arXiv subjects

Yuecai Han

Publications and source records attributed to Yuecai Han.

17 recordsLinked to original sources

arXiv2026

Fractional Stochastic Neural Networks

In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint recursion. For deterministic network parameters, we prove mean square convergence of projected samplewise stochastic gradient descent. Numerical experiments include a closed form convergence test, noisy regression with uncertainty quantification, long memory time series generation and image classification under structured perturbations. The results identify settings in which fractional drivers improve long memory recovery or robustness relative to Brownian and deterministic baselines.

math.OC↗

arXiv2025

Maximum principle for optimal control of infinite horizon stochastic difference equations driven by fractional noises

In this paper, infinite horizon stochastic difference equations and backward stochastic difference equations with fractional noises are studied. The main difficulty comes from fractional noises on infinite horizon. Motivated by discrete-time optimal control problem driven by fractional noises and on infinite horizon, the stochastic maximum principle for discrete-time control problem driven by fractional noises in infinite horizon is proved. As an application, an optimal investment problem is solved.

math.OC↗

arXiv2025

Maximum principle for optimal control of interacting particle system: stochastic flow model

In this paper, we consider the stochastic optimal control problem for the interacting particle system. We obtain the stochastic maximum principle of the optimal control system by introducing a generalized backward stochastic differential equation with interaction. The existence and uniqueness of the solution of this type of equation is proved. We derive the necessary condition that the optimal control should satisfy. As an application, the linear quadratic case is investigated to illustrate the main results.

math.PR↗

arXiv2024

Maximum Principle for Control System driven by Mixed Fractional Brownian Motion

In this paper, we investigate the optimal control problem for systems driven by mixed fractional Brownian motion (including a fractional Brownian motion with Hurst parameter $H>1/2$ and the standard Brownian motion). By using Malliavin calculus and introducing a disturbance control region, we obtain a modified maximum principle. Through martingale representation theorem, we obtain the adjoint backward stochastic differential equation in a natural way. Furthermore, corresponding to [1], a significant result is that the necessary condition is simplified by only containing one equality. As an application, the linear quadratic case is investigated to illustrate the main results.

math.OC↗

arXiv2024

Maximum principle for discrete-time control systems driven by fractional noises and related backward stochastic difference equations

In this paper, the optimal control for discrete-time systems driven by fractional noises is studied. A stochastic maximum principle is obtained by introducing a backward stochastic difference equation contains both fractional noises and the constructed white noises. The solution of the backward stochastic difference equations is also investigated. As an application, the linear quadratic case is considered to illustrate the main results.

math.OC↗

arXiv2024

Linear Filtering for Discrete Time Systems Driven by Fractional Noises

In this paper, we study the discrete time filtering problems for linear systems driven by fractional noises. The main difficulty comes from the non-Markovian of the noises. We construct the difference equation of the covariance process through the properties of the noises and transform the filtering problem to an optimal control problem. We obtain the necessary condition that the coefficients of the optimal filter should satisfy and show there is no consistent optimal filter, which is a significant difference from the classical Kalman filter. Finally, a simple example is considered to illustrate the main results. Further more, our method could also deal with systems driven by any other colored noises, as long as the self-covariation function is known.

math.OC↗

arXiv2024

Optimal Filtering for Interacting Particle Systems

In this paper, we study the optimal filtering problem for a interacting particle system generated by stochastic differential equations with interaction. By using Malliavin calculus, we construct the differential equation of the covariance process and transform the filter problem to an optimal control problem. Finally we give the necessary condition that the coefficient of the optimal filter should satisfy

math.OC↗

arXiv2024

Predicting path-dependent processes by deep learning

In this paper, we investigate a deep learning method for predicting path-dependent processes based on discretely observed historical information. This method is implemented by considering the prediction as a nonparametric regression and obtaining the regression function through simulated samples and deep neural networks. When applying this method to fractional Brownian motion and the solutions of some stochastic differential equations driven by it, we theoretically proved that the $L_2$ errors converge to 0, and we further discussed the scope of the method. With the frequency of discrete observations tending to infinity, the predictions based on discrete observations converge to the predictions based on continuous observations, which implies that we can make approximations by the method. We apply the method to the fractional Brownian motion and the fractional Ornstein-Uhlenbeck process as examples. Comparing the results with the theoretical optimal predictions and taking the mean square error as a measure, the numerical simulations demonstrate that the method can generate accurate results. We also analyze the impact of factors such as prediction period, Hurst index, etc. on the accuracy.

stat.ML↗

arXiv2024

Statistical inference for multi-regime threshold Ornstein-Uhlenbeck processes

In this paper, we investigate the parameter estimation for threshold Ornstein$\mathit{-}$Uhlenbeck processes. Least squares method is used to obtain continuous-type and discrete-type estimators for the drift parameters based on continuous and discrete observations, respectively. The strong consistency and asymptotic normality of the proposed least squares estimators are studied. We also propose a modified quadratic variation estimator based on the long-time observations for the diffusion parameters and prove its consistency. Our simulation results suggest that the performance of our proposed estimators for the drift parameters may show improvements compared to generalized moment estimators. Additionally, the proposed modified quadratic variation estimator exhibits potential advantages over the usual quadratic variation estimator with relatively small sample sizes. In particular, our method can be applied to the multi-regime cases ($m>2$), while the generalized moment method only deals with the two regime cases ($m=2$). The U.S. treasury rate data is used to illustrate the theoretical results.

math.ST↗

arXiv2024

Stochastic Maximum Principle for Control System with Time-varying delay

In this paper, we study the stochastic optimal control problem for control system with time-varying delay. The corresponding stochastic differential equation is a kind of stochastic differential delay equation. We prove the existence and uniqueness of the solution of this equation. We obtain the stochastic maximum principle of the control system with time-varying delay by introducing a kind of generalized anticipated backward stochastic differential equations. We prove the existence and uniqueness of the solution of this adjoint equation. As an application, the linear quadratic moving average control problem is investigated to illustrate the main result.

math.OC↗

arXiv2023

Stochastic Maximum Principle for a generalized Volterra Control System

In this paper, we consider the stochastic optimal control problem for a generalized Volterra control system. The corresponding state process is a kind of a generalized stochastic Volterra integral differential equations. We prove the existence and uniqueness of the solution of this type of equations. We obtain the stochastic maximum principle of the optimal control system by introducing a kind of generalized anticipated backward stochastic differential equations. We prove the existence and uniqueness of the solution of this adjoint equation, which may be singular at some points. As an application, the linear quadratic control problem is investigated to illustrate the main results.

math.OC↗

arXiv2023

Similarity Between Two Stochastic Differential Systems

The main focus of this paper is to explore how much similarity between two stochastic differential systems. Motivated by the conjugate theory of stochastic dynamic systems, we study the relationship between two systems by finding homeomorphic mappings $K$. Particularly, we use the minimizer $K^*$ to measure the degree of similarity. Under appropriate assumptions, we give sufficient and necessary conditions for the existence of the minimizer $K^*$. The former result can be regarded as a strong law of large numbers, while the latter is a stochastic maximum principle. Finally, we provide different examples of stochastic systems and an application to stochastic Hartman Grobman theorem. Thus, the results illustrate what is the similarity, extending the conjugacy in stochastic dynamical systems.

math.DS↗

arXiv2023

Similarity Between Two Dynamical Systems

The main focus of this paper is to explore how much similarity between two dynamical systems. Analogous to the classical Hartman-Grobman theorem, the relationship between two systems can be linked by a homeomorphic map $K$, and the core is to study the minimizer $K^*$ to measure the degree of similarity. We prove the sufficient conditions and necessary conditions (the maximum principle) for the existence of the minimizer $K^*$. Further, we establish similarity theorem based on the Takens embedding theorem. As applications, Lorenz system, Chua's circuit system and Chen's system are simulated and tested. The results illustrate what is the similarity, which extends the conjugacy in dynamical systems.

math.DS↗

arXiv2022

Approximate Pricing of Derivatives Under Fractional Stochastic Volatility Model

We investigate the problem of pricing derivatives under a fractional stochastic volatility model. We obtain an approximate expression of the derivative price where the stochastic volatility can be composed of deterministic functions of time and fractional Ornstein-Uhlenbeck process. Numerical simulations are given to illustrate the feasibility and operability of the approximation, and also demonstrate the effect of long-range on derivative prices.

q-fin.PR↗

arXiv2019

Exit problem as the generalized solution of Dirichlet problem

This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $α$-stable processes, which help verify the solvability of the original HJB equation.

math.PR↗

arXiv2018

Asian Option Pricing under Uncertain Volatility Model

In this paper, we study the asymptotic behavior of Asian option prices in the worst case scenario under an uncertain volatility model. We give a procedure to approximate the Asian option prices with a small volatility interval. By imposing additional conditions on the boundary condition and cutting the obtained Black-Scholes-Barenblatt equation into two Black-Scholes-like equations, we obtain an approximation method to solve the fully nonlinear PDE.

q-fin.PR↗

arXiv2012

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter $H>1/2$). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motion and the corresponding underlying standard Brownian motion. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motion while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.

math.OC↗