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Small Noise Perturbations in Multidimensional Case

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 Added by Andrey Pilipenko
 Publication date 2021
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and research's language is English




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In this paper we study zero-noise limits of $alpha -$stable noise perturbed ODEs which are driven by an irregular vector field $A$ with asymptotics $% A(x)sim overline{a}(frac{x}{leftvert xrightvert })leftvert xrightvert ^{beta -1}x$ at zero, where $overline{a}>0$ is a continuous function and $beta in (0,1)$. The results established in this article can be considered a generalization of those in the seminal works of Bafico cite% {Ba} and Bafico, Baldi cite{BB} to the multi-dimensional case. Our approach for proving these results is inspired by techniques in cite% {PP_self_similar} and based on the analysis of an SDE for $tlongrightarrow infty $, which is obtained through a transformation of the perturbed ODE.



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The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered.
In this paper we solve a selection problem for multidimensional SDE $d X^varepsilon(t)=a(X^varepsilon(t)) d t+varepsilon sigma(X^varepsilon(t)), d W(t)$, where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H$. It is assumed that $X^varepsilon(0)=x^0in H$, the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H$, and the limit ODE $d X(t)=a(X(t)), d t$ does not have a unique solution. We show that if the drift pushes the solution away of $H$, then the limit process with certain probabilities selects some extreme solutions to the limit ODE. If the drift attracts the solution to $H$, then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.
135 - D. Crisan , J. Diehl , P. K. Friz 2012
In the late seventies, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff] pointed out that it would be natural for $pi_t$, the solution of the stochastic filtering problem, to depend continuously on the observed data $Y={Y_s,sin[0,t]}$. Indeed, if the signal and the observation noise are independent one can show that, for any suitably chosen test function $f$, there exists a continuous map $theta^f_t$, defined on the space of continuous paths $C([0,t],mathbb{R}^d)$ endowed with the uniform convergence topology such that $pi_t(f)=theta^f_t(Y)$, almost surely; see, for example, Clark [In Communication Systems and Random Process Theory (Proc. 2nd NATO Advanced Study Inst., Darlington, 1977) (1978) 721-734, Sijthoff & Noordhoff], Clark and Crisan [Probab. Theory Related Fields 133 (2005) 43-56], Davis [Z. Wahrsch. Verw. Gebiete 54 (1980) 125-139], Davis [Teor. Veroyatn. Primen. 27 (1982) 160-167], Kushner [Stochastics 3 (1979) 75-83]. As shown by Davis and Spathopoulos [SIAM J. Control Optim. 25 (1987) 260-278], Davis [In Stochastic Systems: The Mathematics of Filtering and Identification and Applications, Proc. NATO Adv. Study Inst. Les Arcs, Savoie, France 1980 505-528], [In The Oxford Handbook of Nonlinear Filtering (2011) 403-424 Oxford Univ. Press], this type of robust representation is also possible when the signal and the observation noise are correlated, provided the observation process is scalar. For a general correlated noise and multidimensional observations such a representation does not exist. By using the theory of rough paths we provide a solution to this deficiency: the observation process $Y$ is lifted to the process $mathbf{Y}$ that consists of $Y$ and its corresponding L{e}vy area process, and we show that there exists a continuous map $theta_t^f$, defined on a suitably chosen space of H{o}lder continuous paths such that $pi_t(f)=theta_t^f(mathbf{Y})$, almost surely.
We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain probabilities $p_+$ and $p_-=1-p_+$, respectively. We propose a space-time transformation that reduces the investigation of the original problem to the study of the exact growth rate of a solution to a certain SDE with self-similar noise. This problem is interesting in itself. Moreover, the probabilities $p_+$ and $p_-$ coincide with probabilities that the solution of the transformed equation converges to $+infty$ or $-infty$ as $ttoinfty,$ respectively.
In this paper we prove the existence of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<1/2. Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a local time variational calculus argument.
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