Two-parameter Martingales Research Articles

Littlewood–Paley–Rubio De Francia Inequality for the Two-Parameter Walsh System

A version of Littlewood–Paley–Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $$ {I}_k={I}_k^1\times {I}_k^2 $$ in ℤ+ × ℤ+ and a family of functions fk with Walsh spectrum inside Ik the following is true: $$ {\left\Vert \sum_k{f}_k\right\Vert}_{L^p}\le {C}_p{\left\Vert {\left(\sum_k{\left|{f}_k\right|}^2\right)}^{1/2}\right\Vert}_{L^p},\kern2em 1<p\le 2, $$ where Cp does not depend on the choice of rectangles {Ik} or functions {fk}. The arguments are based on the atomic theory of two-parameter martingale Hardy spaces. In the course of the proof, a two-parameter version of Gundy’s theorem on the boundedness of operators taking martingales to measurable functions is formulated, which might be of independent interest.

Newton’s method for nonlinear stochastic wave equations

Abstract We consider nonlinear stochastic wave equations driven by time-space white noise. The existence of solutions is proved by the method of successive approximations. Next we apply Newton’s method. The main result concerning its first-order convergence is based on Cairoli’s maximal inequalities for two-parameter martingales. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

Properties of set-valued integrals and set-valued stochastic equations driven by two-parameter martingales

In the paper we study properties of Aumann's stochastic integral driven by a two-parameter increasing process and set-valued Itô's integral with respect to the two-parameter martingale. Both types of integrals are understood as set-valued processes. Next, the existence, uniqueness and convergence properties of solutions to set-valued stochastic integral equations with respect to such integrators are investigated. We present new types of such equations that generalize those studied earlier. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in single-valued deterministic and stochastic cases.

Atomic subspaces of $${L_1}$$ L 1 -martingale spaces

We introduce and study some new martingale spaces \({\mathcal{H}_r}\) between martingale Hardy spaces \({H_1}\) and \({L_1}\) for \({0 < r \leqq \infty}\). We investigate the dual spaces of \({\mathcal{H}_r}\) for \({1 < r \leqq \infty}\) and some embeddings for \({0 < r \leqq \infty}\). As applications, we obtain the atomic decomposition of \({L_1}\)-martingales, and measure the distance from \({L_1}\)-martingale space to the Hardy space \({H_1}\). We also obtain some similar results for two-parameter martingales when the \({\sigma}\)-algebra filtration \({\mathcal{F} = ({\mathcal{F}}_{n,m}, \, (n, \, m) \in {\mathbb{N}}^{2})}\) is generated by finitely many atoms.

Two-Parameter Fuzzy-Valued Stochastic Integrals and Equations

This article is concerned with notions of fuzzy-valued stochastic integrals driven by two-parameter martingales and increasing processes. We present their main properties and formulate next two-parameter fuzzy-valued stochastic integral equations. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.

Set-Valued Stochastic Integrals and Equations with Respect to Two-Parameter Martingales

This article is concerned with notions of set-valued stochastic integrals driven by two-parameter martingales and increasing processes. We investigate their main properties and we consider next multivalued stochastic integral equations in the plane. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.

Regularities of local times of two-parameter martingales

We prove that, as Wiener functionals, local times of two-parameter martingales belong to some fractional Sobolev spaces over the Wiener space and, therefore, exist (2, ∞)-quasi surely.

Exponential inequalities for two-parameter martingales

In this paper, we establish an exponential estimate for the tail probability of the supremum of a two-parameter continuous martingale vanishing on the axes, assuming that the quadratic variation is bounded.

Atomic Decompositions for Two-Parameter Vector-Valued Martingales and Two-parameter Vector-valued Martingale Spaces

In this paper atomic decompositions for two-parameter vector-valued martingales are given. With the help of the atomic decompositions the relations between the mutual embedding of two-parameter vector-valued martingale spaces and geometric properties of Banach spaces are investigated. Our study shows that geometric properties of Banach spaces determine the embedding of martingale spaces and conversely the latter can characterize the former.

Two-parameter vector-valued martingales and geometrical properties of Banach spaces

We obtained a number of inequalities and laws of large numbers for two-parameter vector-valued martingales. In the other direction we characterizedp-smoothness andq-convexity of Banach spaces by using these inequalities and laws of large numbers for two-parameter vector-valued martingales.

Asymptotic inference for near unit roots in spatial autoregression

Asymptotic inference for estimators of $(\alpha_n, \beta_n)$ in the spatial autoregressive model $Z_{ij}(n) = \alpha_n Z_{i-1, j}(n) + \beta_n Z_{i, j-1}(n) - \alpha_n \beta_n Z_{i-1, j-1}(n) + \varepsilon_{ij}$ is obtained when $\alpha_n$ and $\beta_n$ are near unit roots. When $\alpha_n$ and $\beta_n$ are reparameterized by $\alpha_n = e^{c/n}$ and $\beta_n = e^{d/n}$, it is shown that if the "one-step Gauss-Newton estimator" of $\lambda_1 \alpha_n + \lambda_2 \beta_n$ is properly normalized and embedded in the function space $D([0, 1]^2)$, the limiting distribution is a Gaussian process. The key idea in the proof relies on a maximal inequality for a two-parameter martingale which may be of independent interest. A simulation study illustrates the speed of convergence and goodness-of-fit of these estimators for various sample sizes.

Cesàro Summability of Two-Parameter Walsh–Fourier Series

A new atomic decomposition of the two-parameter dyadic martingale Hardy spacesHpdefined by the quadratic variation is given. We introduceHp-quasi-local operators and prove that if a sublinear operatorVisHp-quasi-local and bounded fromL2toL2then it is also bounded fromHptoLp(0<p⩽1). By an interpolation theorem we get thatVis of weak type (H#1, L1) where the Hardy spaceH#1is defined by the hybrid maximal function. As an application it is shown that the maximal operator of the Cesàro means of a two-parameter martingale is bounded fromHptoLp(4/5<p⩽∞) and is of weak type (H#1, L1). So we obtain that the Cesàro means of a functionf∈H#1converge a.e. to the function in question. Finally, it is verified that if the supremum is taken over all two-powers, only, then the maximal operator of the Cesàro means is bounded fromHptoLpfor every 2/3<p⩽∞.

Inequalities and duality results with respect to two-parameter strong martingales

In the mar t ingale theory usually the following three mart ingale Hardy spaces are considered: the spaces Hp resp. Hp s resp. H i (0 < p < oc) generated by the maximal funct ion f* resp. by the quadrat ic variation S(f) resp. by s(f) called the condit ional quadrat ic variation wi th respect to (Svn_]). For one-parameter mart ingales it was proved by Davis ([9], p = 1) and by BURKHOLDER and GUNDY ([4], [6], 1 < p < oc) tha t H~ is equivalent to H i . Fur thermore , GARSlA [11] and HERZ [13], [14] proved tha t the duals of H~', H~ and H i (1 < p < oc) are BMO~, BM02 and H~ (1/p + 1/q = 1), respectively. The equivalence of the BMOq (1 < q < ~ ) spaces was shown also by GARSlA [11] and HERZ [13]. We recall tha t the BMO~ and BMOq norms are defined by [[f[[BMO; := sup [[(En[f En-lf]q)l/ql[~ hEN and IlfllBMoq := sup I I (En l f Enflq)l/qll~, hEN respectively, where En denotes the condit ional expectat ion operator with respect to ~-n. Some of these results are known for two-parameter mart ingales, too. The B u r k h o l d e r G u n d y inequality was proved by METRAUX [20]. In case the stochastic basis is regular, Davis' inequality was verified by BROSSARD [2], [3], while for general stochastic basis, it is yet unknown. Up to this time the B~/IOq spaces have been introduced only (see BERNARD [i], WEISZ [29]), nevertheless, the BMO~ spaces have not been considered as yet. The

Duality results and inequalities with respect to Hardy spaces containing function sequences

With the help of two-parameter martingales and strong martingales Hardy spaces consisting of adapted function sequences are considered. The Hardy spaces generated by the square and by the conditional square functions and their dual spaces are investigated. An inequality due to Stein and Lepingle is extended to two parameters.

$U$-Statistic Processes: A Martingale Approach

For i.i.d. data $X_1,\cdots, X_n$ and a kernel $h$, the associated $U$-statistic process is defined as $U_n (u, v) = \frac{1}{n(n-1)} \sum_{1\leq i\neq j\leq n} h(X_i, X_j)1_{\{X_i\leq u,X_j\leq \nu\}}.$ Variants of these processes occur, for example, in the representation of the product-limit estimator of a lifetime distribution for censored/truncated data or in trimmed $U$-statistics. We derive an almost sure representation of $U_n$ under weak moment assumptions on $h$. Proofs rely on a proper decomposition of the remainder term into strong two-parameter martingales.

On integral representations of two-parameter martingales

On integral representations of two-parameter martingales

An application of two-parameter martingales in harmonic analysis

An application of two-parameter martingales in harmonic analysis

A class of two-parameter stochastic integrators

Let M be a continuous square integrable two-parameter martingale. Then the quadratic i-variations [M]i appear as integrators of terms of the second differential order in Ito's formula, whereas terms of the third differential order are described by mixed variations Ni which behave like [M]i in parameter direction i and like M in the complementary direction. We prove that both [M]sup:i$esup and Nsup:i$esup,=l,2, are stochastic integrators the integrals of which are defined on some vector space of 1- resp. 2-previsible processes. On one hand, this result shows that non-continuous previsible processes are integrable and is therefore basic for an Ito formula for non-continuous two-parameter martingales. On the other hand, the way it is derived may give a hint what multi-parameter semimartingales (martingale-like processes) are

Stochastic integrals of point processes and the decomposition of two-parameter martingales

Let M be a square integrable martingale indexed by [0, 1] 2 with respect to a filtration which possesses the property of conditional independence. Assume that M has trajectories which are continuous for approach from the right upper quadrant and possess limits for the remaining three. M can have three kinds of jumps. A point t is a 0-jump if Δ t M = lim s↑ t [ M t − M ( t 1, s 2) − M ( s 1, t 2) + M s ] ≠ 0, a 1-jump if Δ t M = 0 and lim s 1↑ t 1 [ M t − M ( s 1, t 2) ] ≠ 0. Analogously, 2-jumps are defined. With the 0-jumps associate the two-parameter point process μ M which assigns unit point mass to nontrivial ( t, Δ t M), with the 1-jumps the one-parameter point process μ 1 M which puts unit mass to nontrivial ( t 1, Δ t 1 M (+, 1)), and with the 2-jumps a corresponding μ 2 M . We define stochastic integrals with respect to the compensated μ M , μ i M , i = 1, 2, with the help of which we can describe the jump components associated with the respective jumps in the orthogonal decomposition of M by discontinuous and continuous parts.

R-variations for two-parameter continuous martingales and itô's formula

Different kinds of variations associated with a continuous two-parameter martingale bounded in L 4 are studied. As an application a “compact” Itô formula and a version of a two-parameter Tanaka formula are proved.