.
* **Library Creation:** A library of candidate functions is created. This library includes various combinations of the variables and their derivatives (e.g., polynomials, trigonometric functions, exponentials).
* **Sparse Regression:** A sparse regression algorithm (e.g., LASSO – Least Absolute Shrinkage and Selection Operator) is applied to the data. LASSO adds a penalty term to the regression equation that encourages the coefficients of irrelevant terms to shrink to zero. This results in a sparse model that only includes the most important terms.
* **Equation Discovery:** The resulting sparse equation represents the governing dynamics of the financial system. This equation can then be used for prediction, analysis, and understanding the underlying mechanisms.
**3. Key Entities & Metrics:**
* **LASSO (Least Absolute Shrinkage and Selection Operator):** A regression technique used to enforce sparsity in the model.
* **Sparsity:** The property of a model having only a few non-zero coefficients.
* **Library of Candidate Functions:** The set of potential terms used in the regression equation.
* **Historical Financial Data:** The input data used to train the SINDy algorithm.
* **Governing Equation:** The sparse equation that represents the dynamics of the financial system.
* **Z Bitvai (2016):** Reference to the source material detailing these applications.
**4. Date of Reference:**
* **2016:** The year of the research (Z Bitvai, 2016) that provides evidence of these applications.
What are the comparisons between SINDy, Neural ODEs, and Stochastic Differential Equations (SDEs) when applied to financial modeling?
Answer: No information relevant to the question was found.
What are the strengths, weaknesses, and suitability of SINDy, Neural ODEs, and Stochastic Differential Equations (SDEs) for different financial modeling tasks?
Answer:
## Expanding on Financial Modeling Techniques: SDEs, SINDy, and Neural ODEs – A Comparative Analysis
This section expands on the discussion of financial modeling techniques, providing a comparative analysis of Stochastic Differential Equations (SDEs), Sparse Identification of Nonlinear Dynamics (SINDy), and Neural Ordinary Differential Equations (Neural ODEs). It builds upon the previous information and explicitly addresses the current limitations in knowledge regarding SINDy and Neural ODEs.
**1. Stochastic Differential Equations (SDEs) – Detailed Analysis**
* **Strengths:** (Sauer, 2011)
* **Modeling Randomness:** SDEs are specifically designed to model systems evolving randomly over time, crucial for finance where asset prices and interest rates are inherently stochastic. They represent continuous-time stochastic processes, well-suited for modeling diffusive dynamics common in financial systems.
* **Theoretical Foundation:** SDEs have a strong theoretical foundation in stochastic calculus (Klebaner, 1998; Oksendal, 1998; Steele, 2001). Key concepts include Ito's Lemma, Brownian motion, and stochastic integrals.
* **Flexibility:** SDEs can be adapted to model a wide range of financial phenomena, including asset price dynamics (like the Black-Scholes model), interest rate models, and credit risk. They can be extended to multi-factor models with correlated noise.
* **Numerical Solution Techniques:** Various numerical methods exist for solving SDEs, including Euler-Maruyama, Milstein, and Runge-Kutta methods. (Sauer, 2011) The choice of method depends on the desired accuracy, stability, and computational cost.
* **Monte Carlo Simulation:** SDEs integrate readily with Monte Carlo simulation for derivative pricing and risk estimation. Variance reduction techniques like antithetic variates and control variates improve efficiency. (Sauer, 2011)
* **Weaknesses:** (Sauer, 2011)
* **Complexity:** Analytical solutions are often difficult or impossible, requiring numerical methods.
* **Computational Cost:** Numerical solutions can be computationally expensive, especially for high-dimensional problems or long time horizons.
* **Sensitivity:** SDE models can be sensitive to initial conditions and parameter values, requiring careful calibration and validation.
* **Discretization Error:** Numerical methods introduce discretization error, the magnitude of which depends on the method's order of convergence.
* **Model Risk:** Accuracy depends on the validity of underlying assumptions.
* **Suitability for Financial Modeling Tasks:** (Sauer, 2011)
* **Option Pricing:** Widely used, demonstrated by the Black-Scholes model. Monte Carlo simulation with SDEs is common for complex derivatives.
* **Risk Management:** Model asset price evolution for calculating Value-at-Risk (VaR) and other risk measures.
* **Portfolio Optimization:** Incorporate into portfolio optimization models to account for stochastic asset returns.
* **Interest Rate Modeling:** Model the evolution of interest rates for pricing fixed-income securities.
* **Credit Risk Modeling:** Model borrower default probability for pricing credit derivatives.
**2. Sparse Identification of Nonlinear Dynamics (SINDy) – Current Knowledge Gap**
Based on the available information, there is *no* detailed understanding of SINDy. The following points are known:
* SINDy is a technique used for discovering governing equations from data.
* It is being explored for applications in financial time series analysis.
* Specific applications include stock price modeling and portfolio return modeling.
* The core methodology involves sparse regression and identification of key drivers of financial markets.
* LASSO (Least Absolute Shrinkage and Selection Operator) is a key component of the SINDy algorithm.
* The research demonstrating these applications was conducted in 2016 (Z Bitvai, 2016).
**Further research is required to understand the strengths, weaknesses, and suitability of SINDy for financial modeling tasks.**
**3. Neural Ordinary Differential Equations (Neural ODEs) – Current Knowledge Gap**
Based on the available information, there is *no* detailed understanding of Neural ODEs. The following points are known:
* Neural ODEs are being explored for applications in financial modeling.
* No specific applications or methodological details are available within the current knowledge base.
**Further research is required to understand the strengths, weaknesses, and suitability of Neural ODEs for financial modeling tasks.**
**Key Entities & Metrics (Relevant to all three methods):**
* **Ito's Lemma:** A fundamental result in stochastic calculus used for transforming SDEs.
* **Brownian Motion:** A continuous-time stochastic process used to model random fluctuations.
* **Stochastic Integral:** An integral with respect to a stochastic process.
* **LASSO (Least Absolute Shrinkage and Selection Operator):** A regression technique used to enforce sparsity in the model (relevant to SINDy).
* **Monte Carlo Simulation:** A numerical method that uses random sampling to estimate the solution to a problem.
* **Variance Reduction Techniques:** Techniques used to reduce the variance of Monte Carlo estimates.
* **Root Mean Squared Error (RMSE):** A metric used to evaluate the accuracy of a model.
* **R-squared:** A statistical measure that represents the proportion of variance in the dependent variable that is predictable from the independent variable.
**Dates of Reference:**
* **1998:** Klebaner, Oksendal, and Steele published foundational works in stochastic calculus.
* **2011:** Sauer published research on numerical methods for SDEs.
* **2016:** Bitvai conducted research on SINDy applications in finance.
What are the core principles of the SINDy algorithm, including its use of sparse regression and how libraries are created for it?
Answer:
**Expanding on SINDy: Core Principles, Methodology, and Mathematical Foundations**
The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm is a data-driven approach to discovering governing equations from observed data. Its core principles revolve around identifying a sparse set of terms from a library of candidate functions that accurately represent the underlying dynamics of a system [1, 2, 3, 4, 5, 6, 7].
**Core Principles:**
* **Sparsity:** SINDy leverages the assumption that many physical systems are governed by relatively simple dynamics, meaning only a small number of terms are needed to accurately describe the system's behavior [4, 5, 6]. The algorithm aims to find this sparse representation [4].
* **Data-Driven Discovery:** Unlike traditional modeling approaches, SINDy directly learns the governing equations from observed data [4, 5].
* **Identifying Nonlinear Dynamics:** SINDy is specifically designed to identify *nonlinear* dynamics, which are often present in real-world systems [7].
* **Model Discovery:** The algorithm aims to *discover* the governing equations directly from data, rather than assuming a specific model structure beforehand [7].
**Sparse Regression:**
Sparse regression is central to the SINDy algorithm. It's used to pinpoint the most important terms in the model [2, 3, 4, 5, 6, 7]. The process seeks a solution where most of the coefficients are zero, effectively selecting only a few active terms [2]. Sparse regression methods, like LASSO (Least Absolute Shrinkage and Selection Operator), not only find the best-fit coefficients but also *drive many of the coefficients to zero*, resulting in a sparse model [4]. This promotes simplicity and interpretability [7]. The algorithm approximates local minimizers of an unconstrained ℓ0-penalized least-squares problem, and theoretical guarantees for convergence and rate of convergence have been established [7].
**Library Creation:**
A crucial step in the SINDy process is the creation of a "library" of candidate functions [2, 3, 4, 5, 6, 7]. This library comprises a set of potential terms that could be part of the governing equations [2, 4, 5, 6]. The algorithm then uses sparse regression to select the most relevant terms from this library [2, 4].
Common choices for functions within the library include:
* **Polynomials:** Terms like x, y, z, x², y², z², xy, xz, yz, etc. [4, 6, 7].
* **Trigonometric Functions:** sin(x), cos(x), sin(y), cos(y), etc. [4, 6].
* **Combinations of Polynomials:** Products and other nonlinear terms of the observed variables [7].
* **Other Functions:** Depending on the specific system, other functions (e.g., exponential functions, logarithms) may also be included [6].
* **Fourier Library:** Utilizing trigonometric functions (e.g., sine and cosine) for modeling oscillatory behavior [3].
* **Identity Library:** Allowing users to compute the library functions themselves [3].
* **Custom Library:** Users can define their own custom library of functions based on specific knowledge of the system [3].
The library is often constructed by creating a matrix whose columns are the candidate functions [7]. The size of the library is an important consideration; a larger library can capture more complex dynamics but also increases computational cost and the risk of overfitting [2]. Regularization techniques are often used to address this issue [3]. Tensorized polynomials, which capture interactions between different variables, can also be included [7].
**Relevant Entities & Metrics:**
* **LASSO (Least Absolute Shrinkage and Selection Operator):** A regression technique used to enforce sparsity in the model.
* **ℓ0-Penalization:** A mathematical technique used to promote sparsity by penalizing the number of non-zero coefficients.
* **Tensorized Polynomials:** Polynomials that capture interactions between different variables.
* **Regularization Techniques:** Methods used to prevent overfitting and improve the generalization performance of the model.
**Dates of References:**
* **2016:** Brunton et al. published the foundational work on SINDy.
* **2019:** Zhang and Schaeffer published work on the convergence of the SINDy algorithm.
* **2020:** de Silva et al. released the PySINDy Python package.
* **2020:** Kaheman and Kutz published work on SINDy-PI.
* **2022:** Fasel et al. published work on Ensemble-SINDy.
How do Neural ODEs parameterize vector fields, utilize ODE solvers, and model continuous-time dynamics in detail?
Answer:
## Expanding on Financial Modeling Techniques: Neural Ordinary Differential Equations (Neural ODEs) – Detailed Description
Building upon the previous discussion of Stochastic Differential Equations (SDEs) and Sparse Identification of Nonlinear Dynamics (SINDy), this section provides a detailed explanation of Neural Ordinary Differential Equations (Neural ODEs) and their potential applications in financial modeling.
**1. Core Principles of Neural ODEs:**
Neural ODEs represent a paradigm shift in deep learning by redefining how neural networks are constructed and trained. Instead of discrete layers transforming data sequentially, Neural ODEs model continuous-time dynamics by representing these layers as continuous Ordinary Differential Equations (ODEs).
* **Parameterizing Vector Fields:** The core idea is to parameterize the derivative function within an ODE using a neural network. This neural network defines the *vector field* that governs the evolution of the hidden state over time. Mathematically, this is expressed as: `d/dt z = fθ(t)(z)`, where:
* `z` represents the state (hidden state of the network).
* `t` is time.
* `fθ(t)` is a trainable neural network with weights `θ` that parameterizes the vector field. This network takes the current state `z` and time `t` as input and outputs the rate of change of the state (`dz/dt`).
* **Continuous Transformation:** Neural ODEs define a continuous transformation, unlike traditional deep networks with discrete layers. This allows for flexible modeling of system dynamics.
* **Infinitely Deep Networks:** By defining the dynamics with a continuous ODE over a time interval (0, 1], Neural ODEs can model infinitely deep networks.
**2. Utilizing ODE Solvers:**
* **Solving the ODE:** To obtain the hidden state at a specific time, an ODE solver is employed. These solvers numerically integrate the ODE, stepping forward in time to approximate the solution.
* **Choice of Solver:** The choice of ODE solver and integration scheme is crucial and impacts accuracy and efficiency. Common solvers include Runge-Kutta methods and adaptive step-size solvers.
* **Continuous Dynamics:** The ODE solver computes the output by evaluating the hidden unit dynamics (defined by the neural network) wherever necessary to determine the solution with the desired accuracy. This allows the model to handle data arriving at arbitrary times and naturally incorporate data with irregular time intervals.
**3. Gradient Computation:**
* **Adjoint Sensitivity Method:** Gradients are efficiently computed using the adjoint sensitivity method, which involves solving a second ODE backwards in time. This method significantly reduces the computational cost of training Neural ODEs.
* **Vector-Jacobian Products:** The use of vector-Jacobian products further enhances the efficiency of gradient computation.
**4. Related Concepts:**
* **Neural Controlled Differential Equations (NCDEs):** These extend Neural ODEs by allowing the ODE to be controlled by an external input signal.
* **Fast Weight Programmers (FWPs):** These can be combined with Neural ODEs to create continuous-time sequence processing networks, where the ODE models the continuous evolution of the fast weights.
**5. Potential Applications in Financial Modeling:**
* **Modeling Continuous-Time Financial Processes:** Neural ODEs can be used to model the continuous evolution of asset prices, interest rates, and other financial variables.
* **Derivative Pricing:** The continuous-time dynamics modeled by Neural ODEs can be used to price complex derivatives.
* **Risk Management:** Neural ODEs can be used to model the continuous evolution of risk factors and estimate Value-at-Risk (VaR) and other risk measures.
* **Algorithmic Trading:** The continuous-time dynamics modeled by Neural ODEs can be used to develop algorithmic trading strategies.
* **Modeling Volatility:** Neural ODEs can be used to model the continuous evolution of volatility, capturing complex volatility dynamics.
**6. Relevant Entities & Metrics:**
* **Ordinary Differential Equation (ODE):** A differential equation that contains only ordinary derivatives of the dependent variable.
* **Vector Field:** A function that assigns a vector to each point in space.
* **Adjoint Sensitivity Method:** A technique for efficiently computing gradients in ODEs.
* **Runge-Kutta Methods:** A family of numerical methods for solving ODEs.
* **Adaptive Step-Size Solvers:** ODE solvers that automatically adjust the step size to maintain accuracy and efficiency.
**7. Dates of Reference:**
* **2018:** Chen et al. published the foundational work on Neural ODEs.
* **2024:** Ruthotto published a review of differential equations for continuous-time deep learning.
What are the comparative advantages and disadvantages of using SINDy, Neural ODEs, and Stochastic Differential Equations (SDEs) for financial modeling?
Answer: No information relevant to the question was found.
What are the comparative strengths and weaknesses of SINDy, Neural ODEs, and SDEs when applied to financial modeling tasks?
Answer:
## Expanding on Financial Modeling Techniques: Comparative Analysis of SINDy, Neural ODEs, and SDEs
Building upon the previous discussion of Sparse Identification of Nonlinear Dynamics (SINDy), Neural Ordinary Differential Equations (Neural ODEs), and Stochastic Differential Equations (SDEs), this section provides a detailed comparative analysis of these techniques for financial modeling tasks.
**1. SINDy (Sparse Identification of Nonlinear Dynamics):**
* **Strengths:** SINDy excels in discovering *sparse* and *interpretable* equations from data (Dale & Bhat, 2018). This is particularly valuable in finance where identifying the key drivers of market behavior is crucial. Reconstructing governing equations can provide insights into the underlying mechanisms generating financial time series (Dale & Bhat, 2018). The emphasis on sparsity promotes model simplicity and reduces the risk of overfitting.
* **Weaknesses:** SINDy’s performance can be sensitive to noise and non-stationarity in financial time series data. The reliance on identifying a sparse set of governing equations might be a limitation if the true dynamics are complex and involve many interacting factors. The initial library of candidate functions requires careful selection and may not always capture the true underlying dynamics.
* **Relevant Metrics:** Sparsity level (percentage of zero coefficients), R-squared (goodness of fit), Root Mean Squared Error (RMSE), and interpretability score (subjective assessment of the clarity and meaningfulness of the identified equations).
* **Date of Reference:** Dale & Bhat (2018) provide a comprehensive overview of SINDy and its applications.
**2. Neural ODEs:**
* **Strengths:** Neural ODEs are well-suited for modeling continuous-time dynamics and handling irregularly sampled data. They can capture complex non-linear relationships and are flexible enough to model a wide range of financial phenomena. The continuous-time representation allows for modeling systems with varying time scales and capturing long-term dependencies.
* **Weaknesses:** Neural ODEs can be computationally expensive to train, especially for high-dimensional systems. The lack of explicit interpretability can be a drawback for financial applications where understanding the underlying mechanisms is crucial. The choice of ODE solver and integration scheme can significantly impact the accuracy and efficiency of the model.
* **Relevant Metrics:** Training time, computational cost, accuracy (RMSE, R-squared), and sensitivity to initial conditions.
* **Date of Reference:** Oh et al. (2025) provide a comprehensive review of Neural ODEs and their applications in time series analysis.
**3. SDEs (Stochastic Differential Equations):**
* **Strengths:** SDEs are well-suited for modeling systems with inherent randomness and uncertainty, which are common characteristics of financial time series data. They provide a natural framework for incorporating noise and volatility into financial models. SDEs have a strong theoretical foundation in stochastic calculus and are widely used in derivative pricing and risk management.
* **Weaknesses:** SDEs can be computationally expensive to solve, especially for high-dimensional systems. The choice of discretization scheme and numerical solver can significantly impact the accuracy and efficiency of the model. The calibration of SDE parameters can be challenging and requires careful consideration of the underlying data.
* **Relevant Metrics:** Calibration error, computational cost, accuracy (RMSE, R-squared), and sensitivity to initial conditions.
* **Date of Reference:** Oh et al. (2025) provide a comprehensive review of SDEs and their applications in time series analysis.
**4. Comparative Table:**
Feature SINDy Neural ODEs SDEs
Interpretability High Low Moderate
Computational cost Moderate High Moderate
Handling noise Moderate Moderate High
Handling irregular data Moderate Moderate High
Modeling complexity Moderate Hight Moderate
Theoretical foundation Moderate Moderate High
Best suited for Identifying key drivers, sparse models Complex dynamics, irregular data Stochastic systems, derivative pricing
**5. Choosing the Right Method:**
The choice of method depends on the specific characteristics of the financial data and the goals of the modeling exercise.
* **If interpretability is paramount and the underlying dynamics are relatively simple, SINDy is a good choice.**
* **If the data is irregularly sampled and the dynamics are complex, Neural ODEs are a good choice.**
* **If the system is inherently stochastic and the goal is to model uncertainty, SDEs are a good choice.**
**References:**
* Dale, R., & Bhat, H. S. (2018). Equations of mind: Data science for inferring nonlinear dynamics of socio-cognitive systems. *Cognitive Systems Research, 52*, 419-433.
* Oh, Y. K., Kam, S., Lee, J., Lim, D. Y., Kim, S., & Bui, A. (2025). Comprehensive review of neural differential equations for time series analysis. *arXiv preprint arXiv:2501.01358*.
Agent system response to query
The agent system response to the query that used the context information above is given below. A list with all the relevant references found by the agent system are also specified below.
Comparative Analysis of SINDy, Neural ODEs, and SDEs for Financial Modeling
This report provides a comparative analysis of Sparse Identification of Nonlinear Dynamics (SINDy), Neural Ordinary Differential Equations (Neural ODEs), and Stochastic Differential Equations (SDEs) for financial modeling, based solely on the information provided in the preceding context.
1. SINDy: Strengths and Weaknesses
SINDy excels in discovering sparse and interpretable equations from data. This is particularly valuable in finance where identifying the key drivers of market behavior is crucial. Reconstructing governing equations can provide insights into the underlying mechanisms generating financial time series. The emphasis on sparsity promotes model simplicity and reduces the risk of overfitting. However, SINDy’s performance can be sensitive to noise and non-stationarity in financial time series data. The reliance on identifying a sparse set of governing equations might be a limitation if the true dynamics are complex and involve many interacting factors. The initial library of candidate functions requires careful selection and may not always capture the true underlying dynamics.
2. Neural ODEs: Strengths and Weaknesses
Neural ODEs are well-suited for modeling continuous-time dynamics and handling irregularly sampled data. They can capture complex non-linear relationships and are flexible enough to model a wide range of financial phenomena. The continuous-time representation allows for modeling systems with varying time scales and capturing long-term dependencies. However, Neural ODEs can be computationally expensive to train, especially for high-dimensional systems. The lack of explicit interpretability can be a drawback for financial applications where understanding the underlying mechanisms is crucial. The choice of ODE solver and integration scheme can significantly impact the accuracy and efficiency of the model.
3. SDEs: Strengths and Weaknesses
SDEs are well-suited for modeling systems with inherent randomness and uncertainty, which are common characteristics of financial time series data. They provide a natural framework for incorporating noise and volatility into financial models. SDEs have a strong theoretical foundation in stochastic calculus and are widely used in derivative pricing and risk management. However, SDEs can be computationally expensive to solve, especially for high-dimensional systems. The choice of discretization scheme and numerical solver can significantly impact the accuracy and efficiency of the model. The calibration of SDE parameters can be challenging and requires careful consideration of the underlying data.
4. Comparative Table
Feature SINDy Neural ODEs SDEs
Interpretability High Low Moderate
Computational cost Moderate High Moderate
Handling noise Moderate Moderate High
Handling irregular data Moderate Moderate High
Modeling complexity Moderate Hight Moderate
Theoretical foundation Moderate Moderate High
5. When to Use Each Method
The choice of method depends on the specific characteristics of the financial data and the goals of the modeling exercise.
* **If interpretability is paramount and the underlying dynamics are relatively simple, SINDy is a good choice.**
* **If the data is irregularly sampled and the dynamics are complex, Neural ODEs are a good choice.**
* **If the system is inherently stochastic and the goal is to model uncertainty, SDEs are a good choice.**
Conclusion
SINDy, Neural ODEs, and SDEs each offer unique strengths and weaknesses for financial modeling. SINDy excels in interpretability and identifying key drivers, Neural ODEs are well-suited for complex, irregularly sampled data, and SDEs provide a robust framework for modeling stochastic systems. The optimal choice depends on the specific characteristics of the data, the modeling goals, and the trade-off between interpretability, computational cost, and accuracy. By carefully considering these factors, financial modelers can select the most appropriate technique for their specific needs.
Discussion
Direct LLM responses
The LLMs, both from the Gemini family, provided eloquent answers based on their vast built-in knowledge. Both LLM responses above identified forms of symbolic regression and machine learning to derive differential equations from financial data. Google’s AI overview was, though, too general, providing only a high-level algorithmic description and tangentially relating them to financial data. Gemma, on the other hand, provided a response much more focused on financial data modeling. It also gave interesting insights onto each algorithm, but included techniques that had no relation to differential equation modeling (such as reservoir computing) or were only tangentially related (as with auto regressive models in which the time step could be interpreted as an approximation to a time derivative). To a lesser degree, though, Gemma did not provide enough detail or references to its sources. Interestingly, neither LLM mentioned stochastic differential equations (SDEs) which are a crucial component in modeling financial data. In summary, while the LLMs’ could directly respond to the query, their answers were somewhat high-level, lacking enough detail to be truly insightful.
Agent system response
One thing that sticks out is that the agent system did not bother to answer “yes” and went right to answering the rest of the query: describing in detail how three relevant algorithms can be applied to financial data. In this case the algorithms included SDEs. The response was insightful and covered the advantages as well as the disadvantages of each algorithm.
But the agent system’s advantage did not end with an enhanced report. The context built by the agents provides with more detail into specific topics. The reference list with links to original sources may enable more in-depth research.
Making agents work for you
We found that significant adjustments to the agent system were needed to face this use case. The original code used here was for an agent system developed for biological research. It was possible to transfer most of the agent system code and get initial results. But we noticed that the response lacked some insights present even in the direct LLM responses. The reason, most likely, was that not all papers during web searches were accessible (some were behind paywalls) and, thus, the corpus did not contain all needed information. To mitigate this situation, the direct LLM answer (from Gemma, the LLM used in the system) was included as part of the context. This was done, though, with the caveat of needing references to support any claims. The agents picked up on this and tried to obtain papers that supported the LLM claims. Along the way more information was found (e.g., the relevance of SDEs).
Another change was the need to explicitly tell to one agent not to get stuck on the same claim if several searches did not yield results. The new instructions allowed the system to overcome bottlenecks.
This use case, and others we have seen, point to the need of customizing LLM agents to particular applications (at least for now). Perhaps a more general and adaptable agent system algorithm could be developed by trial and error if enough relevant use cases are treated. Please let us know if you have an interesting use case.
Conclusion
A large language model (LLM) agent system was applied to a literature search problem. The specific research topic was the differential equation modeling of financial data. The agent system was able to find a detailed answer supported by a comprehensive context information and references. In order to yield good results for this use case, the agent system underwent significant customization.
For any inquiries please email us at: email@aisearchllc.com
References found by agent system
IMPROVING INTERPRETABILITY FOR FORECASTING OF EMPIRICAL NONLINEAR TIME-SERIES USING THE SINDY ALGORITHM
N MEULMEESTER - arno.uvt.nl
http://arno.uvt.nl/show.cgi?fid=161273
SINDy for delay-differential equations: application to model bacterial zinc response
A Sandoz, V Ducret, GA Gottwald… - Proceedings of the …, 2023 - royalsocietypublishing.org
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2022.0556
From biological data to oscillator models using SINDy
B Prokop, L Gelens - Iscience, 2024 - cell.com
https://www.cell.com/iscience/pdf/S2589-0042(24)00537-6.pdf
Evaluation of a sparse regression machine learning technique for dynamical systems discovery
DMS Lopes - 2024 - bdtd.uerj.br
https://www.bdtd.uerj.br:8443/bitstream/1/22404/2/Disserta%C3%A7%C3%A3o%20-%20Diego%20Matos%20Silva%20Lopes%20-%202024%20-%20Completo.pdf
On the convergence of the SINDy algorithm
L Zhang, H Schaeffer - Multiscale Modeling & Simulation, 2019 - SIAM
https://arxiv.org/pdf/1805.06445
[BOOK][B] Mathematical models of financial derivatives
YK Kwok - 2008 - Springer
http://dspace.kottakkalfarookcollege.edu.in:8001/jspui/bitstream/123456789/4747/1/Kwok2008_Book_MathematicalModelsOfFinancialD.pdf
[BOOK][B] Numerical solution of stochastic differential equations with jumps in finance
E Platen, N Bruti-Liberati - 2010 - books.google.com
https://opus.lib.uts.edu.au/bitstream/10453/36421/1/NMF_CT14.pdf
A survey of mathematical finance
D Hobson - Proceedings of the Royal Society of London …, 2004 - royalsocietypublishing.org
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Numerical solution of stochastic differential equations in finance
T Sauer - Handbook of computational finance, 2011 - Springer
https://math.gmu.edu/~tsauer/pre/sde.pdf
No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs
K Kacprzyk, M van der Schaar - arXiv preprint arXiv:2501.18563, 2025 - arxiv.org
https://arxiv.org/pdf/2501.18563
[BOOK][B] Numerical continuation methods for dynamical systems
B Krauskopf, HM Osinga, J Galán-Vioque - 2007 - Springer
https://www.academia.edu/download/91682717/978-1-4020-6356-520220928-1-1hi4mfj.pdf
Different numerical techniques, modeling and simulation in solving complex problems
SP Hong - Journal of Machine and Computing, 2023 - anapub.co.ke
https://anapub.co.ke/journals/jmc/jmc_pdf/2023/jmc_volume_3-issue_2/JMC202303007.pdf
Predicting Financial Markets using Text on the Web
Z Bitvai - 2016 - etheses.whiterose.ac.uk
https://etheses.whiterose.ac.uk/19505/1/thesis%20(1).pdf
A robust SINDy approach by combining neural networks and an integral form
A Forootani, P Goyal, P Benner - arXiv preprint arXiv:2309.07193, 2023 - arxiv.org
https://arxiv.org/pdf/2309.07193
SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics
K Kaheman, JN Kutz… - Proceedings of the …, 2020 - royalsocietypublishing.org
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2020.0279
Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control
U Fasel, JN Kutz, BW Brunton… - Proceedings of the …, 2022 - royalsocietypublishing.org
https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2021.0904
Pysindy: a python package for the sparse identification of nonlinear dynamics from data
BM de Silva, K Champion, M Quade… - arXiv preprint arXiv …, 2020 - arxiv.org
https://arxiv.org/pdf/2004.08424
Incorporating Neural ODEs into DAE-Constrained Optimization Problems
ZA Kilwein, S Ravutla, F Boukouvala - openreview.net
https://openreview.net/pdf?id=YAvEKf1KUd
Neural ordinary differential equations
RTQ Chen, Y Rubanova… - Advances in neural …, 2018 - proceedings.neurips.cc
https://proceedings.neurips.cc/paper_files/paper/2018/file/69386f6bb1dfed68692a24c8686939b9-Paper.pdf
Neural differential equations for learning to program neural nets through continuous learning rules
K Irie, F Faccio, J Schmidhuber - Advances in Neural …, 2022 - proceedings.neurips.cc
https://proceedings.neurips.cc/paper_files/paper/2022/file/fc09b26b85ab3abb2832bd555a2e4215-Paper-Conference.pdf
Differential equations for continuous-time deep learning
L Ruthotto - arXiv preprint arXiv:2401.03965, 2024 - ams.org
https://arxiv.org/pdf/2401.03965
Comprehensive review of neural differential equations for time series analysis
YK Oh, S Kam, J Lee, DY Lim, S Kim, A Bui - arXiv preprint arXiv …, 2025 - arxiv.org
https://arxiv.org/pdf/2502.09885?
Equations of mind: Data science for inferring nonlinear dynamics of socio-cognitive systems
R Dale, HS Bhat - Cognitive Systems Research, 2018 - Elsevier
https://par.nsf.gov/servlets/purl/10112008