Searcharxiv⌕ Search

arXiv subjects

Huijie Qiao

Publications and source records attributed to Huijie Qiao.

At least 37 records · Page 2Linked to original sources
arXiv2023

Strong approximation of nonlinear filtering for multiscale McKean-Vlasov stochastic systems

This work concerns the nonlinear filtering problem of multiscale McKean-Vlasov stochastic systems where the whole systems depend on distributions of fast components. First of all, we prove that the slow component of the original system converges to an average system in the $L^{2p}$ ($p\geqslant 1$) sense. Moreover, we obtain the strong convergence order for the $L^2$ case. Then, given an observation process which depends on the slow component and its distribution, we show that the nonlinear filtering of the slow component and its distribution also converges to that of the average system in the $L^{q}$ ($p\geq 8, 1\leq q\leq \frac{p}{8}$) sense.

math.PR↗
arXiv2023

Weak approximation of nonlinear filtering for multiscale McKean-Vlasov stochastic systems

The work concerns the nonlinear filtering problem for a class of multiscale McKean-Vlasov stochastic systems. First of all, by a Poisson equation we prove that the solution of the slow part for a multiscale system weakly converges to the solution of the average equation. Then we define nonlinear filtering of the origin multiscale system and the average equation, and again through the same Poisson equation show the weak approximation between nonlinear filtering of the slow part for the origin multiscale system and that of the average equation.

math.PR↗
arXiv2023

Average principles for forward-backward multivalued stochastic systems and homogenization for systems of nonlinear parabolic PDEs

This work concerns about forward-backward multivalued stochastic systems. First of all, we prove one average principle for general stochastic differential equations in the $L^{2p}$ ($p\geq 1$) sense. Moreover, for $p=1$ a convergence rate is presented. Then combining general stochastic differential equations with backward stochastic variation inequalities, we establish the other average principle for backward stochastic variation inequalities in the $L^{2}$ sense through a time discretization method. Finally, we apply our result to nonlinear parabolic partial differential equations and obtain the homogenization of them.

math.PR↗
arXiv2023

Large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations

In this paper, we study large deviation principles of nonlinear filtering for McKean-Vlasov stochastic differential equations. First of all, we establish the large deviation principle for the space-distribution dependent Zakai equation by a weak convergence approach. Then based on the obtained result and the relationship between the space-distribution dependent Zakai equation and the space-distribution dependent Kushner-Stratonovich equation, the large deviation principle for the latter is proved.

math.PR↗
arXiv2022

Limit theorems of invariant measures for multivalued McKean-Vlasov stochastic differential equations

The work concerns invariant measures for multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the exponential ergodicity of these equations. Then for a sequence of these equations, when their coefficients converge in the suitable sense, the convergence of corresponding strong solutions are presented. Finally, based on the convergence of these solutions, we establish the convergence of corresponding invariant measures.

math.PR↗
arXiv2022

Backward multivalued McKean-Vlasov SDEs and associated variational inequalities

The work concerns a type of backward multivalued McKean-Vlasov stochastic differential equations. First, we prove the existence and uniqueness of solutions for backward multivalued McKean-Vlasov stochastic differential equations. Then, it is presented that their solutions depend continuously on the terminal values. Finally, we give a probabilistic interpretation for viscosity solutions of nonlocal quasi-linear parabolic variational inequalities.

math.PR↗
arXiv2022

Deviation estimates for multivalued McKean-Vlasov stochastic differential equations

The work concerns deviation estimates for multivalued McKean-Vlasov stochastic differential equations. First of all, we prove the large deviation principle for them by the weak convergence approach. Then the central limit theorem for them is shown with the help of a formula for $L$-derivatives. Finally, we establish the moderate deviation principle for them.

math.PR↗
arXiv2022

Uniqueness and superposition of the space-distribution dependent Zakai equations

The work concerns the space-distribution dependent Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First of all, we establish the space-distribution dependent Kushner-Stratonovich equations and the space-distribution dependent Zakai equations. Then, the pathwise uniqueness of their strong solutions is shown. Finally, we prove a superposition principle between the space-distribution dependent Zakai equations and space-distribution dependent Fokker-Planck equations. As a by-product, we give some conditions under which space-distribution dependent Fokker-Planck equations have weak solutions.

math.PR↗
arXiv2022

Limit theorems of stochastic differential equations with jumps

In this paper, we study the convergence for solutions to a sequence of (possibly degenerate) stochastic differential equations with jumps, when the coefficients converge in some appropriate sense. Our main tools are the superposition principles. And then we analyze some special cases and give some concrete and verifiable conditions.

math.PR↗
arXiv2022

The Onsager-Machlup action functional for Mckean-Vlasov SDEs

This paper is devoted to deriving the Onsager-Machlup action functional for Mckean-Vlasov stochastic differential equations in a class of norms that dominate $L^2([0,1], \mathbb{R}^d)$, such as supremum norm $\|\cdot\|_{\infty}$, H$\mathrm{\ddot{o}}$lder norms $\|\cdot\|_α$ with $α<\frac{1}{4}$ and $L^p$-norms with $p>4$ are included. Moreover, the corresponding Euler-Lagrange equation for Onsager-Machlup action functional is derived and a example is given.

math.PR↗
arXiv2020

Coupled McKean-Vlasov stochastic differential equations with jumps

This work concerns a type of coupled McKean-Vlasov stochastic differential equations (MVSDEs in short) with jumps. First, we prove superposition principles for these coupled MVSDEs with jumps and non-local space-distribution dependent Fokker-Planck equations. Since superposition principles are related to the well-posedness of weak solutions for coupled MVSDEs with jumps, then we give some conditions to assure it. After this, we construct space-distribution valued Markov processes associated with these coupled MVSDEs with jumps. Finally, the ergodicity of these coupled MVSDEs with jumps are investigated. As a by-product, we show the exponential ergodicity for a type of MVSDEs with jumps.

math.PR↗
arXiv2020

Superposition principles for the Zakai equations and the Fokker-Planck equations on measure spaces

The work concerns the superposition between the Zakai equations and the Fokker-Planck equations on measure spaces. First, we prove a superposition principle for the Fokker-Planck equations on $\mR^\mN$ under the integrable condition. And then by means of it, we show two superposition principles for the weak solutions of the Zakai equations from the nonlinear filtering problems and the weak solutions of the Fokker-Planck equations on measure spaces. As a by-product, we give some weak conditions under which the Fokker-Planck equations can be solved in the weak sense.

math.PR↗
arXiv2020

Limit theorems of SDEs driven by Lévy processes and application to nonlinear filtering problems

In this paper we study the convergence of solutions for (possibly degenerate) stochastic differential equations driven by Lévy processes, when the coefficients converge in some appropriate sense. First, we prove, by means of a superposition principle, a limit theorem of stochastic differential equations driven by Lévy processes. Then we apply the result to a type of nonlinear filtering problems and obtain the convergence of the nonlinear filterings.

math.PR↗
arXiv2020

Parameter estimation of path-dependent McKean-Vlasov stochastic differential equations

The work concerns a class of path-dependent McKean-Vlasov stochastic differential equations with unknown parameters. First, we prove the existence and uniqueness of these equations under non-Lipschitz conditions. Second, we construct maximum likelihood estimators of these parameters and then discuss their strong consistency. Third, a numerical simulation method for the class of path-dependent McKean-Vlasov stochastic differential equations is offered. Moreover, we estimate the errors between solutions of these equations and that of their numerical equations. Finally, we give an example to explain our result.

math.PR↗
arXiv2020

Nonlinear filtering of stochastic differential equations driven by correlated Lévy noises

The work concerns nonlinear filtering problems of stochastic differential equations with correlated Lévy noises. First, we establish the Kushner-Stratonovich and Zakai equations through martingale representation theorems and the Kallianpur-Striebel formula. Second, we show the pathwise uniqueness and uniqueness in joint law of weak solutions for the Zakai equation. Finally, we investigate the uniqueness in joint law of weak solutions to the Kushner-Stratonovich equation.

math.PR↗
arXiv2020

Stability for Stochastic McKean-Vlasov Equations with Non-Lipschitz Coefficients

In this paper we consider the stability for a type of stochastic McKean-Vlasov equations with non-Lipschitz coefficients. First, sufficient conditions are given for the exponential stability of the second moments for their solutions in terms of a Lyapunov function. Then we weaken the conditions and furthermore obtain exponentially 2-ultimate boundedness of their solutions. After this, the almost surely asymptotic stability of their solutions is proved. Finally we give an example to motivate the choice of Lyapunov functions.

math.PR↗
arXiv2020

Effective Filtering for Multiscale Stochastic Dynamical Systems driven by Lévy processes

The work is about multiscale stochastic dynamical systems driven by Lévy processes. First, we prove that these systems can approximate low-dimensional systems on random invariant manifolds. Second, we establish that nonlinear filterings of multiscale stochastic dynamical systems also approximate that of reduced low-dimensional systems. Finally, we investigate the reduction for $\e=0$ and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.

math.PR↗