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On integral representations of two-parameter martingales
rose, 1993Summary: Sufficient conditions under which every square integrable two-parameter martingale \(N\) can be represented in the form \(N= \int_{R_z} \varphi dM+ \iint_{R_z\times R_z} \psi dM dM\), where \(M\) is a given continuous 4-integrable strong martingale, are derived.
A. Roitgarts
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Proof of existence theorems for the two-parameter martingale problem
Siberian Mathematical Journal, 1995Following \textit{D. W. Stroock} and \textit{S. R. S. Varadhan} [Commun. Pure Appl. Math. 22, 345--400 (1969; Zbl 0167.43903)], the author considers existence theorems for two-parameter martingale problem. Briefly, the measure \(P\) is called a solution of martingale problem for \((z_0,x_0,a,b)\) if \(P\{x(z,\cdot)=x_0(z)\), \(z\in\partial I_{z_0}\}=1\)
V. M. Borodikhin
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On a martingale characterization of two-parameter Wiener process
Statistics & Probability Letters, 1990Abstract A generalization of martingales, named string-martingales in this paper, is introduced for two-dimensional-parameter processes. The two-parameter standard Wiener process is then characterised as a strong-martingale with continuous paths, and an additional property which states that a sort of quadratic variation of the process is ...
E. M. Cabaña
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ON THE RATES OF CONVERGENCE IN THE CENTRAL LIMIT THEOREM FOR TWO-PARAMETER MARTINGALE DIFFERENCES
Acta Mathematica Scientia, 1996Summary: We obtain the uniform bounds on the rate of convergence in the central limit theorem for a class of two-parameter martingale difference sequences under certain conditions.
Hongwei Long
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Theory of Probability & Its Applications, 2012
The aim of this article is to establish the existence of a good version of a parametrized family of two-parameters martingales given by stochastic integrals. More precisely, the main result roughly states that, given a measurable family \(\mu(0,x,\omega)\) of \(L^2\)-martingales (where \(0\) sums some measurable space, \(x\in (\mathbb{R}_+)^2 ...
N. A. Kolodii
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The aim of this article is to establish the existence of a good version of a parametrized family of two-parameters martingales given by stochastic integrals. More precisely, the main result roughly states that, given a measurable family \(\mu(0,x,\omega)\) of \(L^2\)-martingales (where \(0\) sums some measurable space, \(x\in (\mathbb{R}_+)^2 ...
N. A. Kolodii
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Some Laws of the Iterated Logarithm for Two Parameter Martingales
Journal of Theoretical Probability, 1999The author obtains some laws of the iterated logarithm for two parameter martingales \(X=\{X_t: t\in N^2\}\) where \(N\) denotes the set of integers. Put \({\mathcal F}_t= \sigma\{X_s:s\leq t\}\), \(t\in N^2\), where the inequality \(s\leq t\) is meant to hold componentwise, and let \({\mathcal F}_1^{(m)}(0-)\) \((m=0,1,\dots)\) denote the \(\sigma ...
Jiming Jiang
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Two-Parameter Martingales and Their Properties
2013This chapter provides well-known results concerning the properties of two-parametric martingales and stochastic integration on the plane. We begin with the auxiliary chapter, which also contains some facts which are of independent interest. Our standard references for the results below are [20–24, 40, 42, 44, 47, 48, 65, 71].
Pavel S. Knopov, Olena N. Deriyeva
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Russian Mathematical Surveys, 1982
CONTENTSIntroduction § 1. General definitions § 2. Some examples § 3. Inequalities § 4. Martingales of a continuous argument § 5. The characteristics of martingales § 6.
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CONTENTSIntroduction § 1. General definitions § 2. Some examples § 3. Inequalities § 4. Martingales of a continuous argument § 5. The characteristics of martingales § 6.
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Different kinds of two-parameter martingales
Israel Journal of Mathematics, 1985Different kinds of two-parameter martingales were progressively introduced by several authors on a two-parameter stochastic basis (\(\Omega\),\({\mathcal F},P,({\mathcal F}_{st})_{s,t\geq 0}):\) weak martingales, i-martingales, strong martingales, martingales with orthogonal increments, path-invariant martingales and direction-invariant martingales ...
Merzbach, Ely, Nualart, David
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Exponential inequalities for two-parameter martingales
Statistics & Probability Letters, 2001The authors prove the following exponential inequality for two-parameter martingales: Let \(M=\{M_z, z\in\mathbb{R}^2_+\}\) be a two-parameter martingale whose quadratic variation \(([M]_z,\;z\in\mathbb{R}^2_+)\) is bounded by an increasing function \(f(z)\).
Moret, Sílvia, Nualart, David
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