The Experts below are selected from a list of 183 Experts worldwide ranked by ideXlab platform
Alejandro F Ramirez - One of the best experts on this subject based on the ideXlab platform.
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asymptotic expansion of the invariant measure for ballistic Random Walk in the low disorder regime
Annals of Probability, 2017Co-Authors: David Campos, Alejandro F RamirezAbstract:We consider a Random Walk in Random environment in the low disorder regime on ZdZd, that is, the probability that the Random Walk jumps from a site xx to a nearest neighboring site x+ex+e is given by p(e)+eξ(x,e)p(e)+eξ(x,e), where p(e)p(e) is deterministic, {{ξ(x,e):|e|1=1}:x∈Zd}{{ξ(x,e):|e|1=1}:x∈Zd} are i.i.d. and e>0e>0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in ee for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in ee for the invariant measure of Random perturbations of the simple Symmetric Random Walk in dimensions d=2d=2.
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asymptotic expansion of the invariant measure for ballistic Random Walk in the low disorder regime
arXiv: Probability, 2015Co-Authors: David Campos, Alejandro F RamirezAbstract:We consider a Random Walk in Random environment in the low disorder regime on $\mathbb Z^d$. That is, the probability that the Random Walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\epsilon \xi(x,e)$, where $p(e)$ is deterministic, $\{\{\xi(x,e):|e|_1=1\}:x\in\mathbb Z^d\}$ are i.i.d. and $\epsilon>0$ is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in $\epsilon$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in $\epsilon$ for the invariant measure of Random perturbations of the simple Symmetric Random Walk in dimensions $d=2$.
Jan Obloj - One of the best experts on this subject based on the ideXlab platform.
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two explicit skorokhod embeddings for simple Symmetric Random Walk
Stochastic Processes and their Applications, 2019Co-Authors: Jan Obloj, Xun Yu ZhouAbstract:Abstract Motivated by problems in behavioural finance, we provide two explicit constructions of a Randomized stopping time which embeds a given centred distribution μ on integers into a simple Symmetric Random Walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azema–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of μ .
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two explicit skorokhod embeddings for simple Symmetric Random Walk
arXiv: Probability, 2017Co-Authors: Jan Obloj, Xun Yu ZhouAbstract:Motivated by problems in behavioural finance, we provide two explicit constructions of a Randomized stopping time which embeds a given centered distribution $\mu$ on integers into a simple Symmetric Random Walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azema-Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of $\mu$.
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classes of measures which can be embedded in the simple Symmetric Random Walk
Electronic Journal of Probability, 2008Co-Authors: Alexander M G Cox, Jan OblojAbstract:We characterize the possible distributions of a stopped simple Symmetric Random Walk $X_\tau$, where $\tau$ is a stopping time relative to the natural filtration of $(X_n)$. We prove that any probability measure on $\mathbb{Z}$ can be achieved as the law of $X_\tau$ where $\tau$ is a minimal stopping time, but the set of measures obtained under the further assumption that $(X_{n\land \tau}:n\geq 0)$ is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on $\mathbb{Z}$. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the Chacon-Walsh (1976) and Azema-Yor (1979) embeddings and show that they lead to yet smaller sets of achievable measures. Finally, we solve explicitly the Skorokhod embedding problem constructing, for a given measure $\mu$, a minimal stopping time $\tau$ which embeds $\mu$ and which further is uniformly integrable whenever a uniformly integrable embedding of $\mu$ exists.
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Classes of Skorokhod Embeddings for the Simple Symmetric Random Walk
2006Co-Authors: Alexander Cox, Jan OblojAbstract:The Skorokhod Embedding problem is well understood when the underlying process is a Brownian motion. We examine the problem when the underlying is the simple Symmetric Random Walk and when no external Randomisation is allowed. We prove that any measure on Z can be embedded by means of a minimal stopping time. However, in sharp contrast to the Brownian setting, we show that the set of measures which can be embedded in a uniformly integrable way is strictly smaller then the set of centered probability measures: specifically it is a fractal set which we characterise as an iterated function system. Finally, we define the natural extension of several known constructions from the Brownian setting and show that these constructions require us to further restrict the sets of target laws.
David Campos - One of the best experts on this subject based on the ideXlab platform.
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asymptotic expansion of the invariant measure for ballistic Random Walk in the low disorder regime
Annals of Probability, 2017Co-Authors: David Campos, Alejandro F RamirezAbstract:We consider a Random Walk in Random environment in the low disorder regime on ZdZd, that is, the probability that the Random Walk jumps from a site xx to a nearest neighboring site x+ex+e is given by p(e)+eξ(x,e)p(e)+eξ(x,e), where p(e)p(e) is deterministic, {{ξ(x,e):|e|1=1}:x∈Zd}{{ξ(x,e):|e|1=1}:x∈Zd} are i.i.d. and e>0e>0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in ee for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in ee for the invariant measure of Random perturbations of the simple Symmetric Random Walk in dimensions d=2d=2.
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asymptotic expansion of the invariant measure for ballistic Random Walk in the low disorder regime
arXiv: Probability, 2015Co-Authors: David Campos, Alejandro F RamirezAbstract:We consider a Random Walk in Random environment in the low disorder regime on $\mathbb Z^d$. That is, the probability that the Random Walk jumps from a site $x$ to a nearest neighboring site $x+e$ is given by $p(e)+\epsilon \xi(x,e)$, where $p(e)$ is deterministic, $\{\{\xi(x,e):|e|_1=1\}:x\in\mathbb Z^d\}$ are i.i.d. and $\epsilon>0$ is a parameter which is eventually chosen small enough. We establish an asymptotic expansion in $\epsilon$ for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in $\epsilon$ for the invariant measure of Random perturbations of the simple Symmetric Random Walk in dimensions $d=2$.
Bruno Schapira - One of the best experts on this subject based on the ideXlab platform.
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capacity of the range in dimension 5
Annals of Probability, 2020Co-Authors: Bruno SchapiraAbstract:We prove a Central limit theorem for the capacity of the range of a Symmetric Random Walk on Z 5 , under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension d ≥ 5, for the probability that the ranges of two independent Random Walks intersect. The latter are then used for computing covariances of some intersection events, at the leading order.
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capacity of the range in dimension 5
arXiv: Probability, 2019Co-Authors: Bruno SchapiraAbstract:We prove a Central limit theorem for the capacity of the range of a Symmetric Random Walk on $\mathbb Z^5$, under only a moment condition on the step distribution. The result is analogous to the central limit theorem for the size of the range in dimension three, obtained by Jain and Pruitt in 1971. In particular an atypical logarithmic correction appears in the scaling of the variance. The proof is based on new asymptotic estimates, which hold in any dimension $d\ge 5$, for the probability that the ranges of two independent Random Walks intersect. The latter are then used for computing covariances of some intersection events, at the leading order.
Elena Zhizhina - One of the best experts on this subject based on the ideXlab platform.
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scaling limit of Symmetric Random Walk in high contrast periodic environment
Journal of Statistical Physics, 2017Co-Authors: Andrey Piatnitski, Elena ZhizhinaAbstract:The paper deals with the asymptotic properties of a Symmetric Random Walk in a high contrast periodic medium in $$\mathbb Z^d$$ , $$d\ge 1$$ . From the existing homogenization results it follows that under diffusive scaling the limit behaviour of this Random Walk need not be Markovian. The goal of this work is to show that if in addition to the coordinate of the Random Walk in $$\mathbb Z^d$$ we introduce an extra variable that characterizes the position of the Random Walk inside the period then the limit dynamics of this two-component process is Markov. We describe the limit process and observe that the components of the limit process are coupled. We also prove the convergence in the path space for the said Random Walk.
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large time behaviour of Symmetric Random Walk in high contrast periodic environment
arXiv: Probability, 2016Co-Authors: Andrey Piatnitski, Elena ZhizhinaAbstract:The paper deals with the asymptotic properties of a Symmetric Random Walk in a high contrast periodic medium in $\mathbb Z^d$, $d\geq 1$. We show that under proper diffusive scaling the Random Walk exhibits a non-standard limit behaviour. In addition to the coordinate of the Random Walk in $\mathbb Z^d$ we introduce an extra variable that characterizes the position of the Random Walk in the period and show that this two-component process converges in law to a limit Markov process. The components of the limit process are mutually coupled, thus we cannot expect that the limit behaviour of the coordinate process is Markov. We also prove the convergence in the path space for the said Random Walk.
