Ito Calculus Research Articles

First passage time of discretized plates with geometrical nonlinearity

The first passage time of discretized plate structures subjected to transversal and in-plane intensive transient loadings treated as nonstationary random processes is studied. The mean rate of various crossings for the nonstationary random response is evaluated using the definitions of Vanmarcke and Roberts. The definition of Roberts is extended to include clump size and clustering effects in similar spirit to that of Vanmarcke. Approximated first passage probabilities based on these definitions of mean rate of various crossings are obtained for some square steel plate structures whose response statistics are derived by an approach employing the Ito's calculus.

Three Books on Stochastic Integration

What do the terms and calculus connote? In the past they would refer to the Ito integral, a mathematically rigorous way to make sense of with respect to a Brownian motion. Times have changed. Now that the Ito integral has become a familiar if not ubiquitous object, stochastic integration has come to refer to an imposing panoply of abstract French probability theory, only now becoming accessible to the nonspecialist. The of today, of course, has its roots in Brownian motion. The Wiener process, the mathematical model of Brownian motion, is indeed the wellspring of much of modem probability theory, perhaps due to its triple role of martingale, strong Markov process, and Gaussian process. It is the interplay of the martingale and Markov process properties that underlies the history of integration. By developing his integral in 1944 with processes as integrands, It6 [11] was able to study multidimensional diffusions with purely probabilistic techniques, an improvement over the analytic methods of Feller. Many diffusions can be represented as solutions of systems of differential equations of the form:

On a class of the universally integrable random functions

([0, 1]).The purpose of this paper is to show that if we replace the usualIto integral by the extended Ito integral (E.I.I., for short) in the defini-tion of the quasi-martingale (1), we also obtain an analogous theorem.For the reader's convenience, we recall the basic facts concerning theE.I.I, following [6], which gives a rather systematic survey of recentdevelopments in this theory, mainly due to the Soviet school (such asDaletskii-Paramonova, Shevljakov, Skorokhod etc.).Let

Continuous-Time Red and Black: How to Control a Diffusion to a Goal

A player starts at x in (0, 1) and tries to reach 1. The process (Xt, t ≥ 0) of his positions moves according to a diffusion process (or, more generally, an Ito process) whose infinitesimal parameters μ, σ are chosen by the player at each instant of time from a set depending on his current position. To maximize the probability of reaching 1, the player should choose the parameters so as to maximize μ/σ2, at least when the maximum is achieved by bounded, measurable functions. This implies that bold (timid) play is optimal for subfair (superfair), continuous-time red-and-black. Furthermore, in superfair red-and-black, the strategy which maximizes the drift coefficient of {log Xt} minimizes the expected time to reach 1.

Itô calculus for σ-models and Yang-Mills theories

It is pointed out that the effective continuum action for σ-models and Yang-Mills theories may differ from the naive continuum action by terms of order g 2 or higher, which are nonsymmetric. The modifications are produced by a generalization of the Ito calculus to dimensions higher than one.

Information structure and equilibrium asset prices

In a continuous trading economy, it is shown that if information is revealed continuously and if agents' preferences are continuous in a certain topology, then equilibrium asset price processes must have continuous sample paths. Except for uninteresting cases, the sample paths of price processes will be of unbounded variation. In particular, if the information is generated by a Brownian motion,then equilibrium asset price processes are Ito integrals. When information is not revealed continuously, the times (which may be random) at which prices can have jumps are identified.

Explosion time of second-order Ito processes

In this paper a sufficient condition is given for the existence of the global solution as is a sufficient condition for the non-existence of the global solution of the second-order stochastic differential equation with the random disturbance of the so-called white noise.

Asymptotic normality of the continuous-time stochastic approximation algorithm

In this paper the continuous-time stochastic approximation algorithm seeking for the zero of a regression function is considered when the measurement error is a stochastic process generated by an Ito integral as the input of a linear system. The conditions are given to guarantee the asymptotic normality of the algorithm which is modified from the Robbins-Monro procedure proposed for the case where the measurement error is a process of independent increment.

Approximating Ito integrals of differential forms and geodesic deviation

Suppose X is a semimartingale on a differential manifold M with a linear connection Γ. The main purpose of this paper is to show that the „Ito integral“ (with respect to Γ) of a differential form along the path of X is the limit in probability of certain Riemann sums, constructed in a natural way using the exponential map in differential geometry. For this, we study the deviation between the stochastic development of X in the tangent space at some point, and the image of X under the inverse of the exponential map at the point.

The partial malliavin calculus and its application to non-linear filtering

In this article, the Malliavin calculus is used to derive regularity properties of the conditional distribution of one ltd process given a second Ito process. The relation between the processes involved is the usual one assumed in the study of filtering theory. The non-degeneracy which we require is stated in terms of Malliavins covariance matrix in Theorem (3.15). More practical conditions are given in Lemma (3.19) for general Tto processes and in Lemma (3.29) for diffusions. Finally, in Theorem (4.6) a “localized” version of these results is given for diffusions.

The physical implications of two forms of stochastic calculi

The physical implications of Ito and Stratonovich stochastic calculi are analysed in terms of the transformation of a Poisson multiplicative stochastic process into a Gaussian process.

Ito versus stratonovich revisited

It is shown that a digital simulation of a noise induced phase transition using an algorithm consistent with the Ito stochastic calculus is in agreement with the predictions of that theory, whereas experiments with an analogue simulator yield measured results in agreement with the predictions of the Stratonovich theory.

A finitely additive white noise approach to nonlinear filtering

An approach to nonlinear filtering theory is developed in which finitely additive white noise replaces the Wiener process in the observation process model. The important case when the signal is a Markov process independent of the noise is investigated in detail. The theory turns out to be simpler than the current theory based on the stochastic calculus. Stochastic partial differential equations are replaced by partial differential equations in which the observation (in the finitely additive set up) occurs as a parameter. Theorems on existence and uniqueness of solutions are obtained. The white noise approach has the advantage that it provides a robust solution to the filtering problem. Furthermore, the robust theory based on the Ito calculus can be recovered from the results of this paper.

The Central Limit Theorem for Stochastic Integrals with Respect to Levy Processes

Let $M$ be a symmetric independently scattered random measure on $\lbrack 0, 1\rbrack$ with control measure $m$ which is uniformly in the domain of normal attraction of a stable measure of index $p \in (0, 2\rbrack$. Let $f$ be a non-anticipating process with respect to $X(t) = M\lbrack 0, t\rbrack$ if $m$ is continuous, and a previsible process in general, satisfying $\int^1_0 E|f|^p dm < \infty$. Then the stochastic integral $\int^t_0 f dM$ can be defined as a process in $D\lbrack 0, 1\rbrack$ and is in the domain of normal attraction of a stable process of order $p$ in $D\lbrack 0, 1\rbrack$ in the sense of of weak convergence of probability measures. If $M$ is Gaussian and continuous in probability then the central limit theorem holds in $C\lbrack 0, 1\rbrack$; in particular, Ito and diffusion processes satisfy the CLT. Our main tool is an upper bound for the weak $L^p$ norm of $\sup_{0 \leq t \leq 1} |\int^t_0 f dM|$ in terms of the $L^p(P \times m)$ norm of $f$.

Environmental stochasticity of gompertz model with time delay

We consider the environmental stochasticity of the Gompertz model with time delay when the parameters are assumed to be described by correlated Gaussian white noise processes. The delay kernel is assumed to be exponentially decaying and the Fokker-Planck equation is obtained in both the Stratonovich and Ito calculi. The exact expressions for the first and second order moments of the logarithm of population size are evaluated and the stability of the system in mean as well as In mean-square is discussed.

Multiple stochastic integrals: Projection and iteration

Multiple stochastic integrals are defined relative to a class of sets. The classic cases of multiple Wiener integral and Ito integral (as well as its generalization by Wong-Zakai-Yor) are recovered by specializing the class of sets appropriately. Any square-integrable functional of the Wiener process has a canonical representation in terms of the integrals. Formulas are given for projecting a stochastic integral onto the space of Wiener functionals and for representing multiple stochastic integrals as iterated integrals. Applications to a change in probability measure arising in a signal detection problem are given.

Asymmetric Wiener Control

Abstract : A one-dimensional Wiener control problem with integral discounted quadratic cost function and asymmetric bounds on the control is considered, with infinite horizon. The optimal control is explicitly found. Bellman equations and Ito integrals are used to show optimality. (Author)

Stochastic equations and krylov's estimates for semimartingales

In 1969 N. V. Krylov obtained the following estimate where X is an Ito process in is the exit time of X from a bounded region is Ld-norm of a measurable nonnegative function , and N is a constant. We generalize estimates of this type to semimartingales and give applications to the theory of stochastic equations with respect to semimartingales. The questions of the existence, uniqueness, convergence and comparison of solutions of these equations are also studied.

A Singular Perturbation Approach to Kramers’ Diffusion Problem

We consider Kramers’ diffusion problem, which seeks to calculate the rate of escape of a particle from one potential well over a barrier, to another presumably deeper and therefore more stable well. Though Kramers introduced the problem as a model for chemical reactions, it applies to numerous rate processes, including atomic migration and ionic conductivity in crystals, and transitions due to noise, between stable states of dynamical systems with multi-stable states, to name but a few. We propose a new approach, not based on a Fokker–Planck equation, but rather on the solution of a singularly perturbed boundary value problem. Specifically, we relate the rate of escape to the first passage time from the domain of attraction of the stable point corresponding to the first well. The first passage time is then characterized via the Ito calculus, as a solution of an elliptic partial differential equation of singular perturbation type. Finally this equation is solved asymptotically by methods previously developed by the authors. We obtain some new results on the rate of escape, which reduce to those of Kramers for the cases he considered, and in addition discuss the validity of the various results derived by Kramers. Finally, in contrast to other approaches, our methods readily extend to higher dimensions.

Diffusion Across Characteristic Boundaries

We consider the motion of a particle acted on by the deterministic force vector b${\bf b}({\bf x}( t ))$ and perturbed by random forces of white noise type. Such a particle will leave any bounded domain $\Omega $ in finite time. We consider the case where b is such that $\partial \Omega $ consists of a trajectory or trajectories of the sytem $\dot x( b( t ) )$. Thus we consider the cases of an unstable limit cycle and a center. We observe that these problems are such that ${\bf b}$ is not derivable from a potential. For each problem we derive expressions for (i) the mean first passage time to $\partial \Omega $, and (ii) the probability distribution of exit points on $\partial \Omega $. Our method is to employ the Ito calculus to characterize the quantities (i) and (ii) as solutions of singularly perturbed elliptic boundary value problems, and then to derive asymptotic representations of the solutions of those problems. The results obtained are new and are of importance in a variety of applications includ...