Martingale Spaces Research Articles

Interpolation theorems on weighted Lorentz martingale spaces

In this paper several interpolation theorems on martingale Lorentz spaces are given. The proofs are based on the atomic decompositions of martingale Hardy spaces over weighted measure spaces. Applying the interpolation theorems, we obtain some inequalities on martingale transform operator.

VIRASORO MODULE STRUCTURE OF LOCAL MARTINGALES OF SLE VARIANTS

Martingales often play an important role in computations with Schramm–Loewner Evolutions (SLEs). The purpose of this article is to provide a straightforward approach to the Virasoro module structure of the space of local martingales for variants of SLEs. In the case of ordinary chordal SLE, it has been shown in Bauer and Bernard's Phys. Lett. B557 that polynomial local martingales form a Virasoro module. We will show for more general variants that the module of local martingales has a natural submodule [Formula: see text] that has the same interpretation as the module of polynomial local martingales of chordal SLE, but it is in many cases easy to find more local martingales than that. We discuss the surprisingly rich structure of the Virasoro module [Formula: see text] and construction of the "SLE state" or "martingale generating function" by Coulomb gas formalism. In addition, Coulomb gas or Feigin–Fuchs integrals will be shown to transparently produce candidates for multiple SLE pure geometries.

A note on absorption probabilities in one-dimensional random walk via complex-valued martingales

Let { X n , n ⩾ 1 } be a sequence of i.i.d. random variables taking values in a finite set of integers, and let S n = S n - 1 + X n for n ⩾ 1 and S 0 = 0 be a random walk on Z , the set of integers. By using the zeros, together with their multiplicities, of the rational function f ( x ) = E ( x X ) - 1 , x ∈ C , we characterize the space U of all complex-valued martingales of the form { g ( S n ) , n ⩾ 0 } for some function g : Z → C . As an application we calculate the absorption probabilities of the random walk { S n , n ⩾ 0 } by applying the optional stopping theorem simultaneously to a basis of the martingale space U. The advantage of our method over the classical approach via the Markov chain techniques (cf. Kemeny and Snell [1960. Finite Markov Chains. Van Nostrand, Princeton, NJ.]) is in the size of the matrix that is needed to be inverted. It is much smaller by our method. Some examples are presented.

SLE local martingales, reversibility and duality

We study Schramm–Loewner evolutions (SLEs) reversibility and duality using the Virasoro structure of the space of local martingales. For both problems we formulate a setup where the questions boil down to comparing two processes at a stopping time. We state algebraic results showing that local martingales for the processes have enough in common. When one has in addition integrability, the method gives reversibility and duality for any polynomial expected value.

GUNDY'S DECOMPOSITION FOR NON-COMMUTATIVE MARTINGALES AND APPLICATIONS

We provide an analogue of Gundy's decomposition for $L_1$-bounded non-commutative martingales. An important difference from the classical case is that for any $L_1$-bounded non-commutative martingale, the decomposition consists of four martingales. This is strongly related with the row/column nature of non-commutative Hardy spaces of martingales. As applications, we obtain simpler proofs of the weak type $(1,1)$ boundedness for non-commutative martingale transforms and the non-commutative analogue of Burkholder's weak type inequality for square functions. A sequence $(x_n)_{n \ge 1}$ in a normed space $\mathrm{X}$ is called 2-co-lacunary if there exists a bounded linear map from the closed linear span of $(x_n)_{n \ge 1}$ to $l_2$ taking each $x_n$ to the $n$th vector basis of $l_2$. We prove (using our decomposition) that any relatively weakly compact martingale difference sequence in $L_1 (\mathcal{M}, \tau)$ whose sequence of norms is bounded away from zero is 2-co-lacunary, generalizing a result of Aldous and Fremlin to non-commutative $L_1$-spaces.

Convergence of Riesz space martingales

We prove a martingale convergence for sub and super martingales on Riesz spaces. As a consequence we can form Krickeberg and Riesz like decompositions. The minimality of the Krickeberg decomposition yields a natural ordered lattice structure on the space of convergent martingales making this space into a Dedekind complete Riesz space. Finally we show that the Riesz space of convergent martingales is Riesz isomorphic to the order closure of the union of the ranges of the conditional expectations in the filtration. Consequently we can characterize the space of order convergent martingales both in Riesz spaces and in the setting of probability spaces.

Notes on matrix valued paraproducts

Denote by Mn the algebra of nn matrices. We consider the dyadic paraproducts b associated with Mn valued functions b, and show that the L 1 NMnO norm of b does not dominatekbk L2N' 2O!L2N' 2O uniformly overn. We also consider paraproducts associated with noncommutative martingales and prove that their boundedness on bounded noncommutative L p - martingale spaces implies their boundedness on bounded non- commutativeL q -martingale spaces for all 1<p<q<1 .

THE MINIMAL OPERATOR AND WEIGHTED INEQUALITIES FOR MARTINGALES

In this article the authors introduce the minimal operator on martingale spaces, discuss some one-weight and two-weight inequalities for the minimal operator and characterize the conditions which make the inequalities hold.

Martingale generating functions for Markov chains

The martingales on the transient states of a finite-state Markov chain belong to the linear manifold whose basis is the absorption probabilities on the absorbing states of the chain; the dimensionality of the space of martingales is therefore that of the number of absorbing states. Martingale generating functions are defined, summarizing the martingales of the Markov chain. Their use in studying the properties of the chain are exemplified for Markov chains modelling conflict processes and the simple epidemic process. Martingale generating functions for time-dependent martingales are derived in some particular cases, which assist in the description of the behaviour of the process.

B-valued martingale spaces with small index and atomic decompositions

Some atomic decomposition theorems for Banach-space-valued martingales are established. Using them, the embedding relationships between martingale spaces with small index are discussed. The results obtained here are connected closely with thep-uniform smoothness and q-uniform convexity of Banach space in which the martingales take values.

On the maximal inequalities for martingales involving two functions

Let Φ(t) and Ψ(t) be nonnegative convex functions, and let φ φ and ψ be the right continuous derivatives of Φ and Ψ, respectively. In this paper, we prove the equivalence of the following three conditions: (i) ∥f * ∥ Φ ≤ c∥f∥ Ψ , (ii) L Ψ ⊆ H Φ and (iii) ∫ t s0 φ(s)/s ds ≤ cψ(ct), ∀t > s 0 , where L Ψ and H Φ are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.

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.

Some clifford martingale spaces

We study the spacesH p, Sp, Kp, Σ p and Open image in new window of martingales with measure du=ϕd∪, where ϕ is a function with Clifford algebra values and satisfiesb p + -condition. We proved they are equivalent and isomorphic to the corresponding spaces of martingales with real measure d∪, respectively (1≤p<∞). In the end, we give a martingalef such that Open image in new window .

Two-Dimensional Conjugate Martingale Transforms

Characterizations of the two-dimensional H1, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

Real interpolation betweenB-valued martingale Hardy spaces

The interpolation spaces between Banach-space-valued martingale Hardy-spaces, between Hardy and BMO spaces are identified respectively. Some results obtained here are connected closely with the convexity and smoothness of the Banach space which the martingales take values in.

Atomic decompositions of Banach-spacevalued martingales

Several theorems for atomic decompositions of Banach-space-valued martingales are proved. As their applications, the relationship among some martingale spaces such asH α(X) andρ H α in the case 0< α⩽ are studied. It is shown that there is a close connection between the results and the smoothness and convexity of the value spaces.

Atomic Decompositions and Inequalities for Vector-Valued Discrete-Time Martingales

We consider martingales with discrete time taking values in a Banach lattice X that has UMD-property (UMD means unconditionality of martingale differences). We suppose that the UMD-lattice X consists of real-valued functions. Two notions of maximal value for such martingales are introduced (in the case of real-valued martingales these notions are the same and also coincide with the notion of usual maximal value). We also introduce the notion of quadratic variation and both usual and predictable classes of martingale spaces corresponding to maximal values and quadratic variation. The equivalence of these classes is established. In particular, Davis inequalities are proved with the help of atomic decompositions. The case of a regular stochastic basis is considered separately.

Haar Systems and Some Results on the Basis in a Martingale Space withMixed Norm

For martingale spaces with mixed norm defined with respect to diadic flow of $\sm$-algebras, we find a condition on summation characteristics, which implies no unconditional basis in these spaces (a generalization of the classical result of Pelczynski proved for $L_1$-spaces). In these spaces (under other conditions on characteristics) generalized Haar systems are considered; the test of the existence of an unconditional basis in terms of the Paley function is obtained and the convergence theorem for almost all choices of signs is proved.

On spaces of estimating functions

This paper is concerned with quasi-likelihood parameter estimation and it presents a unified picture of the relationship between the score function, quasi-score function, martingale space and general linear space. Several examples are used to show how to apply the quasi-likelihood method flexibly, based on knowledge about the observed process, through the construction of suitable orthogonal spaces.

Some results related to interpolation on hardy spaces of regular martingales

Let (Ω,F, P) be a probability space and {F n}n≥0 a regular increasing sequence of sub-σ-fields ofF. LetH 1(Ω) be the usual Hardy space ofF n-martingales. We show that the couple (H 1(Ω),L ∞(Ω)) is a partial retract of (L 1(Ω),L ∞(Ω)). It is also proved that (L p(Ω),BMO(Ω)) is a partial retract of (L p(Ω),L ∞(Ω)) for all 1<p<∞.