Strong Martingale Research Articles

On Parameter-Measurability of the Stochastic Integral with Respect to the Two-Parameter Strong Martingale

This paper contains sufficient conditions of measurability with respect to a parameter of the limit of a sequence of random fields with trajectories in ${\mathbb D}$. Measurability with respect to a parameter of the quadratic variation of the two-parameter strong martingale was proved. We obtain sufficient conditions of measurability with respect to a parameter of the stochastic integral with respect to the two-parameter strong martingale.

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.

Set indexed strong martingales and path independent variation

In this paper, I extend the result that any strong martingale shows path independent variation, which was introduced by Cairoli and Walsh for the plane, to set indexed strong martingales, and I also analyse the connection between path independent variation and independent increment set indexed processes.

The proportional hazards regression model with staggered entries: A strong martingale approach

The proportional hazards regression model, when subjects enter the study in a staggered fashion, is studied. A strong martingale approach is used to model the two-time parameter counting processes. It is shown that well-known univariate results such as weak convergence and martingale inequalities can be extended to this two-dimensional model. Strong martingale theory is also used to prove weight convergence of a general weighted goodness-of-fit process and its weighted bootstrap counterpart.

Spatial autoregression model: strong consistency

Let ( α ̂ n, β ̂ n) denote the Gauss–Newton estimator of the parameter ( α, β) in the autoregression model Z ij = αZ i−1, j + βZ i, j−1 − αβZ i−1, j−1 + ε ij . It is shown in an earlier paper that when α=β=1, {n 3/2( α ̂ n−α, β ̂ n−β)} converges in distribution to a bivariate normal random vector. A two-parameter strong martingale convergence theorem is employed here to prove that n r( α ̂ n−α, β ̂ n−β)→ 0 ̄ almost surely when r< 3 2 .

Strong Martingales: Their Decompositions and Quadratic Variation

Set-indexed strong martingales and a form of predictability for set-indexed processes are defined. Under a natural integrability condition, we show that any set-indexed strong submartingale can be decomposed in the Doob–Meyer sense. A form of predictable quadratic variation for square-integrable set-indexed strong martingales is defined and sufficient conditions for its existence are given. Under a conditional independence assumption, these reduce to a simple moment condition and, if the strong martingale has continuous sample paths, the resulting quadratic variation can be approximated in the L 2-sense by sums of conditional expectations of squared increments.

Martingales on Random Sets and the Strong Martingale Property

Let $X$ be a process defined on an optional random set. The paper develops two different conditions on $X$ guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that $X$ is the restriction of some special semimartingale $Z$ with canonical decomposition $Z=M+A$. The first condition, which is both necessary and sufficient, is an absolute continuity condition on $A$. Under additional hypotheses, the existence of a martingale extension can be characterized by a strong martingale property of $X$. Uniqueness of the extension is also considered.

A martingale characterization of the set-indexed Brownian motion

We characterize a Brownian motion indexed by a semilattice of sets, using the theory of set-indexed martingales: a square integrable continuous set-indexed strong martingale is a Brownian motion if and only if its compensator is deterministic and continuous.

Estimates on moments of the solutions to stochastic differential equations with respect to martingales in the plane

Let M = { M z , z ϵ R 2 +} be a two-parameter strong martingale, A be a two-parameter increasing process on R 2 + = [0, + ∞) × [0, + ∞). Consider the following stochastic differential equations in the plane: X z = X 0 + ∞ R z a(ξ,X) dM ξ + ∞ R z b(ξ,X) dA ξ for z ϵ R 2 +. Under some assumptions on the coefficients a, b and the integrators M, A, we prove the existence and uniqueness of solutions for the equations, and obtain some estimates on moments of solution.

Strong approximations for multiparameter martingales

We prove a strong approximation of a strong martingale difference array by a corresponding array of independent normal variables. The rate of approximation is sufficiently quick to prove a functional central theorem and a functional law of the iterated logarithm for these martingales. In the course of the proof we develop an exponential inequality for rectangular sums of strong martingale difference arrays

A martingale characterization of the set-indexed poisson process

A martingale characterization of the set-indexed Poisson process is proved: a set-indexed point process is a Poisson process if and only if there exists a deterministic and increasing process such that the difference is a strong martingale

Martingale Convergence of Generalized Conditional Expectations

We prove that the martingale convergence theorem for generalized conditional expectations in von Neumann algebras holds in the weak topology without restrictions. The situation is therefore different fom the strong topology case, where there are restrictive conditions which distinguish between increasing and decreasing sequences of von Neumann algebras. Moreover, known couterexamples show that in the decreasing case the strong martingale convergence theorem might not hold.

On a problem in the statistics of strong martingales in the plane

We study the problem of finding an explicit form of the conditional mathematical expectation M{θ|f }, whereθ is a function of an unknown strong martingale on the plane andf is theσ-algebra generated by the observed strong martingale.

Compensator approximations for point processes on the plane

If a planar point process N is adapted to a filtration endowed with the property of conditional independence [denoted by (F4)], a compensator A* may be defined such that N — A* is a strong martingale. It is shown that, under certain conditions, there exists a discrete strong LA-approximation for A*. This approximation is used to find an upper bound for the total variation distance between N(s, t) and a Poisson random variable in terms of A*. If (F4) is not satisfied, a bound may be expressed in terms of two one-dimensional compensators. Examples are given, and it is shown that the A* bound may improve on bounds available through existing one-dimensional techniques.

A Characterization of the Spatial Poisson Process and Changing Time

Watanabe proved that if $X_t$ is a point process such that $X_t - t$ is a martingale, then $X_t$ is a Poisson process and this result was generalized by Bremaud for doubly stochastic Poisson processes. Here we define two-parameter point processes and extend this property without needing the strong martingale condition. Using this characterization, we study the problem of transforming a two-parameter point process into a two-parameter Poisson process by means of a family of stopping lines as a time change. Nualart and Sanz gave conditions in order to transform a square integrable strong martingale into a Wiener process. Here, we do the same for the Poisson process by a similar method but under more general conditions.

Local times of continuous N-parameter strong martingales

Let M be a 4 N-integrable, real-valued continuous N-parameter strong martingale. By extending Itô-type formulas for M to a function whose 2 Nth derivative is Dirac's δ-distribution, Tanaka-type formulas for M are obtained. They represent local time of M with respect to occupation time scaled by the N-fold product of the Stieltjes measure defined by the quadratic variation of M and its kth derivatives in space, where k ≤ N − 1. Applications of Doob's and Burkholder's inequalities give continuity properties: space time continuity for local time, space continuity for the derivatives. In case N is even, for the continuity of the ( N − 1)st derivative an additional condition on M is needed which may have a relation to the existence of local times of M w.r.t. different occupation time scales.

A stochastic calculus for continuous N-parameter strong martingales

Let M be a 4 N-integrable, real-valued continuous N-parameter strong martingale. Burkholder's inequalities prove to be an adequate tool to control the quadratic oscillations of M and the integral processes associated with it (i.e. multiple 1-stochastic integrals with respect to M and its quadratic variation) such that a 1-stochastic calculus for M can be designed. As the main results of this calculus, several Ito-type formulas are established: one in terms of the integral processes associated with M, another one in terms of the so-called ‘variations’, i.e. stochastic measures which arise as the limits of straightforward and simple approximations by Taylor's formula; finally, a third one which is derived from the first by iterated application of a stochastic version of Green's formula and which may be the strong martingale form of a prototype for general martingales.

An Extension of Stochastic Integrals in the Plane

For a Wiener process with a two-dimensional parameter $\{W_z, z \in R_+^2\}$, four types of stochastic integrals: $\int \phi dW, \int \psi dW dW, \int \psi dW dz, \int \psi dz dW$, have been defined under the condition $$E \int \phi^2 dz < \infty \quad \text{and} \quad E \int \psi^2 dz dz' < \infty.$$ The main purpose of this note is to extend the definition of these stochastic integrals by replacing $E(\bullet) < \infty$ with $(\bullet) < \infty$ a.s. in these conditions. Our results are in fact even more general, allowing $W$ to be replaced by a strong martingale with appropriate properties.

ECONOMETRIC FORECASTING OF COMMON STOCK INVESTMENT RETURNS: A NEW METHODOLOGY USING FUNDAMENTAL OPERATING DATA

The Journal of FinanceVolume 27, Issue 2 p. 371-380 Session Topic: Contributed Papers I ECONOMETRIC FORECASTING OF COMMON STOCK INVESTMENT RETURNS: A NEW METHODOLOGY USING FUNDAMENTAL OPERATING DATA Walt McKibben, Walt McKibben School of Business Administration, The University of Western Ontario.Search for more papers by this author Walt McKibben, Walt McKibben School of Business Administration, The University of Western Ontario.Search for more papers by this author First published: May 1972 https://doi.org/10.1111/j.1540-6261.1972.tb00966.xCitations: 5 Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume27, Issue2May 1972Pages 371-380 RelatedInformation