Alexander Shaposhnikov, Lukas Wresch
We construct a series of stochastic differential equations of the form $dX_t = b(t, X_t) dt + dB_t$ which exhibit nonuniqueness in the path-by-path sense while having a unique adapted solution in the sense of stochastic processes, i.e. pathwise uniqueness holds.
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We extend the Yamada-Watanabe condition for pathwise uniqueness to stochastic differential equations with jumps, in the special case where small jumps are summable.
A new proof of pathwise uniqueness for SDEs with Sobolev diffusion and integrable drift term is introduced by extending a method from E. Fedrizzi and F. Flandoli (Pathwise uniqueness and continuous dependence of SDEs with non-regular drift, Stochastics 83 (2011), pp. 241--257) to the case of nonconstant diffusion.
We give a simple proof that in a Lipschitz domain in two dimensions with Lipschitz constant one, there is pathwise uniqueness for the Skorokhod equation governing reflecting Brownian motion.
We show pathwise uniqueness for a class of degenerate Itô-SDE among all of its weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Consequently, by the Yamada-Watanabe Theorem and a weak existence result, the pathwise unique solutions can be shown to be strong and to exist. The main tools to show pathwise uniqueness are inequalities associated with maximal functions and a Krylov type estimate derived from elliptic regularity and uniqueness in law.
For $α\in (0,1)$, we consider stochastic differential equations driven by one-sided stable processes of order $α$: \[dX_t= ϕ(X_{t-})\ dZ_t.\] We prove that pathwise uniqueness holds for this equation under the assumptions that $ϕ$ is continuous, non-decreasing and positive on $\R$. A counterexample is given to show that the positivity of $ϕ$ is crucial for pathwise uniqueness to hold.
Based on the weak existence and weak uniqueness, we study the pathwise uniqueness of the solutions for a class of one-dimensional stochastic differential equations driven by pure jump processes. By using Tanaka's formula and the local time technique, we show that there is no gap between the strong uniqueness and weak uniqueness when the coefficients of the Poisson random measures satisfy a suitable condition
We prove pathwise uniqueness for solutions of parabolic stochastic pde's with multiplicative white noise if the coefficient is Hölder continuous of index $γ>3/4$. The method of proof is an infinite-dimensional version of the Yamada-Watanabe argument for ordinary stochastic differential equations.
Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. For pathwise uniqueness, the drift is assumed to be Dini-continuous in the state variable, while the diffusion must be Lipschitz, continuous in time and uniformly nondegenerate. The setting is classical McKean--Vlasov, that is, coefficients of the equation are represented as integrals over the marginal distributions of the process.
Here we study stochastic differential equations with a reflecting boundary condition. We provide sufficient conditions for pathwise uniqueness and non-explosion property of solutions in a framework admitting non-Lipschitz continuous coefficients and non-smooth domains.
We show the pathwise uniqueness for stochastic partial differential equation driven by a cylindrical $α$-stable process with Hölder continuous drift, thus obtaining an infinite dimensional generalization of the result of Priola [Osaka J. Math., 2012] in the case $H=\mathbb{R}^d$. The proof is based on an infinite dimensional Kolmogorov equation with non-local operator.
Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
研究具有赫尔德连续系数的退化随机微分方程的路径唯一性问题，利用马利瓦因紧致性准则构造强解，通过对偶山田-渡边论证及弱唯一性得出路径唯一性。
We consider the existence and pathwise uniqueness of the stochastic heat equation with a multiplicative colored noise term on IR^d for d greater or equal to 1. We focus on the case of non-Lipschitz noise coefficients and singular spatial noise correlations. In the course of the proof a new result on Hoelder continuity of the solutions near zero is established.
We consider one-dimensional stochastic differential equations with jumps in the general case. We introduce new technics based on local time and we prove new results on pathwise uniqueness and comparison theorems. Our approach are very easy to handled and don't need any approximation approach. Similar equations without jumps were studied in the same context by \cite{Le Gall}, \cite{Ouknine} and others authors. As an application we get a new condition on the pathwise uniqueness for the solutions to stochastic differential equations driven by a symmetric stable Lévy processes.
Pathwise uniqueness holds for the Skorokhod stochastic differential equation in $C^{1+γ}$ domains in $\mathbb{R}^d$ for $γ>1/2$ and $d\geq3$.
In this paper we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.
We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $α$ with drift and diffusion coefficients $b,σ$. When $α\in (1,2)$, we investigate pathwise uniqueness for this equation. When $α\in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $α\in (0,1)$ or $α\in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $σ…
A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.
We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is Hölder continuous of index $γ>1-\fracη{2(η+1)}$. Here $η\in(0,1)$ is a constant that defines the spatial regularity of the noise.
We study a class of stochastic differential equations with non-Lipschitzian coefficients.A unique strong solution is obtained and a large deviation principle of Freidln-Wentzell type has been established.
We introduce a new method of proving pathwise uniqueness, and we apply it to the degenerate stochastic differential equation \[dX_t=|X_t|^α dW_t,\] where $W_t$ is a one-dimensional Brownian motion and $α\in(0,1/2)$. Weak uniqueness does not hold for the solution to this equation. If one restricts attention, however, to those solutions that spend zero time at 0, then pathwise uniqueness does hold and a strong solution exists. We also consider a class of stochastic differential equations with reflection.
Our aim in this paper is to establish some strong stability properties of a solution of a stochastic differential equation driven by a fractional Brownian motion for which the pathwise uniqueness holds. The results are obtained using Skorokhod's selection theorem.
On the pathwise uniqueness of stochastic 2D Euler equations with Kraichnan noise and $L^p$-data
本文将随机二维Euler方程涡量形式解的路径唯一性从$p>3/2$推广到任意$p>1$，去除了Coghi和Maurelli结果中对$L^p$指数的限制。
Pathwise uniqueness for stochastic heat and damped equations with Hölder continuous drift
本文在不要求结构条件成立的前提下，证明了无穷维随机微分方程的逐路径唯一性，并覆盖最高到三维的随机热方程、一维随机阻尼波动方程及最高到三维的随机Euler-Bernoulli阻尼梁方程。
We study the solutions of the stochastic heat equation with multiplicative space-time white noise. We prove a comparison theorem between the solutions of stochastic heat equations with the same noise coefficient which is Hölder continuous of index $γ>3/4$, and drift coefficients that are Lipschitz continuous. Later we use the comparison theorem to get sufficient conditions for the pathwise uniqueness for solutions of the stochastic heat equation, when both the white noise and the drift coefficients are Hölder continuous.
We consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term. Here, the noise is white in time and colored in q dimensional space ($q \geq 1$) with a singular correlation kernel. The noise coefficient is Hölder continuous in the solution. We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the Hölder coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) who treat the case of str…
In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with Hölder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ΔX_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $α$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $β$-Hölder continuous for $1<α<2$ and $0<β<1$. For $G\equiv0$ and $H(x)=x^β$, in Mytnik (2002) a weak solution t…
Strong solutions to degenerate SDEs and uniqueness for degenerate Fokker-Planck equations
本文通过受限Yamada-Watanabe定理，结合Fokker-Planck方程的存在性与新的受限路径唯一性结果，证明了退化SDE强解的存在性，并给出了Fokker-Planck方程的新唯一性结论。
Pathwise Uniqueness for Multiplicative Young and Rough Differential Equations Driven by Fractional Brownian Motion
该研究证明了任意维度下由赫斯特参数$H\in(1/3,1)$的分数布朗运动驱动的乘性随机微分方程的轨道唯一性，通过结合随机平均估计与改进版随机缝合引理，放宽了波动率系数的赫尔德连续阶数要求，改进了已有经典结果。
Pathwise uniqueness for singular stochastic Volterra equations with Hölder coefficients
本文证明了一类由布朗运动驱动、带奇异核和Hölder连续扩散系数的一维随机Volterra方程的道路唯一性，进而得到唯一强解的存在性。
Strong existence, pathwise uniqueness and chains of collisions in infinite Brownian particle systems
研究一类无限维奇异随机微分方程的强存在性与路径唯一性，其状态空间为特定锥。通过参数$p$控制碰撞局部时间分裂比例，在一定条件下建立解的性质，关键在于路径唯一性与碰撞链有限性的联系，并运用多种论证和技术进行证明。
A BSDEs approach to pathwise uniqueness for stochastic evolution equations
本文利用无穷维正倒向随机微分方程系统，证明了一类Hilbert空间中漂移项为Hölder连续的随机发展方程的路径唯一性，并证明解对初值具有Lipschitz依赖性。
We investigate pathwise uniqueness for the squared Bessel and Cox-Ingersoll-Ross processes with additional reflection term that is multiplied by some real number strictly between minus one and one. The reflection term is the symmetric local time of the corresponding processes at a deterministic time dependent curve.
Pathwise uniqueness by noise for singular stochastic PDEs
本文通过噪声正则化方法，解决了奇异随机偏微分方程中具有微分形式漂移项的路径唯一性问题，为流体动力学和相分离模型等提供了新的唯一性结果。
We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $α$-stable Lévy processes with values in $\R^d$ having a bounded and $β$-Hölder continuous drift term. We assume $β> 1 - \fracα{2} $ and $α\in [ 1, 2)$. The proof requires analytic regularity results for associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.
We prove the Ito-Tanaka formula and the existence of pathwise stochastic integrals for a wide class of Gaussian processes. Motivated by financial applications, we define the stochastic integrals as forward-type pathwise integrals introduced by Föllmer and as pathwise generalized Lebesgue-Stieltjes integrals introduced by Zähle. As an application, we illustrate the importance of Ito-Tanaka formula for pricing and hedging of financial derivatives.
We extend some results about Föllmer's pathwise Itô calculus that have only been derived for continuous paths to càdlàg paths with quadratic variation. We study some fundamental properties of pathwise Itô integrals with respect to càdlàg integrators, especially associativity and the integration by parts formula. Moreover, we study integral equations with respect to pathwise Itô integrals. We prove that some classes of integral equations, which can be explicitly solved in the usual stochastic calculus, can also be solved within the framework of Föllmer's calculus.
We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity, i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic i…
Existence and uniqueness for singular stochastic differential equations with piecewise well-behaved coefficients
本文研究了一维广义随机微分方程中存在性和唯一性问题，特别是在分布漂移和退化、可能不连续的扩散系数下的情况。提出路径唯一性条件和粘贴方法，为广义随机微分方程提供了首个显式粘贴定理。
We study stochastic volatility models in which the volatility process is a positive continuous function of a continuous Volterra stochastic process. We state some pathwise large deviation principles for the scaled log-price.
We develop a Fourier approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate, and it leads to recursive algorithms for the calculation of pathwise stochastic integrals, both of Itô and of Stratonovich type. We apply it to solve stochastic differential equations in a pathwise manner.
White-noise case stochastic heat equation was derived from Dawson-Watanabe superprocess. The pathwise uniqueness of their solutions with non-Lipschitz coefficients has attracted wide attention and in particular, the square-root coefficient case was listed as a long standing open problem in the famous literature (i.e. Perkins \cite{Perkins02} p217.) This short note gives an affirmative answer to this open problem. Our idea is using the $φ_k(x)$ function constructed by Yamada-Watanable to prove that \beqlb \lab{key} \E_μ| X_{t}(z)-Y_{t}(z) | = \lim_{k \ra \infty} \E_μ|\int_{\r}\frac{1}{\sqrt{ε_k…
In this paper we establish the pathwise Taylor expansions for random fields that are "regular" in the spirit of Dupire's path-derivatives \cite{Dupire}. Our result is motivated by but extends the recent result of Buckdahn-Bulla-Ma \cite{BBM}, when translated into the language of pathwise calculus. We show that with such a language the pathwise Taylor expansion can be naturally carried out to any order and for any dimension, and it coincides with the existing results when reduced to these special settings. More importantly, the expansion can be both "forward" and "backward" (i.e., the temporal …
We study the existence and uniqueness of the stochastic viscosity solutions of fully nonlinear, possibly degenerate, second order stochastic pde with quadratic Hamiltonians associated to a Riemannian geometry. The results are new and extend the class of equations studied so far by the last two authors.
Pathwise central limit theorem and moderate deviations via rough paths for SPDEs with multiplicative noise
本研究借助粗糙路径理论提出通用框架，推导小乘性线性噪声扰动的随机偏微分方程的路径中心极限定理与中偏差原理，给出最优收敛速度，并应用于多类典型随机偏微分方程。
We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We investigate the degeneracy phenomenon induced by directly controlling the coefficient of the noise term, and propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As an application, we use pathwise stochastic control in the context of stochastic filtering to construct filters which are robust to parameter uncertainty, demo…
Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness imply strong existence for stochastic differential equations with jumps.
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…
Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs: a multidimensional Yamada-Watanabe approach
该研究将Yamada-Watanabe假设推广至多维，在不构造Lyapunov函数的情况下，证明了带随机非Lipschitz系数的McKean-Vlasov SDE解的稳定性、唯一性及确定性情形下强解的存在性，还显式算出Lyapunov指数。
We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the usual (stochastic) local times a.s. for paths of a continuous semimartingale. We establish pathwise version of the Itô-Tanaka, change of variables and change of time formulae. We provide equivalent conditions for existence of pathwise local time. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choic…
In this article we study existence of pathwise stochastic integrals with respect to a general class of $n$-dimensional Gaussian processes and a wide class of adapted integrands. More precisely, we study integrands which are functions that are of locally bounded variation with respect to all variables. Moreover, multidimensional Itô formula is derived.
The Tanaka equation $dX_t={\operatorname{sign}}(X_t)\,dB_t$ is an example of a stochastic differential equation (SDE) without strong solution. Hence pathwise uniqueness does not hold for this equation. In this note we prove that if we modify the right-hand side of the equation, roughly speaking, with a strong enough additive noise, independent of the Brownian motion B, then the solution of the obtained equation is pathwise unique.
We consider the classical estimation problem of an unknown drift parameter within classes of nondegenerate diffusion processes. Using rough path theory (in the sense of T. Lyons), we analyze the Maximum Likelihood Estimator (MLE) with regard to its pathwise stability properties as well as robustness toward misspecification in volatility and even the very nature of the noise. Two numerical examples demonstrate the practical relevance of our results.
We show that a pathwise stochastic integral with respect to fractional Brownian motion with an adapted integrand $g$ can have any prescribed distribution, moreover, we give both necessary and sufficient conditions when random variables can be represented in this form. We also prove that any random variable is a value of such integral in some improper sense. We discuss some applications of these results, in particular, to fractional Black--Scholes model of financial market.
We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.
For any real-valued stochastic process $X$ with cŕdlŕg paths we define non-empty family of processes which have locally finite total variation, have jumps of the same order as the process $X$ and uniformly approximate its paths on compacts. The application of the defined class is the definition of stochastic integral with semimartingale integrand and integrator as a limit of pathwise Lebesgue-Stieltjes integrals. This construction leads to the stochastic integral with some correction term (different from the Stratonovich integral). We compare the obtained result with classical results of Wong-…
For any real-valued stochastic process X with càdlàg paths we define non-empty family of processes, which have finite total variation, have jumps of the same order as the process X and uniformly approximate its paths: This allows to decompose any real-valued stochastic process with càdlàg paths and infinite total variation into a sum of uniformly close, finite variation process and an adapted process, with arbitrary small amplitude but infinite total variation. Another application of the defined class is the definition of the stochastic integral with respect to the process X as a limit of path…
In this article we study the existence of pathwise Stieltjes integrals of the form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation functions $f$ and Hölder continuous random processes $X$ and $Y$. We discuss a notion of sufficient variability for the process $X$ which ensures that the paths of the composite process $t \mapsto f(X_t)$ are almost surely regular enough to be integrable. We show that the pathwise integral can be defined as a limit of Riemann-Stieltjes sums for a large class of discontinuous evaluation functions of locally finite variation, and provide new es…
Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs in arbitrary moments
该研究针对带随机/确定性系数的McKean-Vlasov随机微分方程，基于伊藤公式推导非利普希茨漂移下的解的存在唯一性、矩估计及带显式指数的稳定性，无需依赖李雅普诺夫函数。
This paper is devoted to the study of hyperbolic systems of linear partial differential equations perturbed by a Brownian motion. The existence and uniqueness of solutions are proved by an energy method. The specific features of this class of stochastic partial differential equations are highlighted and the comparison with standard existence results for SPDEs is discussed. The small perturbations problem is studied and a large deviation principle is stated. A pathwise approximation result, similar to the stochastic differential equations case, is established, with an application to a support t…
Pathwise uniqueness in infinite dimension under weak structure conditions
该研究在弱结构条件下，证明了无穷维希尔伯特空间中一类带线性有界算子的SDE弱温和解对初值利普希茨依赖，即轨道唯一性成立，该条件可覆盖一维随机阻尼波动方程及三维内随机阻尼Euler–Bernoulli Beam方程等双曲情形。
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