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Chunrong Feng

Publications and source records attributed to Chunrong Feng.

27 records · Page 2Linked to original sources
arXiv2017

Numerical Approximation of Random Periodic Solutions of Stochastic Differential Equations

In this paper, we discuss the numerical approximation of random periodic solutions (r.p.s.) of stochastic differential equations (SDEs) with multiplicative noise. We prove the existence of the random periodic solution as the limit of the pull-back flow when the starting time tends to $-\infty$ along the multiple integrals of the period. As the random periodic solution is not explicitly constructible, it is useful to study the numerical approximation. We discretise the SDE using the Euler-Maruyama scheme and moldiflied Milstein scheme. Subsequently we obtain the existence of the random periodic solution as the limit of the pull-back of the discretised SDE. We prove that the latter is an approximated random periodic solution with an error to the exact one at the rate of $\sqrt {Δt}$ in the mean-square sense in Euler-Maruyama method and $Δt$ in the Milstein method. We also obtain the weak convergence result for the approximation of the periodic measure.

math.NA↗
arXiv2016

Anticipating Random Periodic Solutions--I. SDEs with Multiplicative Linear Noise

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify them as solutions of coupled forward-backward infinite horizon stochastic integral equations (IHSIEs), using the "substitution theorem" of stochastic differential equations with anticipating initial conditions. In general, random periodic solutions and the solutions of IHSIEs, are anticipating. For the linear noise case, with the help of the exponential dichotomy given in the multiplicative ergodic theorem, we can identify them as the solutions of infinite horizon random integral equations (IHSIEs). We then solve a localised forward-backward IHRIE in $C(\mathbb{R}, L^2_{loc}(Ω))$ using an argument of truncations, the Malliavin calculus, the relative compactness of Wiener-Sobolev spaces in $C([0, T], L^2(Ω))$ and Schauder's fixed point theorem. We finally measurably glue the local solutions together to obtain a global solution in $C(\mathbb{R}, L^2(Ω))$. Thus we obtain the existence of a random periodic solution and a periodic measure.

math.PR↗
arXiv2015

Pathwise Random Periodic Solutions of Stochastic Differential Equations

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω))$ and generalized Schauder's fixed point theorem to prove the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations. The condition on $F$ is then further weakened by applying the coupling method of forward and backward Gronwall inequalities. The results are also valid for stationary solutions as a special case when the period $τ$ can be an arbitrary number.\

math.PR↗
arXiv2015

Random Periodic Solutions of SPDEs via Integral Equations and Wiener-Sobolev Compact Embedding

In this paper, we study the existence of random periodic solutions for semilinear SPDEs on a bounded domain with a smooth boundary. We identify them as the solutions of coupled forward-backward infinite horizon stochastic integral equations on $L^2(D)$ in general cases. For this we use Mercer's Theorem and eigenvalues and eigenfunctions of the second order differential operators in the infinite horizon integral equations. We then use the argument of the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω\times D))$ and generalized Schauder's fixed point theorem to prove the existence of a solution of the integral equations. This is the first paper in literature to study random periodic solutions of SPDEs. Our result is also new in finding semi-stable stationary solution for non-dissipative SPDEs, while in literature the classical method is to use the pull-back technique so researchers were only able to find stable stationary solutions for dissipative systems.

math.PR↗
arXiv2009

On the Borel-Cantelli Lemma and its Generalization

Let $\{A_n\}_{n=1}^{\infty}$ be a sequence of events on a probability space $(Ω,\mathcal{F},\mathbf{P})$. We show that if $\lim_{m\to\infty}\sum_{n=1}^{m}w_n\mathbf{P}(A_n)=\infty$ where each $w_n\in\mathbb{R}$, then \[{\mathbf{P}}(\limsup A_n)\geq\limsup_{n\to\infty} \frac{\displaystyle\big(\sum_{k=1}^n{w_k\mathbf{P}}(A_k)\big)^2}{\displaystyle\sum_{i=1}^n\sum_{j=1}^nw_iw_j{\mathbf{P}}(A_i\cap A_j)}.\]

math.PR↗
arXiv2009

Variation and Rough Path Properties of Local Times of Lévy Processes

In this paper, we will prove that the local time of a Lévy process is of finite $p$-variation in the space variable in the classical sense, a.s. for any $p>2$, $t\geq 0$, if the Lévy measure satisfies $\int_{R\setminus \{0\}}(|y|^{3\over 2}\wedge 1)n(dy)<\infty$, and is a rough path of roughness $p$ a.s. for any $2<p<3$ under a slightly stronger condition for the Lévy measure. Then for any function $g$ of finite $q$-variation ($1\leq q <3$), we establish the integral $\int_{-\infty}^{\infty}g(x)dL_t^x$ as a Young integral when $1\leq q<2$ and a Lyons' rough path integral when $2\leq q<3$. We therefore apply these path integrals to extend the Tanaka-Meyer formula for a continuous function $f$ if $\nabla ^-f$ exists and is of finite $q$-variation when $1\leq q<3$, for both continuous semi-martingales and a class of Lévy processes.

math.PR↗
arXiv2006

Two-parameter $p, q$-variation Paths and Integrations of Local Times

In this paper, we prove two main results. The first one is to give a new condition for the existence of two-parameter $p, q$-variation path integrals. Our condition of locally bounded $p,q$-variation is more natural and easy to verify than those of Young. This result can be easily generalized to multi-parameter case. The second result is to define the integral of local time $\int_{-\infty}^\infty\int_0^t g(s,x)d_{s,x}L_s(x)$ pathwise and then give generalized It$\hat {\rm o}$'s formula when $\nabla^-f(s,x)$ is only of bounded $p,q$-variation in $(s,x)$. In the case that $g(s,x)=\nabla^-f(s,x)$ is of locally bounded variation in $(s,x)$, the integral $\int_{-\infty}^\infty\int_0^t \nabla^-f(s,x)d_{s,x}L_s(x)$ is the Lebesgue-Stieltjes integral and was used in Elworthy, Truman and Zhao \cite{Zhao}. When $g(s,x)=\nabla^-f(s,x)$ is of only locally $p, q$-variation, where $p\geq 1$,$q\geq 1$, and $2q+1>2pq$, the integral is a two-parameter Young integral of $p,q$-variation rather than a Lebesgue-Stieltjes integral. In the special case that $f(s,x)=f(x)$ is independent of $s$, we give a new condition for Meyer's formula and $\int_{-\infty}^\infty L_t(x)d_x\nabla^-f(x)$ is defined pathwise as a Young integral. For this we prove the local time $L_t(x)$ is of $p$-variation in $x$ for each $t\geq 0$, for each $p>2$ almost surely ($p$-variation in the sense of Lyons and Young, i.e. $\sup\limits_{E: \ a finite partition of [-N,N]} \sum\limits_{i=1}^m|L_t(x_i)-L_t(x_{i-1})|^p<\infty$).

math.PR↗
arXiv2005

A Generalized It$\hat {\rm o}$'s Formula in Two-Dimensions and Stochastic Lebesgue-Stieltjes Integrals

A generalized It${\hat {\rm o}}$ formula for time dependent functions of two-dimensional continuous semi-martingales is proved. The formula uses the local time of each coordinate process of the semi-martingale, left space and time first derivatives and second derivative $\nabla_1^- \nabla_2^-f$ only which are assumed to be of locally bounded variation in certain variables, and stochastic Lebesgue-Stieltjes integrals of two parameters.The two-parameter integral is defined as a natural generalization of the It${\hat {\rm o}}$ integral and Lebesgue-Stieltjes integral through a type of It${\hat {\rm o}}$ isometry formula.

math.PR↗