Theory Of Stochastic Integration Research Articles

About A. N. Kolmogorov's work on the entropy of dynamical systems

In the fall of 1957 A. N. Kolmogorov started lecturing on the theory of dynamical systems and supervised a seminar on the same theme at the Mechanical–Mathematical Department of Moscow State University. He began his lectures with the theory of systems with a pure point spectrum, which he approached from a probabilistic point of view. This approach, undoubtedly, has many advantages. In the seminar we studied Ito's theory of multiple stochastic integrals and, under the supervision of A. N. Kolmogorov, I. V. Girsanov constructed an example of a Gaussian dynamical system with a simple continuous spectrum. At one of the meetings of the seminar, still before the advent of entropy, Kolmogorov suggested a proof of an assertion, which today would read: the unitary operator induced by a K-automorphism has a countable Lebesgue spectrum. At this point Kolmogorov was studying Shannon's theory of information and the concept of the capacity of functional spaces. Judging by his well known article, we can say that the first of these, and all that is related to it, played a big role in the development of information theory in our country. The investigation of capacity is connected with Kolmogorov's work on Hilbert's 13th problem and was summarized in his well-known survey co-authored by V. M. Tihomirov.

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.

A non-commutative martingale representation theorem for non-Fock quantum Brownian motion

A non-commutative theory of stochastic integration is constructed in which the integrators are the components of the quantum Brownian motion with non-unit variance. Unlike the unit variance (Fock) case, there is a Kunita-Watanabe type representation theorem for processes which are martingales with respect to the generated filtration.

Quantum Ito's formula and stochastic evolutions

Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.

Stochastic integration w.r.t. continuous local martingales

In this note we develop the theory of stochastic integration w.r.t. continuous local martingales using a simple time change technique. We allow progressively measurable integrands.

Quasi-free quantum stochastic integrals for the CAR and CCR

A theory of quantum martingales and quantum stochastic integrals in quasi-free representations of the CAR and CCR is presented. For the CAR, the results generalize some of those developed in Barnett, Streater, and Wilde ( J. Funct. Anal. 48 (1982), 172–212, J. London Math. Soc. 27 (1983), 373–384) and for the CCR, the results contain the standard Itô theory of stochastic integration with respect to Brownian motion as a special case.

Stochastic Integration and $L^p$-Theory of Semimartingales

If $X$ is a bounded left-continuous and piecewise constant process and if $Z$ is an arbitrary process, both adapted, then the stochastic integral $\int X dZ$ is defined as usual so as to conform with the sure case. In order to obtain a reasonable theory one needs to put a restriction on the integrator $Z$. A very modest one suffices; to wit, that $\int X_n dZ$ converge to zero in measure when the $X_n$ converge uniformly or decrease pointwise to zero. Daniell's method then furnishes a stochastic integration theory that yields the usual results, including Ito's formula, local time, martingale inequalities, and solutions to stochastic differential equations. Although a reasonable stochastic integrator $Z$ turns out to be a semimartingale, many of the arguments need no splitting and so save labor. The methods used yield algorithms for the pathwise computation of a large class of stochastic integrals and of solutions to stochastic differential equations.

Nonparametric estimation based on censored observations of a Markov renewal process

Uniform consistency and weak convergence is proved of estimators of the transition probabilities of an arbitrary finite state space Markov renewal process, based on n independent and identically distributed “right censored” realizations of the process. The approach uses the theory of stochastic integrals and counting processes. It is shown how the results may be extended to the non-identically distributed case and to general censorship under suitable conditions.

Nonparametric Inference for a Family of Counting Processes

Let $\mathbf{B} = (N_1, \cdots, N_k)$ be a multivariate counting process and let $\mathscr{F}_t$ be the collection of all events observed on the time interval $\lbrack 0, t\rbrack.$ The intensity process is given by $\Lambda_i(t) = \lim_{h \downarrow 0} \frac{1}{h}E(N_i(t + h) - N_i(t) \mid \mathscr{F}_t)\quad i = 1, \cdots, k.$ We give an application of the recently developed martingale-based approach to the study of $\mathbf{N}$ via $\mathbf{\Lambda}.$ A statistical model is defined by letting $\Lambda_i(t) = \alpha_i(t)Y_i(t), i = 1, \cdots, k,$ where $\mathbf{\alpha} = (\alpha_1, \cdots, \alpha_k)$ is an unknown nonnegative function while $\mathbf{Y} = (Y_1, \cdots, Y_k),$ together with $\mathbf{N},$ is a process observable over a certain time interval. Special cases are time-continuous Markov chains on finite state spaces, birth and death processes and models for survival analysis with censored data. The model is termed nonparametric when $\mathbf{\alpha}$ is allowed to vary arbitrarily except for regularity conditions. The existence of complete and sufficient statistics for this model is studied. An empirical process estimating $\beta_i(t) = \int^t_0 \alpha_i(s) ds$ is given and studied by means of the theory of stochastic integrals. This empirical process is intended for plotting purposes and it generalizes the empirical cumulative hazard rate from survival analysis and is related to the product limit estimator. Consistency and weak convergence results are given. Tests for comparison of two counting processes, generalizing the two sample rank tests, are defined and studied. Finally, an application to a set of biological data is given.

Filtering, Prediction and Smoothing for Counting Process Observations, a Martingale Approach

In this paper we consider the filtering, prediction and smoothing problem for counting process observations. The method used is the theory of martingales and stochastic integrals. We define two equations in terms of martingales, forming the stochastic system. The observation equation associates the counting process with a rate process. The second equation expresses in general terms the evolution of the unobserved process that is to be estimated. We consider the filtering, prediction and smoothing problem using the least squares error criterion. The stochastic differential equations for the optimal estimates are derived. We discuss the resulting filter and give some examples.

A converse to some inequalities and approximations in the theory of Stieltjes and stochastic integrals, and for nth derivatives

A converse to some inequalities and approximations in the theory of Stieltjes and stochastic integrals, and for nth derivatives

An Inequality in the Theory of Stochastic Integrals

An Inequality in the Theory of Stochastic Integrals

Martingales with Independent Increments

In section one, $L_1$ bounds are obtained for transforms of discrete parameter martingales having independent increments. Section two develops the theory of quadratic variation for continuous parameter martingales having independent increments. An application of the results of these two sections to the theory of stochastic integrals appears in section three.

Correlation Problems in the Study of the Brightness of the Milky way.

view Abstract Citations References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Correlation Problems in the Study of the Brightness of the Milky way. Ramakrishnan, Alladi ; Srinivasan, S. K. Abstract The problem of the fluctuations in the brightness of the Milky Way is studied from a new point of view-the correlation between the contributions to brightness from different parts of the astrophysical system. Besides extending the results obtained by Mttnch, this paper points out the relevance to the phenomenological theory of stochastic integration recently developed by one of us (R). Publication: The Astrophysical Journal Pub Date: May 1956 DOI: 10.1086/146183 Bibcode: 1956ApJ...123..479R full text sources ADS |