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Jinniao Qiu

Publications and source records attributed to Jinniao Qiu.

At least 37 records · Page 2Linked to original sources
arXiv2022

Zero-Inertia Limit: from Particle Swarm Optimization to Consensus Based Optimization

Recently a continuous description of the particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in arXiv:2012.05613 where the authors formally showed the link between PSO and the consensus based optimization (CBO) through zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in arXiv:2012.05613 by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space and the results are illustrated with some numerical examples.

math.DS↗
arXiv2021

Controlled Ordinary Differential Equations with Random Path-Dependent Coefficients and Stochastic Path-Dependent Hamilton-Jacobi Equations

This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases, the value function turns out to be a random field on the path spaces and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

math.OC↗
arXiv2021

Mean-field particle swarm optimization

In this work we survey some recent results on the global minimization of a non-convex and possibly non-smooth high dimensional objective function by means of particle based gradient-free methods. Such problems arise in many situations of contemporary interest in machine learning and signal processing. After a brief overview of metaheuristic methods based on particle swarm optimization (PSO), we introduce a continuous formulation via second-order systems of stochastic differential equations that generalize PSO methods and provide the basis for their theoretical analysis. Subsequently, we will show how through the use of mean-field techniques it is possible to derive in the limit of large particles number the corresponding mean-field PSO description based on Vlasov-Fokker-Planck type equations. Finally, in the zero inertia limit, we will analyze the corresponding macroscopic hydrodynamic equations, showing that they generalize the recently introduced consensus-based optimization (CBO) methods by including memory effects. Rigorous results concerning the mean-field limit, the zero-inertia limit, and the convergence of the mean-field PSO method towards the global minimum are provided along with a suite of numerical examples.

math.OC↗
arXiv2021

On the mean-field limit for the consensus-based optimization

This paper is concerned with the large particle limit for the consensus-based optimization (CBO), which was postulated in the pioneering works [6,28]. In order to solve this open problem, we adapt a compactness argument by first proving the tightness of the empirical measures $\{μ^N\}_{N\geq 2}$ associated to the particle system and then verifying that the limit measure $μ$ is the unique weak solution to the mean-field CBO equation. Such results are extended to the model of particle swarm optimization (PSO).

math.OC↗
arXiv2021

Stochastic Differential Games with Random Coefficients and Stochastic Hamilton-Jacobi-Bellman-Isaacs Equations

In this paper, we study a class of zero-sum two-player stochastic differential games with the controlled stochastic differential equations and the payoff/cost functionals of recursive type. As opposed to the pioneering work by Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp.~293--314] and the seminal work by Buckdahn and Li [SIAM J. Control Optim., 47 (2008), pp.~444--475], the involved coefficients may be random, going beyond the Markovian framework and leading to the random upper and lower value functions. We first prove the dynamic programming principle for the game, and then under the standard Lipschitz continuity assumptions on the coefficients, the upper and lower value functions are shown to be the viscosity solutions of the upper and the lower fully nonlinear stochastic Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, respectively. A stability property of viscosity solutions is also proved. Under certain additional regularity assumptions on the diffusion coefficient, the uniqueness of the viscosity solution is addressed as well.

math.PR↗
arXiv2020

Numerical Approximations of Coupled Forward-Backward SPDEs

We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725--2755] by considering general nonlinear and nonlocal FBSPDEs with more inclusive coupling; self-contained proofs are provided and different numerical techniques for the resulting finite-dimensional equations are adopted. For such FBSPDEs, we first prove the existence and uniqueness of the strong solution as well as of the weak solution. Then the finite element method in the spatial domain leads to approximations of FBSPDEs by finite-dimensional forward-backward stochastic differential equations (FBSDEs) which are numerically computed by using some deep learning-based schemes. The convergence analysis is addressed for the spatial discretization of FBSPDEs, and the numerical examples, including both decoupled and coupled cases, indicate that our methods are quite efficient.

math.NA↗
arXiv2020

The microscopic derivation and well-posedness of the stochastic Keller-Segel equation

In this paper, we propose and study a stochastic aggregation-diffusion equation of the Keller-Segel (KS) type for modeling the chemotaxis in dimensions $d=2,3$. Unlike the classical deterministic KS system, which only allows for idiosyncratic noises, the stochastic KS equation is derived from an interacting particle system subject to both idiosyncratic and common noises. Both the unique existence of solutions to the stochastic KS equation and the mean-field limit result are addressed.

math.AP↗
arXiv2020

Pricing Options Under Rough Volatility with Backward SPDEs

In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic partial differential equation (BSPDE). The existence and uniqueness of weak solution is proved for general nonlinear BSPDEs with unbounded random leading coefficients whose connections with certain forward-backward stochastic differential equations are derived as well. These BSPDEs are then used to approximate American option prices. A deep leaning-based method is also investigated for the numerical approximations to such BSPDEs and associated non-Markovian pricing problems. Finally, the examples of rough Bergomi type are numerically computed for both European and American options.

q-fin.MF↗
arXiv2020

Stochastic Path-Dependent Hamilton-Jacobi-Bellman Equations and Controlled Stochastic Differential Equations with Random Path-Dependent Coefficients

In this paper, we propose and study the stochastic path-dependent Hamilton-Jacobi-Bellman (SPHJB) equation that arises naturally from the optimal stochastic control problem of stochastic differential equations with path-dependence and measurable randomness. Both the notions of viscosity solution and classical solution are proposed, and the value function of the optimal stochastic control problem is proved to be the viscosity solution to the associated SPHJB equation. A uniqueness result about viscosity solutions is also given for certain superparabolic cases, while the uniqueness of classical solution is addressed for general cases. In addition, an Itô-Kunita-Wentzell-Krylov formula is proved for the compositions of random fields and stochastic differential equations in the path-dependent setting.

math.PR↗
arXiv2019

Optimal Liquidation in Target Zone Models and Neumann Problem of Backward SPDEs with Singular Terminal Condition

We study the optimal liquidation problems in target zone models using dynamic programming methods. Such control problems allow for stochastic differential equations with reflections and random coefficients. The value function is characterized with a Neumann problem of backward stochastic partial differential equations (BSPDEs) with singular terminal conditions. The existence and the uniqueness of strong solution to such BSPDEs are addressed, which in turn yields the optimal feedback control. In addition, the unique existence of strong solution to Neumann problem of general semilinear BSPDEs in finer functions space, a comparison theorem, and a new link between forward-backward stochastic differential equations and BSPDE are proved as well.

math.OC↗
arXiv2019

Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations

This paper is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.

math.OC↗
arXiv2019

Controlled Reflected SDEs and Neumann Problem for Backward SPDEs

We solve the optimal control problem of a one-dimensional reflected stochastic differential equation, whose coefficients can be path dependent. The value function of this problem is characterized by a backward stochastic partial differential equation (BSPDE) with Neumann boundary conditions. We prove the existence and uniqueness of sufficiently regular solution for this BSPDE, which is then used to construct the optimal feedback control. In fact we prove a more general result: The existence and uniqueness of strong solution for the Neumann problem for general nonlinear BSPDEs, which might be of interest even out of the current context.

math.PR↗
arXiv2018

Viscosity Solutions of Stochastic Hamilton-Jacobi-Bellman Equations

In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well.

math.OC↗
arXiv2016

A Functional Limit Theorem for Limit Order Books with State Dependent Price Dynamics

We consider a stochastic model for the dynamics of the two-sided limit order book (LOB). Our model is flexible enough to allow for a dependence of the price dynamics on volumes. For the joint dynamics of best bid and ask prices and the standing buy and sell volume densities, we derive a functional limit theorem, which states that our LOB model converges in distribution to a fully coupled SDE-SPDE system when the order arrival rates tend to infinity and the impact of an individual order arrival on the book as well as the tick size tends to zero. The SDE describes the bid/ask price dynamics while the SPDE describes the volume dynamics.

q-fin.MF↗
arXiv2016

Maximum Principle for Quasi-linear Reflected Backward SPDEs

This paper establishes a maximum principle for quasi-linear reflected backward stochastic partial differential equations (RBSPDEs for short). We prove the existence and uniqueness of the weak solution to RBSPDEs allowing for non-zero Dirichlet boundary conditions and, using a stochastic version of De Giorgi's iteration, establish the maximum principle for RBSPDEs on a general domain. The maximum principle for RBSPDEs on a bounded domain and the maximum principle for backward stochastic partial differential equations (BSPDEs for short) on a general domain can be obtained as byproducts. Finally, the local behavior of the weak solutions is considered.

math.AP↗
arXiv2015

A Constrained Control Problem with Degenerate Coefficients and Degenerate Backward SPDEs with Singular Terminal Condition

We study a constrained optimal control problem with possibly degenerate coefficients arising in models of optimal portfolio liquidation under market impact. The coefficients can be random in which case the value function is described by a degenerate backward stochastic partial differential equation (BSPDE) with singular terminal condition. For this degenerate BSPDE, we prove existence and uniqueness of a nonnegative solution. Our existence result requires a novel gradient estimate for degenerate BSPDEs.

q-fin.MF↗
arXiv2015

Weak Solution for a Class of Fully Nonlinear Stochastic Hamilton-Jacobi-Bellman Equations

This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation with controlled leading coefficients, which is a type of fully nonlinear backward stochastic partial differential equation (BSPDE for short). In order to formulate the weak solution for such kind of BSPDEs, the classical potential theory is generalized in the backward stochastic framework. The existence and uniqueness of the weak solution is proved, and for the partially non-Markovian case, we obtain the associated gradient estimate. As a byproduct, the existence and uniqueness of solution for a class of degenerate reflected BSPDEs is discussed as well.

math.OC↗