Huijie Qiao, Jiang-Lun Wu
In the paper, we consider a type of stochastic differential equations driven by G-Lévy processes. We prove that a kind of their additive functionals has path independence and extend some known results.
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The path independence of additive functionals for SDEs driven by the G-Brownian motion is characterized by nonlinear PDEs. The main result generalizes the existing ones for SDEs driven by the standard Brownian motion.
In this article, the path independent property of additive functionals of McKean-Vlasov stochastic differential equations with jumps is characterised by nonlinear partial integro-differential equations involving $L$-derivatives with respect to probability measures introduced by P.-L. Lions. Our result extends the recent work [16] by Ren and Wang where their concerned McKean-Vlasov stochastic differential equations are driven by Brownian motions.
Path independence for the additive functionals of stochastic Volterra equations with singular kernels and Hölder continuous coefficients
本文针对奇异核、Hölder 连续系数随机 Volterra 方程，先证明适定性，再借助抛物型 SPDE 的 mild 表述与随机发展方程结果刻画可加泛函路径无关性，并应用于分数 Brown 运动相关核。
Let $X$ be a symmetric strong Markov process on a Luzin space. In this paper, we present criteria of the $L^p$-independence of spectral bounds for generalized non-local Feynman-Kac semigroups of $X$ that involve continuous additive functionals of $X$ having zero quadratic variations and discontinuous additive functionals of $X$.
We define a product of algebraic probability spaces equipped with two states. This product is called a conditionally monotone product. This product is a new example of independence in non-commutative probability theory and unifies the monotone and Boolean products, and moreover, the orthogonal product. Then we define the associated cumulants and calculate the limit distributions in central limit theorem and Poisson's law of small numbers. We also prove a combinatorial moment-cumulant formula using monotone partitions. We investigate some other topics such as infinite divisibility for the addit…
Let P2(Rd) be the space of probability measures on Rd with finite second moment. The path independence of additive functionals of McKean-Vlasov SDEs is characterized by PDEs on the product space Rd*P2(Rd) equipped with the usual derivative in space variable and Lions derivative in distribution. These PDEs are solved by using probabilis- tic arguments developed from [2]. In particular, the path independence of the Girsanov transformation killing the drift term is identified with a nonlinear PDE on Rd*P2(Rd), which includes corresponding results derived earlier for the classical SDEs as special …
Based on a recent result on characterising the path-independence of the Girsanov transformation for non-Lipschnitz stochastic differential equations (SDEs) with jumps on $R^d$, in this paper, we extend our consideration of characterising the path-indpendent property from finite-dimensional SDEs with jumps to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in the manner that one could then link the characterisation of the path-independence for finit…
By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.
The paper introduces and investigates the natural extension to the path-dependent setup of the usual concept of canonical Markov class introduced by Dynkin and which is at the basis of the theory of Markov processes. That extension, indexed by starting paths rather than starting points will be called path-dependent canonical class. Associated with this is the generalization of the notions of semi-group and of additive functionals to the path-dependent framework. A typical example of such family is constituted by the laws $({\mathbb P}^{s,$η$})\_{(s,η) \in{\mathbb R} \times Ω}$ , where for fixe…
Sample-path large deviations for a class of heavy-tailed Markov additive processes
本文针对由仿射递归驱动的重尾马尔可夫可加过程，在$M_1'$拓扑下发展了样本路径大偏差原理，并揭示了最可能路径为含正负跳的阶跃函数。
This paper deals with the drift estimation in linear stochastic evolution equations (with emphasis on linear SPDEs) with additive fractional noise (with Hurst index ranging from 0 to 1) via least-squares procedure. Since the least-squares estimator contains stochastic integrals of divergence type, we address the problem of its pathwise (and robust to observation errors) evaluation by comparison with the pathwise integral of Stratonovich type and using its chain-rule property. The resulting pathwise LSE is then defined implicitly as a solution to a non-linear equation. We study its numerical pr…
Central limit theorems for additive functionals in patricia tries
本文研究了帕特里克树中加法函数在独立同分布情形下的渐近正态性，推导了随机边树分布的渐近正态性，并给出了渐近均值和方差的公式，以及边树中k个键的比例渐近表达式。
We consider additive functionals of stationary Markov processes and show that under Kipnis-Varadhan type conditions they converge in rough path topology to a Stratonovich Brownian motion, with a correction to the Levy area that can be described in terms of the asymmetry (non-reversibility) of the underlying Markov process. We apply this abstract result to three model problems: First we study random walks with random conductances under the annealed law. If we consider the Ito rough path, then we see a correction to the iterated integrals even though the underlying Markov process is reversible. …
This paper develops a systematic treatment of monotonicity-based pathwise dualities for Markov processes taking values in partially ordered sets. We show that every Markov process that takes values in a finite partially ordered set and whose generator can be represented in monotone maps has a pathwise dual process. In the special setting of attractive spin systems this has been discovered earlier by Gray. We show that the dual simplifies a lot when the state space is a lattice (in the order-theoretic meaning of the word) and all monotone maps satisfy an additivity condition. This leads to a un…
Tests of independence for pairs of paths of non-stationary Gaussian processes
本文为非平稳高斯过程（标准布朗运动与分数布朗运动）路径对的独立性检验提供了理论结果。
Uniform pathwise stability of additive singular SDEs driven by fractional Brownian motion
本文研究了分数布朗运动驱动的加性奇异随机微分方程的长期行为，通过统一路径估计方法证明了解的均匀路径稳定性与不变测度的唯一性。
Let $X_t$ solve the multidimensional Itô's stochastic differential equations on $\R^d$ $$dX_t=b(t,X_t)dt+σ(t,X_t)dB_t$$ where $b:[0,\infty)\times\R^d\to\R^d$ is smooth in its two arguments, $σ:[0,\infty)\times\R^d\to\R^d\otimes\R^d$ is smooth with $σ(t,x)$ being invertible for all $(t,x)\in[0,\infty)\times\R^d$, $B_t$ is $d$-dimensional Brownian motion. It is shown that, associated to a Girsanov transformation, the stochastic process $$\int^t_0\langle(σ^{-1}b)(s,X_s),dB_t\rangle+\frac{1}{2}\int^t_0|σ^{-1}b|^2(s,X_s)ds$$ is a function of the arguments $t$ and $X_t$ (i.e., path-independent) if a…
Sample-path large deviations for unbounded additive functionals of the reflected random walk
本文研究 Lindley 递归 Markov 链无界泛函的样本路径大偏差，利用再生周期分解在 M_1' 拓扑下证明次线性速度 LDP，并揭示 MRW 繁忙期面积的重尾特性。
In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.
In this paper we analyse the pathwise approximation of stochastic differential equations by polynomial splines with free knots. The pathwise distance between the solution and its approximation is measured globally on the unit interval in the $L_{\infty}$-norm, and we study the expectation of this distance. For equations with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary spline approximation methods, which use $k$ free knots. The optimal order is achieved by an approximation method $\hat{X}_{k}^{\dagger}$, which combines an Euler scheme on…
We consider a jump-diffusion process on a bounded domain with reflection at the boundary, and establish long-term results for a general additive process of its path. This includes the long-term behaviour of its occupation time in the interior and on the boundaries. We derive a characterization of the large deviation rate function which quantifies the rate of exponential decay of probabilities of rare events for paths of the process. The characterization relies on a solution of a partial integro-differential equation (PIDE) with boundary constraints. We develop a practical implementation of our…
We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post- and pre-splitting processes using time reversal. These results offer valuable insight into behavior of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener-Hopf factorization, where time is counted in each…
Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with Hölder continuous paths
本文研究了由加性噪声驱动的反应-扩散方程中行波解的稳定性，得出了不同时间尺度下的误差界及稳定性条件。
The slow drift (with speed $\eps$) of a parameter through a pitchfork bifurcation point, known as the dynamic pitchfork bifurcation, is characterized by a significant delay of the transition from the unstable to the stable state. We describe the effect of an additive noise, of intensity $σ$, by giving precise estimates on the behaviour of the individual paths. We show that until time $\sqrt\eps$ after the bifurcation, the paths are concentrated in a region of size $σ/\eps^{1/4}$ around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a t…
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