Quadratic Variation Process Research Articles

Generalized Integration and Stochastic ODEs

Stochastic forward integrals for processes more general than semimartingales are shown to exist, generalized forms of Ito-Wentzell formula and covariation formula are proved, and one-dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved. This generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift.

Moderate deviations for martingale differences and applications to φ -mixing sequences

For a R d -valued sequence of martingale differences { m k } k S 1 , we obtain a moderate deviation principle for the sequence of partial sums { Z n ( t ) 1 ~ k =1 [ nt ] m k / b n , t ] [0,1]}, in the space of càdlàg functions equipped with the Skorohod topology, under the following conditions: a Chen-Ledoux type condition, an exponential convergence in probability of the associated quadratic variation process of the martingale, and a condition of "Lindeberg" type. For the small jumps of Z n (·), we apply the general result of Puhalskii [Puhalskii, A. (1994). "Large deviations of semimartingales via convergence of the predictable characteristics". Stoch. Stoch. Rep. , 49 , pp. 27-85]. Following the method of Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280] and Arcones [Arcones, A. (1999). "The large deviation principle for stochastic processes", Submitted for publication], we prove that the large jumps part of Z n (·) is negligible in the sense of the moderate deviations. One can regard our result as an extension to martingale differences, of the beautiful characterization of moderate deviations for i.i.d.r.v. case due to Chen [Chen, X. (1991). "The moderate deviations of independent vectors in Banach space". Chin. J. Appl. Probab. Stat. , 7 , pp. 124-32] and Ledoux [Ledoux, M. (1992). "Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi". Ann. Inst. H. Poincaré , 28 , pp. 267-280]. Using the Gordin [Gordin, M.I. (1969). "The central limit theorem for stationary processes". Soviet Math. Dokl. , 10 , pp. 1174-1176] decomposition, the martingale result is applied to prove the moderate deviation principle for a wide class of stationary { -mixing sequences of random variables.

A relationship between Brownian motions with opposite drifts via certain enlargements of the Brownian filtration

which is the quadratic variation process of the corresponding geometric Brownian motion (μ) = { (μ) = exp ( (μ)) ≧ 0}. These Wiener functionals play important roles in many fields; mathematical finance (see, e.g., Yor [33], [34], Geman-Yor [10], [11], Leblanc [17]), diffusion processes in random environment (Comtet-Monthus [5], Comtet-Monthus-Yor [6], Kawazu-Tanaka [16]), probabilistic studies related to Laplacians on hyperbolic spaces (Gruet [12], Ikeda-Matsumoto [13]) and so on. The reader will find more related topics and references in [37]. However, even for fixed , the joint law of ( (μ) (μ)) or, equivalently, that of ( = (0) ) due to the Cameron-Martin relationship between (μ) and is fairly complicated, although it is quite tractable (see [33]). As an example of the description of this law, we present the following conditional Laplace transform ([2], [13], [18]): [ exp ( − 2

Stochastic calculus with respect to continuous finite quadratic variation processes

The quadratic variation of a continuous process (when it exists) is defined through a regularization procedure. A large class of finite quadratic variation processes is provided, with a particular emphasis on Gaussian processes. For such processes a calculus is developed with application to the study of some stochastic differential equations.

On the Convergence of Dirichlet Processes

For a given weakly convergent sequence {Xn} of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the Xn values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of {Xn} is a Dirichlet process, too.

A version of Pitman's 2 M — X theorem for geometric Brownian motions

We show that geometric Brownian motion with parameter μ, i.e., the exponential of linear Brownian motion with drift μ, divided by its quadratic variation process is a diffusion process. Taking logarithms and an appropriate scaling limit, we recover the Rogers-Pitman extension to Brownian motion with drift of Pitman's representation theorem for the three-dimensional Bessel process. Time inversion and generalized inverse Gaussian distributions play crucial roles in our proofs.

Covariation de convolution de martingales

We evaluate the quadratic variation process in the sense of [5] and [6], which coincides with the classical quadratic variation in the case of semimartingales, for processes of the type {X t = ∫ 0 t G(t, s) dM(s), t ≥ 0}, where {G(t, s), t ≥ s ≥ 0} is a continuous deterministic function and M is a continuous square integrable martingale. Moreover, X admits an orthogonal representation. If G(t, s) = G (t − s), where G is a real function, then X coincides with a convolution of martingales.

The linear-Quadratic Control Problem Revisited

A long-run, average-cost, stochastic, linear-quadratic control problem that incorporates different time scales is considered. The system dynamics and the cost functional are modeled with the help of a locally square-integrable semimartingale process with independent increments and the corresponding predictable quadratic variation process. The solution of the control problem is given in terms of the solution of certain system of algebraic and differential Riccati equations. The model considered here embodies as particular cases the "traditional" continuous-time and discrete-time linear quadratic control problems, and is applicable, for example, to certain hybrid control problems that cannot be treated using existing control methods.

Spectral characterization of the optional quadratic variation process

In this paper we show how the periodogram of a semimartingale can be used to characterize the optional quadratic variation process.

On solutions of stochastic differential equations with drift

We study stochastic differential equations of the formdX t=σ(X t)dMt+b(Xt)d t whereM is a continuous local martingale and stands for its quadratic variation process. The conditions introduced by Engelbert and Schmidt, which ensure the existence and uniqueness in law of solutions of SDE's driven by the Wiener process without drift (or with generalized drift) are shown to be no longer valid.

A functional central limit theorem for the quadratic variation of a class of gaussian random fields

AbstractWe prove a central limit theorem for the quadratic variation process of some Lévy‐Baxter‐type Gaussian random fields.

Stochastic integration without tears

Stochastic integration theory is developed by axiomatizing the concept of semi-martingale in terms of a continuity property of integrals of simple functions. Using this approach, stochastic integration for left-continuous integrands, the change of variables formula and properties of the quadratic variation process are established in an elementary way. Submartingale decomposition theorems are introduced at a late stage in order to extend the results to general predictable integrands.

On the quadratic variation process of a continuous Martingale

In this article we give a simple proof of the existence of the quadratic variation process of a continuous local martingale by providing an explicit expression for it.