1. Introduction

The theoretical evolution in the field of financial engineering has always kept pace with market complexity through iterative upgrades. The traditional stochastic model system, represented by the Black Scholes model, has pioneered a mathematical analysis paradigm for asset pricing. However, its linear assumption framework has shown fundamental limitations in characterizing nonlinear financial phenomena such as jump risk and volatility surface variability. The current financial innovation presents a multidimensional breakthrough trend: the path dependence of structured products such as multi asset production rights has given rise to difficulties in solving high-dimensional state spaces; The contagion effect of corporate credit default requires risk modeling to have real-time dynamic response capabilities; The instantaneous switching of market microstructure in high-frequency trading scenarios poses sub second feedback requirements for prediction systems. These challenges collectively point to the theoretical boundaries of traditional mathematical financial methods, requiring the establishment of a new analytical framework with higher dimensional adaptability and computational efficiency.

The innovation of machine learning technology provides interdisciplinary solutions to overcome the above bottlenecks. The innovative practice of neural stochastic differential equations embeds deep learning networks into stochastic process modeling, replacing traditional Monte Carlo simulations with nonlinear mapping of feature space, achieving a computational efficiency leap of over 80% while maintaining pricing accuracy. The deep reinforcement learning framework effectively resolves the curse of dimensionality in high-dimensional stochastic optimal control by constructing a dynamic interaction mechanism between intelligent agents and financial environments, opening a technical path for dynamic combinatorial optimization that combines theoretical rigor and engineering feasibility. It is worth noting that these methods are not a simple replacement for traditional theories, but rather form a interpretable hybrid modeling system by integrating the core mathematical architecture of stochastic differential equations, Markov processes, and optimal control theory.

This article constructs a four-dimensional analysis framework of "theory algorithm application challenge", systematically deconstructing the innovative mechanism of interdisciplinary methods in financial engineering. The study first establishes the theoretical convergence of neural differential equations in asset price dynamic modeling, then analyzes the strategy generation mechanism of deep reinforcement learning in dynamic risk hedging, and finally verifies the effective boundary of the methodology through a dual case study of high-frequency trading signal recognition and credit default chain reaction prediction. Research has found that machine learning empowered stochastic models not only significantly expand the modeling dimensions of traditional financial mathematics, but also reconstruct the time response paradigm of risk management through real-time data assimilation mechanisms. The study also reveals the regulatory penetration challenges that may arise from the black box transformation of algorithms, providing important directions for theoretical verification and collaborative development of computational ethics for future research.

2. Basic Theories and Methods

2.1. Financial Modeling of Stochastic Differential Equations

The geometric Brownian motion model based on the Ito process (Equation 1) is the cornerstone of dynamic modeling of financial assets, and its mathematical form is:

\( d{S_{t}}=μ{S_{t}}dt+σ{S_{t}}d{W_{t}}\ \ \ (1) \)

However, commodity prices are often driven by unexpected events (such as policy adjustments or supply chain disruptions), and a jump diffusion model (Equation 2) needs to be introduced for extension:

\( d{S_{t}}= μ{S_{t}}+σ{S_{t}}d{W_{t}}+{J_{t}}d{N_{t}}\ \ \ (2) \)

Among them, JtJt is the jump amplitude, and NtNt is the Poisson process of intensity lambda. The multi factor jump model proposed by Duffie et al. reduced the mean square error of price predictions by 32% in empirical energy markets, validating the practicality of the framework [1].

2.2. Markov Process and Dynamic Prediction

In terms of numerical methods, Monte Carlo simulation has long dominated the field of option pricing, but its computational cost increases exponentially in high-dimensional scenarios. The neural stochastic differential equation solver developed by Han et al. approximates a high-dimensional solution space through a deep network and successfully compresses the pricing time of 100-dimensional American options from 12 hours to 1.5 hours, with an error rate stable within 1.5% [2].

Hidden Markov models (HMMs) capture market system transitions through implicit state sequences. Broadie et al. proposed a regression-based risk estimation framework that integrates Markov state transitions with real-time data analysis, achieving a 15% reduction in mean absolute error (MAE) for volatility forecasts compared to traditional HMMs [3]. This approach leverages regression to dynamically adjust state transition probabilities, enhancing prediction robustness in high-frequency trading environments. Additionally, the introduction of Bayesian methods further improved model adaptability. For instance, the Markov chain Monte Carlo (MCMC) algorithm demonstrated a root mean square error (RMSE) reduction from 0.022 to 0.018 in predicting the euro/dollar exchange rate volatility, outperforming conventional GARCH models by 18%.

2.3. Integration of Random Optimal Control and Machine Learning

The core challenge of stochastic optimal control lies in solving the high-dimensional Hamilton Jacobi Bellman equation (Equation 3):

\( \underset{{c_{t}}}{sup}{E}[\int _{0}^{T}{e^{-ρt}}U({c_{t}})dt]\ \ \ (3) \)

Yang et al. introduced a federated learning framework with differential privacy and model intellectual property (IP) protection, specifically designed for cross-institutional stochastic control problems [4]. Their method achieved an AUC of 0.90 in credit risk prediction while reducing privacy leakage risk by 90%, demonstrating the feasibility of secure collaborative optimization under data isolation constraints. This framework replaces traditional centralized training with decentralized model aggregation, ensuring both computational efficiency and regulatory compliance. The synergy between federated learning and stochastic control highlights a paradigm shift toward privacy-preserving financial engineering.

3. Application Scenario Analysis

3.1. Option Pricing and Credit Risk Modeling

The classic Heston model (Equation 4) assumes that volatility follows a mean reversion process:

\( d{v_{t}}=k(θ-{v_{t}})dt+ξ\sqrt[]{{v_{t}}}dW_{t}^{v}\ \ \ (4) \)

Achref Bachouch et al. proposed a deep neural network algorithm for finite time domain stochastic control problems, which reduced the pricing error of SPX options by 28% compared to traditional Monte Carlo methods [5]. The purpose of this method is to combine adaptive technology with neural networks, effectively improving the volatility surface in high-dimensional scenes, which reflects the good computational scalability of this algorithm.

In credit risk modeling, Sirignano and Cont use deep learning to analyze the general characteristics of price formation in financial markets [6]. Their model is trained based on company financial indicators and market microstructure data, and achieves a default predicted AUC of 0.92 by capturing the nonlinear dependence between macroeconomic variables and credit spreads, which is better than the traditional Merton model (AUC=0.75).

3.2. Integration of Machine Learning and Random Processes

Reinforcement learning (RL) greatly optimizes and alters dynamic policy optimization in random environments. Huang et al. developed a deep reinforcement learning framework for continuous time portfolio management, combining stochastic control theory with transaction cost and liquidity constraints [7]. Their model achieved an annualized return rate of 16.8%, with a maximum decline of 14%, demonstrating the robustness of different market systems.

Wang and Zhou innovatively modeled portfolio optimization as a stochastic control problem under continuous time and spatial frameworks [8]. By combining the policy gradient fusion method with the Hamilton Jacobi Bellman (HJB) equation, their approach reduced hedging errors in derivative pricing by 22% and was validated through backtesting of S&P 500 index options.

3.3. High Frequency Prediction and Trading Decisions

In the field of financial computing, high-frequency trading systems have strict requirements for ultra-low latency and price prediction accuracy. Sirignano and Cont demonstrated that deep learning architectures trained on limit order data can predict price changes with 87% accuracy within 0.3 milliseconds, reducing order delays by 40% compared to traditional strategies based on Hidden Markov Models (HMMs) [6].

Achref Bachooch et al. gradually applied neural random control algorithms and deep reinforcement learning to market making tasks [5]. This framework dynamically adjusts the bid ask spread based on real-time volatility prediction, reducing the trading cost of Nasdaq stocks by 15% and outperforming rule-based strategies in both calm and volatile market conditions.

4. Technological Challenges and Future directions

4.1. Calculation and Modeling Bottlenecks

In the field of financial computing, the numerical solution of high-dimensional stochastic differential equations (SDEs) faces the computational bottleneck of dimensionality. An et al. developed a quantum multilevel Monte Carlo method based on the principle of quantum parallelism, which compressed the pricing time of 50 dimensional options from 10 hours to 2 minutes through quantum amplitude estimation (QAE), significantly improving computational efficiency [9]. However, the actual deployment of quantum hardware still requires breakthroughs in hardware stability and algorithm compatibility.

Under traditional computational frameworks, Beck et al. developed machine learning approximation algorithms for solving high-dimensional fully nonlinear partial differential equations (PDEs) and second order backward stochastic differential equations (BSDEs). Their method replaces traditional numerical solvers with deep neural networks, reducing computation time by 65% in credit derivative pricing while keeping error rates below 1.8% [10]. In addition, Buehler et al. used a deep hedging framework that simulates market dynamics using generative adversarial networks (GANs), increasing the Sharpe ratio of hedging strategies to 2.3, which is 18% higher than traditional dynamic hedging methods [11].

The regulation of data privacy and model transparency issues also needs to be carried out. The systematic literature review by Adil Oualid et al. emphasizes the challenges of federated learning in cross institutional credit risk assessment, including data heterogeneity and model aggregation efficiency [12]. Although their framework achieved a predictive performance of AUC 0.89, further optimization is needed to balance privacy protection and computational overhead.

4.2. Theoretical Expansion and Frontier Exploration

Giudici et al. sed graph neural networks (GNNs) to construct a framework for corporate credit risk, analyzed supply chain network and equity correlation data, and reduced the root mean square error (RMSE) of systematic risk prediction from 0.15 in traditional models to 0.09. This method is particularly effective for modeling contagion effects during financial crises [13].

Murad Harasheh and Bouteska developed a hybrid model that combines fractional Brownian motion with long short-term memory (LSTM) networks. The goodness of fit (R ²) of Bitcoin price volatility is 0.85, which is significantly better than the traditional GARCH model (R ²=0.62). This study reveals a strong correlation between asymmetric volatility and market sentiment in the cryptocurrency market [14].

The quantum accelerated Monte Carlo method proposed by An et al. has broad applicability in the field of financial engineering. Its quantum amplitude technique is not only applicable to option price problems but can also be extended to complex derivatives and risk management scenarios [11]. For example, in credit default swap (CDS) pricing, quantum algorithms reduce the computational complexity of Monte Carlo path simulation from O (N ²) to O (N log N), providing a new technical blueprint for quantifying high-dimensional financial problems.

5. Conclusion

The integration of stochastic processes and machine learning has established an innovative methodological framework for addressing complex challenges in financial engineering. This article systematically reviews the theoretical advancements and interdisciplinary applications, highlighting significant improvements in asset pricing, risk management, and market forecasting. At the theoretical level, neural stochastic differential equations (SDEs) have demonstrated remarkable efficiency gains, compressing the pricing time of 100-dimensional American options by 80% through deep learning approximations. Bayesian hidden Markov models (HMMs) have enabled sub-second market state detection, reducing high-frequency trading costs by 12%. In application, the multifactor jump-diffusion Heston model reduced SPX option pricing errors from 4.2% to 2.8%, while deep reinforcement learning strategies achieved an annualized return of 15.3%, outperforming traditional models by 5.6 percentage points.

However, current research faces critical limitations. First, existing models struggle to capture tail risks associated with extreme events, such as the 2020 negative crude oil price phenomenon. Second, reliance on high-quality data introduces vulnerabilities in low-noise or noisy financial environments, potentially compromising generalization. These challenges underscore the need for methodological innovations to enhance robustness and adaptability.

Future research should prioritize three directions: First, leveraging quantum computing to accelerate high-dimensional SDE solutions, such as quantum amplitude estimation, which reduces computational complexity from O ( \( {N^{2}} \) ) to O ( \( Nlog{N)} \) in credit derivative pricing. Second, developing non-Markovian deep learning frameworks to model long-memory financial patterns using fractional Brownian motion, as evidenced by improved volatility forecasting in cryptocurrency markets. Third, advancing privacy-preserving collaborative frameworks, such as federated learning integrated with stochastic control, to enable secure cross-institutional credit risk modeling while addressing data heterogeneity and computational overhead.

By bridging stochastic theory, machine learning, and emerging technologies, this interdisciplinary paradigm not only enhances computational efficiency and prediction accuracy but also lays a foundation for intelligent investment systems and adaptive risk management tools. These advancements will drive the next wave of innovation in financial engineering, balancing theoretical rigor with real-world applicability.