Theory Of Stochastic Integration Research Articles

On Information Processes in Statistical Experiments with Distributed Observations

Using stochastic integration theory in topological vector spaces general formulas for the Hellinger processes are derived. Feynman–Kac type formulas are obtained for the related Hellinger integrals in terms of the Hellinger processes and the geometric mean measures. The expected logarithmic utility from data, characterized as the Shannon information, is also considered.

Lévy area of Wiener processes in Banach spaces

The goal of this paper is to construct canonical Lévy area processes for Banach space valued Brownian motions via dyadic approximations. The significance of the existence of canonical Lévy area processes is that a (stochastic) integration theory can be established for such Brownian motions (in Banach spaces). Existence of flows for stochastic differential equations with infinite dimensional noise then follows via the results of Lyons and Lyons and Qian. This investigation involves a careful analysis on the choice of tensor norms, motivated by the applications to infinite dimensional stochastic differential equations.

Stochastic analysis, rough path analysis and fractional Brownian motions

In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition.

Algebraic theory of product integrals in quantum stochastic calculus

Motivated by the search for solutions of the quantum Yang–Baxter equation, an algebraic theory of quantum stochastic product integrals is developed. The product integrators are formal power series in an indeterminate h whose coefficients are elements of the Lie algebra ℒ labelling the usual integrators of a many-dimensional quantum stochastic calculus. The product integrals are also formal power series in h, whose coefficients are finite iterated additive stochastic integrals which act on the exponential domain in the Fock space of the calculus and which represent elements of the universal enveloping algebra 𝒰 of ℒ. They obey a multiplication rule suggested by the quantum Itô product formula, and are characterized among all such formal power series by a grouplike property.

Outcomes of epidemic models with general infection and removal rate functions at certain stopping times

This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting byx,ythe numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate βxyxyand infectives are removed at rate γxyy, where the ratio βxy/ γxycan be written in the formq(x+y) /xp(x) for appropriate functionsp,q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel–Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.

Outcomes of epidemic models with general infection and removal rate functions at certain stopping times

This paper aims to show how certain known martingales for epidemic models may be derived using general techniques from the theory of stochastic integration, and hence to extend the allowable infection and removal rate functions of the model as far as possible. Denoting by x, y the numbers of susceptible and infective individuals in the population, then we assume that new infections occur at rate β xy xy and infectives are removed at rate γ xy y, where the ratio β xy / γ xy can be written in the form q(x+y) / xp(x) for appropriate functions p,q. Under this condition, we find equations giving the distribution of the number of susceptibles remaining in the population at appropriately defined stopping times. Using results on Abel–Gontcharoff pseudopolynomials we also derive an expression for the expectation of any function of the number of susceptibles at these times, as well as considering certain integrals over the course of the epidemic. Finally, some simple examples are given to illustrate our results.

Robust option replication for a Black‐Scholes model extended with nondeterministic trends

Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and Scholes model. This memory or nondeterministic trend behavior is often seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension of the Black and Scholes model by adding a term to the original Wiener term involving a smoother process which accounts for these effects. The problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing replicating portfolio for a European option without any further knowledge of the extension and that, as a consequence, the classical concept of volatility needs to be re‐interpreted.

The Itô and Stratonovich integrals for stochastic differential equations with poisson white noise

The relationship between the Itô and the Stratonovich integrals used for solving stochastic differential equations with Gaussian white noise is well known. However, this relationship seems to be less clear when dealing with stochastic differential equations driven by Poisson white noise. It is shown that there is no difference between the Itô and the Stratonovich integrals used to define the solution of stochastic differential equations with Poisson white noise. This result is in disagreement with findings of some previous publications but in agreement with the classical definition of the Itô and Stratonovich integrals. Intuitive considerations, arguments based on the theory of stochastic integrals with semimartingales, and examples are used to prove and demonstrate the claimed equality of the Itô and Stratonovich integrals.

The Black-Scholes pricing formula in the quantum context.

A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects. The lack of simultaneous observability of relevant variables and the interference of attempted observation with the values of these variables represent such effects. These characteristics have been noted by traders and economists and appear intrinsic to market dynamics. This explanation is explored here in terms of a corresponding generalization of the Wiener process and its role in the Black-Scholes-Merton theory. The differentiability of the Wiener process as a sesquilinear form on a dense domain in the Hilbert space of square-integrable functions over Wiener space is shown and is extended to the quantum context. This provides a basis for a corresponding generalization of the Ito theory of stochastic integration. An extension of the Black-Scholes option pricing formula to the quantum context is deduced.

Queueing systems on a circle

Consider a ring on which customers arrive according to a Poisson process. Arriving customers drop somewhere on the circle and wait there for a server who travels on the ring. Whenever this server encounters a customer, he stops and serves the customer according to an arbitrary service time distribution. After the service is completed, the server removes the client from the circle and resumes his journey. We are interested in the number and the locations of customers that are waiting for service. These locations are modeled as random counting measures on the circle. Two different types of servers are considered: The polling server and the Brownian (or drunken) server. It is shown that under both server motions the system is stable if the traffic intensity is less than 1. Furthermore, several earlier results on the configuration of waiting customers are extended, by combining results from random measure theory, stochastic integration and renewal theory.

A Continuous Polling System with General Service Times

Consider a ring on which a server travels at constant speed. Customers arrive on the ring according to a Poisson process, at locations independently and uniformly distributed over the circle. Whenever the server encounters a customer, he stops and serves the client according to a general service time distribution. After the service is completed, the server removes the customer from the ring and resumes his round. The model is analyzed by means of point processes and regenerative processes in combination with some stochastic integration theory. This approach clarifies the analysis of the continuous polling model and provides the means for further generalizations. For every time $t$, the locations of customers that are waiting for service and the positions of clients that have been served during the last tour of the server are represented by random counting measures. These measures converge in distribution as $t \rightarrow \infty$. A recursive expression for the Laplace functionals of the limiting random measures is found, from which the corresponding $k$th moment measures can be derived.

Useful martingales for stochastic storage processes with Lévy input

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.

Useful martingales for stochastic storage processes with Lévy input

We apply the general theory of stochastic integration to identify a martingale associated with a Lévy process modified by the addition of a secondary process of bounded variation on every finite interval. This martingale can be applied to queues and related stochastic storage models driven by a Lévy process. For example, we have applied this martingale to derive the (non-product-form) steady-state distribution of a two-node tandem storage network with Lévy input and deterministic linear fluid flow out of the nodes.

Stochastic integration on the Cuntz algebra O∞

We develop a theory of non-commutative stochastic integration with respect to the creation and annihilation process on the full Fock space over L 2( R ). Our theory largely parallels the theories of non-commutative stochastic Itô integration on Boson and Fermion Fock space as developed by R. Hudson and K. R. Parthasarathy. It provides the first example of a non-commutative stochastic calculus which does not depend on the quantum mechanical commutation or anticommutation relations, but it is based on the theory of reduced free products of C ∗-algebras by D. Voiculescu. This theory shows that the creation and annihilation processes on the full Fock space over L 2( R ), which generate the Cuntz algebra O ∞, can be interpreted as a generalized Brownian motion. We should stress the fact that in contrast to the other theories of stochastic integration our integrals converge in the C ∗-norm on O ∞, i.e., uniformly rather than in some state-dependent strong operator topology or (non-commutative) L 2-norm.

An inference procedure for a simple epidemic model: a martingale approach

This paper studies various estimators the infection rate of the simple stochastic epidemic model using the theory of counting processes, martingales, and stochastic integrals. A class of estimators for the infection rate have been obtained which includes the maximum likelihood estimator. Explicit expressions are given for these estimators which involve only simple computation. In the case of incomplete observation of the epidemic, estimators are available by choosing an appropriate weight function. Asymptotic results are given. Asymptotic efficiencies among these estimators are considered.

Stochastic Integration on the Full Fock Space with the Help of a Kernel Calculus

We develop a stochastic integration theory with respect to creation, annihilation and gauge operators on the full Fock space. This is done by using a kernel representation for a large class of bounded operators on the full Fock space. It is shown that the kernels form a Banach algebra. Having established the definition of processes and stochastic integrals we go on to prove an Ito formula and use this for examining stochastic evolutions and constructing dilations of special completely positive semigroups. Explicit solutions of the corresponding stochastic differential equations are given.

Stochastic product integral w.r.t. infinite dimensional semimartingale:ii–uniform operator topology case

While stochastic product integration w.r.t finite rank Hilbert Schmidt operator (K2- ) valued semimartingales has received reasonable attention, there is no work, to date, dealing with infinite dimensional semimartingales. We initiate here a theory of stochastic product integration of K2- predictable processes w.r.t the exponential of a K2- Brownian motion. To understand the estimations involved, we start with a finite rank Brownian motion β while keeping the integrand in K2 and then we continue with infinite dimellsioilal Brownian motion. In order to apply our stochastic product integral to solve Doleans–Dade – Protter type linear stochastic equations, we also need to, and therefore, define the Ito integral in the space HS. We finally give the stochastic product integral construction of the solution of a linear stochastic equation. All this is done in the uniform operator topology case. The more difficult unbounded operator case is the subject matter of the final part of this series

Pathwise stochastic integration and applications to the theory of continuous trading

We develop a pathwise construction of stochastic integrals relative to continuous martingales. The key to the construction is an almost sure approximation technique which associates a sequence of finitely generated filtrations (“skeleton filtrations”) and a sequence of simple stochastic processes (“skeleton processes”) to a given continuous martingale and its underlying filtration (which is also assumed to be “continuous”). The pathwise stochastic integral can then be defined along such a “skeleton approximation” and almost-sure convergence follows from a certain completeness property of the skeleton approximation. The limit is the pathwise stochastic integral and it agrees with the integral obtained through the usual Itô approach. The pathwise stochastic integration theory is applied to the analysis of stochastic models of security markets with continuous trading. A convergence theory for continuous market models with exogenously given equilibrium prices is obtained which enables one to view an idealized economy (i.e. a continuous market model) as an almost sure limit of “real-life” economies (i.e. finite market models). Since finite market models are well understood, this convergence sheds light on features of the continuous market model, such as completeness.

Estimating Jointly System and Component Reliabilities Using a Mutual Censorship Approach

Let $F$ denote the life distribution of a coherent structure of independent components. Suppose that we have a sample of independent systems, all having the same structure. Each system is continuously observed until it fails. For every component in each system, either a failure time or a censoring time is recorded. A failure time is recorded if the component fails before or at the time of system failure; otherwise a censoring time is recorded. We introduce a method for finding estimates for $F(t)$, quantiles and other functionals of $F$, based on the censorship of the component lives by system failure. We use the theory of counting processes and stochastic integrals to obtain limit theorems that enable the construction of confidence intervals for large samples. Our approach extends and gives a novel application of censoring methodology.

Sur l'intégration stochastique par rapport à une martingale hilbertienne de carré intégrable

We give a complete characterization of the operator-valued processes which are integrable, in the sense of [6], with respect to a fixed Hilbertvalued square integrable martingale M. This characterisation allows to complete the theory of stochastic integration with respect to Hilbert-valued martingales. In particular, we give a construction of the process ≪M, N≫ (predictable compensator of M⊗N), as well as a Hilbert-valued version of the Kunita-Watanabe inequality. Finally, we deal with the distributivity of the stochastic integral X·M with respect to the martingale M: this property can be usefully applied to obtain a simple proof of a representation theorem of Gal'tchouk-Metivier.