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Search Results: 1 - 10 of 1826 matches for " Leonid Galtchouk "
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About Stochastic Calculus in Presence of Jumps at Predictable Stopping Times  [PDF]
Leonid Galtchouk
Journal of Mathematical Finance (JMF) , 2016, DOI: 10.4236/jmf.2016.63035
Abstract: In this paper, some basic results of stochastic calculus are revised using the following observation: For any semimartingale, the series of jumps at predictable stopping times converges a.s. on any finite time interval, whereas the series of jumps at totally inaccessible stopping times diverges. This implies that when studying random measures generated by jumps of a given semimartingale, it is naturally to define separately a random measure μ generated by the jumps at totally inaccessible stopping times and an other random measure π generated by the jumps at predictable stopping times. Stochastic integrals f ·μμpare well defined for suitable functions f, where μp is the predictable compensator of μ. Concerning the stochastic integral h·π, it is well defined without any compensating of the integer valued measure π.
Adaptive nonparametric estimation in heteroscedastic regression models. Part 2: Asymptotic efficiency
Leonid Galtchouk,Serguey Pergamenshchikov
Mathematics , 2008,
Abstract: The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper (2007) for estimation of unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk. It means that the asymptotic quadratic risk for this procedure coincides with a sharp lower bound.
Adaptive nonparametric estimation in heteroscedastic regression models. Part 1: Sharp non-asymptotic Oracle inequalities
Leonid Galtchouk,Serguey Pergamenshchikov
Mathematics , 2008,
Abstract: An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle inequality) is constructed.
Adaptive sequential estimation for ergodic diffusion processes in quadratic metric. Part 2: Asymptotic efficiency
Leonid Galtchouk,Serguey Pergamenshchikov
Mathematics , 2008,
Abstract: Asymptotic efficiency is proved for the constructed in part 1 procedure, i.e. Pinsker's constant is found in the asymptotic lower bound for the minimax quadratic risk. It is shown that the asymptotic minimax quadratic risk of the constructed procedure coincides with this constant.
On asymptotic normality of sequential LS-estimates of unstable autoregressive processes
Leonid Galtchouk,Victor Konev
Mathematics , 2008,
Abstract: For estimating the unknown parameters in an unstable autoregressive AR(p), the paper proposes sequential least squares estimates with a special stopping time defined by the trace of the observed Fisher information matrix. The limiting distribution of the sequential LSE is shown to be normal for the parameter vector lying both inside the stability region and on some part of its boundary in contrast to the ordinary LSE. The asymptotic normality of the sequential LSE is provided by a new property of the observed Fisher information matrix which holds both inside the stability region of AR(p) process and on the part of its boundary. The asymptotic distribution of the stopping time is derived.
Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression via model selection
Leonid Galtchouk,Serguey Pergamenshchikov
Mathematics , 2008,
Abstract: The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper, 2007, for estimating a unknown nonparametric regression. We prove that this procedure is asymptotically efficient for a quadratic risk, i.e. the asymptotic quadratic risk for this procedure coincides with the Pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimators.
On some estimates for bounded submartingales and the shift inequality
Leonid Galtchouk,Isaac Sonin
Mathematics , 2010,
Abstract: It is well known that if a submartingale $X$ is bounded then the increasing predictable process $Y$ and the martingale $M$ from the Doob decomposition $% X=Y+M$ can be unbounded. In this paper for some classes of increasing convex functions $f$ we will find the upper bounds for $\lim_n\sup_XEf(Y_n)$, where the supremum is taken over all submartingales $(X_n),0\leq X_n\leq 1,n=0,1,...$. We apply the stochastic control theory to prove these results.
Uniform concentration inequality for ergodic diffusion processes observed at discrete times
Leonid Galtchouk,Serguei Pergamenchtchikov
Mathematics , 2011,
Abstract: In this paper a concentration inequality is proved for the deviation in the ergodic theorem in the case of discrete time observations of diffusion processes. The proof is based on the geometric ergodicity property for diffusion processes. As an application we consider the nonparametric pointwise estimation problem for the drift coefficient under discrete time observations.
Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models
Leonid Galtchouk,Serguei Pergamenchtchikov
Statistics , 2010,
Abstract: An adaptive nonparametric estimation procedure is constructed for heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (oracle inequality) is obtained
Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression
Leonid Galtchouk,Serguei Pergamenchtchikov
Statistics , 2010,
Abstract: The paper deals with asymptotic properties of the adaptive procedure proposed in the author paper, 2007, for estimating an unknown nonparametric regression. %\cite{GaPe1}. We prove that this procedure is asymptotically efficient for a quadratic risk, i.e. the asymptotic quadratic risk for this procedure coincides with the Pinsker constant which gives a sharp lower bound for the quadratic risk over all possible estimates
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