arXiv subjects

Jie Xiong

Publications and source records attributed to Jie Xiong.

At least 73 records · Page 4Linked to original sources

arXiv2023

Stochastic filtering under model ambiguity

In this paper, we study a non-linear filtering problem in the presence of signal model uncertainty. The model ambiguity is characterized by a class of probability measures from which the true one is taken. After interchanging the order of extremum problems by using the mini-max theorem, we find that the uncertain filtering problem can be converted to a weighted conditional mean-field optimal control problem. Further, we characterize the ambiguity filter and prove its unique existence.

math.OC↗

arXiv2023

Stochastic maximum principle for hybrid optimal control problems under partial observation

This paper is concerned with a partially observed hybrid optimal control problem, where continuous dynamics and discrete events coexist and in particular, the continuous dynamics can be observed while the discrete events, described by a Markov chain, is not directly available. Such kind of problem is first considered in the literature and has wide applications in finance, management, engineering, and so on. There are three major contributions made in this paper: First, we develop a novel non-linear filtering method to convert the partially observed problem into a completely observed one. Our method relies on some delicate stochastic analysis technique related to hybrid diffusions and is essentially different from the traditional filtering approaches. Second, we establish a new maximum principle based on the completely observed problem, whose two-dimensional state process consists of the continuous dynamics and the optimal filter. An important advantage of the maximum principle is that it takes a simple form and is convenient to implement. Finally, in order to illustrate the theoretical results, we solve a linear quadratic (LQ) example using the derived maximum principle to get some observable optimal controls.

math.OC↗

arXiv2023

Well-posedness of the martingale problem for super-Brownian motion with interactive branching

In this paper a martingale problem for super-Brownian motion with interactive branching is derived. The uniqueness of the solution to the martingale problem is obtained by using the pathwise uniqueness of the solution to a corresponding system of SPDEs with proper boundary conditions. The existence of the solution to the martingale problem and the local Hölder continuity of the density process are also studied.

math.PR↗

arXiv2022

On the empty balls of a critical or subcritical branching random walk

Let $\{Z_n\}_{n\geq 0 }$ be a critical or subcritical $d$-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on $\mathbb{R}^d$. Denote by $R_n:=\sup\{u>0:Z_n(\{x\in\mathbb{R}^d:|x|<u\})=0\}$ the radius of the largest empty ball centered at the origin of $Z_n$. In this work, we prove that after suitable renormalization, $R_n$ converges in law to some non-degenerate distribution as $n\to\infty$. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk, which completes the results of \cite{reves02} for the critical binary branching Wiener process.

math.PR↗

arXiv2022

A Global Maximum Principle for Controlled Conditional Mean-field FBSDEs with Regime Switching

This paper is devoted to a global stochastic maximum principle for conditional mean-field forward-backward stochastic differential equations (FBSDEs, for short) with regime switching. The control domain is unnecessarily convex and the driver of backward stochastic differential equations (BSDEs, for short) could depend on $Z$. Different from the case of non-recursive utility, the first-order and second-order adjoint equations are both high-dimensional linear BSDEs. Based on the adjoint equations, we reveal the relations among the terms of the first- and second-order Taylor's expansions. A general maximum principle is proved, which develops the work of Nguyen, Yin, and Nguyen [22] to recursive utility. As applications, the linear-quadratic problem is considered and a problem with state constraint is studied.

math.OC↗

arXiv2022

Stochastic maximum principle for weighted mean-field system

We study the optimal control problem for a weighted mean-field system. A new feature of the control problem is that the coefficients depend on the state process as well as its weighted measure and the control variable. By applying variational technique, we establish a stochastic maximum principle. As an application, we investigate the optimal premium policy of an insurance firm for asset-liability management problem.

math.OC↗

arXiv2022

Solvability of a class of mean-field BSDEs with quadratic growth

In this paper, we study the multi-dimensional mean-field backward stochastic differential equations (BSDEs, for short) with quadratic growth. Under small terminal value, the existence and uniqueness are proved for the multi-dimensional situation when the generator f(t,Y,E[Y],Z,E[Z]) is of quadratic growth with respect to the last four items, using some new methods. Besides, a kind of comparison theorem is obtained.

math.PR↗

arXiv2022

Stochastic Linear Quadratic Optimal Control Problems with Random Coefficients and Markovian Regime Switching System

This paper thoroughly investigates stochastic linear-quadratic optimal control problems with the Markovian regime switching system, where the coefficients of the state equation and the weighting matrices of the cost functional are random. We prove the solvability of the stochastic Riccati equation under the uniform convexity condition and obtain the closed-loop representation of the open-loop optimal control using the unique solvability of the corresponding stochastic Riccati equation. Moreover, by applying Itô's formula with jumps, we get a representation of the cost functional on a Hilbert space, characterized as the adapted solutions of some forward-backward stochastic differential equations. We show that the necessary condition of the open-loop optimal control is the convexity of the cost functional, and the sufficient condition of the open-loop optimal control is the uniform convexity of the cost functional. In addition, we study the properties of the stochastic value flow of the stochastic linear-quadratic optimal control problem. Finally, as an application, we present a continuous-time mean-variance portfolio selection problem and prove its unique solvability.

math.OC↗

arXiv2022

Linear quadratic leader-follower stochastic differential games for mean-field switching diffusions

In this paper, we consider a linear quadratic (LQ) leader-follower stochastic differential game for regime switching diffusions with mean-field interactions. One of the salient features of this paper is that conditional mean-field terms are included in the state equation and cost functionals. Based on stochastic maximum principles (SMPs), the follower's problem and the leader's problem are solved sequentially and an open-loop Stackelberg equilibrium is obtained. Further, with the help of the so-called four-step scheme, the corresponding Hamiltonian systems for the two players are decoupled and then the open-loop Stackelberg equilibrium admits a state feedback representation if some new-type Riccati equations are solvable.

math.OC↗

arXiv2022

Forward-backward doubly stochastic systems and classical solutions of path-dependent stochastic PDEs

In this paper, a class of non-Markovian forward-backward doubly stochastic systems is studied. By using the technique of functional Itô (or path-dependent) calculus, the relationship between the systems and related path-dependent quasi-linear stochastic partial differential equations (SPDEs in short) is established, and the well-known nonlinear stochastic Feynman-Kac formula of Pardoux and Peng [Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Relat. Fields 98 (1994), pp. 209--227] is developed to the non-Markovian situation. Moreover, we obtain the differentiability of the solution to the forward-backward doubly stochastic systems and some properties of solutions to the path-dependent SPDEs.

math.PR↗

arXiv2022

Backward doubly stochastic differential equations and SPDEs with quadratic growth

In this paper, we initiate the study of backward doubly stochastic differential equations (BDSDEs, for short) with quadratic growth. The existence, comparison, and stability results for one-dimensional BDSDEs are proved when the generator $f(t,Y,Z)$ grows in $Z$ quadratically and the terminal value is bounded, by introducing some new ideas. Moreover, in this framework, we use BDSDEs to give a probabilistic representation for the solutions of semilinear stochastic partial differential equations (SPDEs, for short) in Sobolev spaces, and use it to prove the existence and uniqueness of such SPDEs, thus extending the nonlinear Feynman-Kac formula.

math.PR↗

arXiv2022

On the empty balls of a critical super-Brownian motion

Let $\{X_t\}_{t\geq0}$ be a $d$-dimensional critical super-Brownian motion started from a Poisson random measure whose intensity is the Lebesgue measure. Denote by $R_t:=\sup\{u>0: X_t(\{x\in\mathbb{R}^d:|x|< u\})=0\}$ the radius of the largest empty ball centered at the origin of $X_t$. In this work, we prove that for $r>0$, $$\lim_{t\to\infty}\mathbb{P}\left(\frac{R_t}{t^{(1/d)\wedge(3-d)^+}}\geq r\right)=e^{-A_d(r)},$$ where $A_d(r)$ satisfies $\lim_{r\to\infty}\frac{A_d(r)}{r^{|d-2|+d\ind_{\{d=2\}}}}=C$ for some $C\in(0,\infty)$ depending only on $d$.

math.PR↗

arXiv2022

On the extinction-extinguishing dichotomy for a stochastic Lotka-Volterra type population dynamical system

We study a two-dimensional process $(X, Y)$ arising as the unique nonnegative solution to a pair of stochastic differential equations driven by independent Brownian motions and compensated spectrally positive Lévy random measures. Both processes $X$ and $Y$ can be identified as continuous-state nonlinear branching processes where the evolution of $Y$ is negatively affected by $X$. Assuming that process $X$ extinguishes, i.e. it converges to $0$ but never reaches $0$ in finite time, and process $Y$ converges to $0$, we identify rather sharp conditions under which the process $Y$ exhibits, respectively, one of the following behaviors: extinction with probability one, extinguishing with probability one or both extinction and extinguishing occurring with strictly positive probabilities.

math.PR↗

arXiv2022

Stochastic Linear-Quadratic Optimal Control with Partial Observation

The paper studies a class of quadratic optimal control problems for partially observable linear dynamical systems. In contrast to the full information case, the control is required to be adapted to the filtration generated by the observation system, which in turn is influenced by the control. The variation method fails in this case due to the fact that the filtration is not fixed. To overcome the difficulty, we use the orthogonal decomposition of the state process to write the cost functional as the sum of two parts: one is a functional of the control and the filtering process and the other part is independent of the choice of the control. The first part possesses a mathematical structure similar to the full information problem. By completing the square, it is shown that the optimal control is given by a feedback representation via the filtering process. The optimal value is also obtained explicitly.

math.OC↗

arXiv2022

General Indefinite Backward Stochastic Linear-Quadratic Optimal Control Problems

A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and the cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite and cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approach, a general procedure for constructing optimal controls is developed and the value function is obtained explicitly.

math.OC↗

arXiv2022

Input-to-State Stable Neural Ordinary Differential Equations with Applications to Transient Modeling of Circuits

This paper proposes a class of neural ordinary differential equations parametrized by provably input-to-state stable continuous-time recurrent neural networks. The model dynamics are defined by construction to be input-to-state stable (ISS) with respect to an ISS-Lyapunov function that is learned jointly with the dynamics. We use the proposed method to learn cheap-to-simulate behavioral models for electronic circuits that can accurately reproduce the behavior of various digital and analog circuits when simulated by a commercial circuit simulator, even when interconnected with circuit components not encountered during training. We also demonstrate the feasibility of learning ISS-preserving perturbations to the dynamics for modeling degradation effects due to circuit aging.

cs.LG↗

arXiv2021

RF-Based Human Activity Recognition Using Signal Adapted Convolutional Neural Network

Human Activity Recognition (HAR) plays a critical role in a wide range of real-world applications, and it is traditionally achieved via wearable sensing. Recently, to avoid the burden and discomfort caused by wearable devices, device-free approaches exploiting RF signals arise as a promising alternative for HAR. Most of the latest device-free approaches require training a large deep neural network model in either time or frequency domain, entailing extensive storage to contain the model and intensive computations to infer activities. Consequently, even with some major advances on device-free HAR, current device-free approaches are still far from practical in real-world scenarios where the computation and storage resources possessed by, for example, edge devices, are limited. Therefore, we introduce HAR-SAnet which is a novel RF-based HAR framework. It adopts an original signal adapted convolutional neural network architecture: instead of feeding the handcraft features of RF signals into a classifier, HAR-SAnet fuses them adaptively from both time and frequency domains to design an end-to-end neural network model. We apply point-wise grouped convolution and depth-wise separable convolutions to confine the model scale and to speed up the inference execution time. The experiment results show that the recognition accuracy of HAR-SAnet outperforms state-of-the-art algorithms and systems.

cs.NI↗

arXiv2021

Statistical Estimation and Nonlinear Filtering in Environmental Pollution

This paper studies a nonlinear filtering problem over an infinite time interval. The signal to be estimated is driven by a stochastic partial differential equation involves unknown parameters. Based on discrete observation, strongly consistent estimators of the parameters are derived at first. With the optimal filter given by Bayes formula, the uniqueness of invariant measure for the signal-filter pair has been verified. The paper then establishes approximation to the optimal filter, showing that the pathwise average distance, per unit time, of the computed approximating filter from the optimal filter converges to zero in probability. Simulation results are presented at last.

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