Meyer-Emery inequalities for norms of stochastic integrals with a parameter

We consider stochastic integrals with a parameter in the components. Inequalities for the functional norms of stochastic integrals are derived, generalizing the known inequalities of M. Emery [1] and P.-A. Meyer [2] to triples of spaces connected by the multiplication operation.Translated from Issledovaniya po Prikladnoi Matematike, Kazan', No. 14, pp. 134–143, 1987.     

Martingales and stochastic integrals in the theory of continuous trading

This paper develops a general stochastic model of a frictionless security market with continuous trading. The vector price process is given by a semimartingale of a certain class, and the general stochastic integral is used to represent capital gains. Within the framework of this model, we discuss the modern theory of contingent claim valuation, including the celebrated option pricing formula of Black and Scholes. It is shown that the security market is complete if and only if its vector price process has a certain martingale representation property. A multidimensional generalization of the Black-Scholes model is examined in some detail, and some other examples are discussed briefly.     

On Bernstein type inequalities for stochastic integrals of multivariate point processes

In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).     

Weak convergence of stochastic integrals related to counting processes

Probability Theory and Related Fields -     

Estimate for stochastic integrals

Lithuanian Mathematical Journal -     

Product and moment formulas for iterated stochastic integrals (associated with Lévy processes)

In this paper, we obtain explicit product and moment formulas for products of iterated integrals generated by families of square integrable martingales associated with an arbitrary Lévy process. We propose a new approach applying the theory of compensated-covariation stable families of martingales. Our main tool is a representation formula for products of elements of a compensated-covariation stable family, which enables us to consider Lévy processes, with both jumps and Gaussian part.     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

Multiple stochastic integrals constructed by special expansions of products of the integrating stochastic processes

The paper deals with problems of constructing multiple stochastic integrals in the case when the product of increments of the integrating stochastic process admits an expansion as a finite sum of series with random coefficients. This expansion was obtained for a sufficiently wide class including centered Gaussian processes. In the paper, some necessary and sufficient conditions are obtained for the existence of multiple stochastic integrals defined by an expansion of the product of Wiener processes. It was obtained a recurrent representation for the Wiener stochastic integral as an analog of the Hu–Meyer formula.     

Near optimality of stochastic control in systems with unknown parameter processes

Stochastic control for systems with an unknown parameter is considered in this paper. The underlying problem is to minimize a functional subject to a system described by a singularly perturbed differential equation with an unknown parameter process driven by fast fluctuating random disturbances. This problem arises in the context of stochastic adaptive control, adaptive signal processing, and failure-prone manufacturing systems. Due to the nature of the wide-bandwidth noise processes, identifying the parameter process for eacht is very hard since the driving noise changes very rapidly. An alternative approach is used, and an auxiliary control problem is introduced to overcome the difficulties. By means of weak convergence methods and comparison control techniques, nearly optimal controls are obtained.This research was supported in part by the National Science Foundation under Grant DMS-9022139 and DMS-9224372.     

The functional iterated logarithm law for stochastic processes represented by multiple Wiener integrals

Summary In a previous paper the authors obtained a functional law of the iterated logarithm for a class of self-similar processes with stationary increments, which are represented by multiple Wiener integrals. This result is extended to a certain class of processes represented by multiple Wiener integrals which converge to with an appropriate normalization. As an application a functional log log law for nonlinear functionals of some stationary Gaussian processes is given.     

Elements of stochastic analysis for the case of Grassmann variables. I. Grassmann stochastic integrals and random processes

This paper is the first part of a study of an approach to the definition of analogs of the concepts of classical stochastic analysis such as a stochastic integral, a random process, a stochastic differential equation, etc., for the case of Grassmann variables in a certain particular situation. Analogs of stochastic integrals and random processes are studied in the first part.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 23–36, July, 1993.     

Martingales,scale functions and stochastic life annuities: a note

In this note we derive the most general conditions under which the probability distribution of a generalized stochastic life annuity can be obtained by using the scale function methodology. Our main result is that the cumulative distribution function (CDF) of the generalized stochastic life annuity will obey the partial differential equation (PDE) satisfied by the scale function whenever the underlying process can be “Markovianized”. The scale function is the mapping which converts a Markov diffusion process into a martingale. In many cases, the resulting PDE can be easily solved to yield a closed form expression for the CDF.     

Stochastic integrators indexed by a multi-dimensional parameter

Summary Multi-parameter stochastic integrators are described and classified according to directions of integrability. Sets of directions are distinguished precisely by the combinatorial dimension of corresponding fractional Cartesian products. The main theorem establishes existence of stochastic processes which are integrators in prescribed directions but not others.