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1
Content available remote Estimation by Stable Motions and its Applications
Das Anirban, Denker Manfred, Levina Anna, Tabacu Lucia
Probability and Mathematical Statistics
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2023
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Vol. 43, Fasc. 1
23--39
EN
We propose a family of confidence intervals for nonparametric moment estimators if the observations have large or infinite variances. The theoretical underpinnings which guarantee the soundness of the method are demonstrated. Extensive numerical simulations show its superiority over bootstrap and normal approximation and its wide applicability. Finally, a confidence interval to estimate the coupling strength in neuronal networks is proposed.
2
Content available remote Stochastic complex integrals associated with homogeneous independently scattered random measures on the line
Yamamuro Kouji
Probability and Mathematical Statistics
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2019
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Vol. 39, Fasc. 1
219--236
EN
Complex integrals associated with homogeneous independently scattered random measures on the line are discussed. Theorems corresponding to Cauchy’s theorem and the residue theorem are given. Furthermore, the converse of Cauchy’s theorem is discussed.
3
Content available remote On pathwise stochastic integration with respect to semimartingales
Łochowski R. M.
Probability and Mathematical Statistics
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2014
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Vol. 34, Fasc. 1
23--43
EN
For any real-valued stochastic process X with càdlàg 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). Using properties of a functional called truncated variation we compare the obtained result with classical results of Wong-Zakai and Bichteler on pathwise stochastic integration.
4
Content available A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes
Alpay D., Kipnis A.
Opuscula Mathematica
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2013
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Vol. 33, no. 3
395--417
EN
Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida’s white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula.
5
Content available White noise based stochastic calculus associated with a class of Gaussian processes
Alpay D., Attia H., Levanony D.
Opuscula Mathematica
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2012
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Vol. 32, no. 3
401-422
EN
Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula.
6
Content available remote Sharp norm inequalities for stochastic integrals in which the integrator is a nonnegative supermartingale
Osękowski A.
Probability and Mathematical Statistics
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2009
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Vol. 29, Fasc. 1
29--42
EN
The paper is devoted to sharp inequalities between moments of nonnegative supermartingales and their strong subordinates. Analogous estimates hold true for stochastic integrals with respect to a nonnegative right-continuous supermartingale. Similar inequalities are established for smooth functions on Euclidean domains.
7
Content available remote Lévy processes and stochastic integrals in Banach spaces
Applebaum D.
Probability and Mathematical Statistics
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2007
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Vol. 27, Fasc. 1
75--88
EN
We review infinite divisibility and Lévy processes in Banach spaces and discuss the relationship with notions of type and cotype. The Lévy-Itô decomposition is described. Strong, weak and Pettis-style notions of stochastic integral are introduced and applied to construct generalised Ornstein-Uhlenbeck processes
8
Content available remote Sharp norm inequalities for martingales and their differential subordinates
Osękowski A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 4
373-385
EN
Suppose ƒ = (ƒn), g = (gn) are martingales with respect to the same filtration, satisfying |ƒn-ƒn-i| ≤ |gn -gn-1|, n = 1,2,..., with probability 1. Under some assumptions on ƒo, go and an additional condition that one of the processes is nonnegative, some sharp inequalities between the pth norms of ƒ and g, 0 < p < ∞, are established. As an application, related sharp inequalities for stochastic integrals and harmonic functions are obtained.
9
Content available remote Existence of solutions of set-valued Ito equation
Motyl J.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 4
419-430
EN
In the paper, the problem of existence of solution to the stochastic functional inclusion x [...] is investigated under some assumption of the dissipativity type on set-valued functions A and B.
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