Ito Calculus Research Articles

Annihilation and creation operators in stochastic quantization.

Annihilation and creation operators in quantum mechanics are constructed in the framework of Nelson's stochastic mechanics. Employing Ito's stochastic calculus, annihilation and creation operators were found to be written as (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B\ifmmode \dot{}\else \.{}\fi{}(t)] and (1/\ensuremath{\surd}2\ensuremath{\omega})[d/dt,\ensuremath{\partial}/\ensuremath{\partial}B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t)], respectively, where B\ifmmode \dot{}\else \.{}\fi{}(t) is a Gaussian white noise and B${\ifmmode \dot{}\else \.{}\fi{}}^{\mathrm{*}}$(t) is a time-reversed Gaussian white noise. The operators were shown to have proper commutation relations and to provide a Hamiltonian which has an eigenvalue equation desired in the {N} representation as should exist in conventional quantum mechanics. Therefore, the present formulation provides a correspondence between stochastic mechanics and conventional quantum mechanics.

Itô's stochastic calculus and the bosonic string

Ito's stochastic calculus is used to derive the critical dimension for the bosonic string. An anomaly in the ghost-number current, related to the Riemann-Roch theorem, is also derived.

An implicit direct integrator for random response of multi-degree-of-freedom systems

An implicit direct integration scheme is proposed for the computation of response of multi-degree-of-freedom (MDOF) systems excited by stationary and nonstationary random disturbances. For linear MDOF systems the proposed scheme is regarded as unconditionally stable. Computed results for a simple linear MDOF system are compared with those obtained earlier by the stochastic central difference method (SCDM) and Ito's calculus approach (ICA). It is concluded that the presented implicit direct integration scheme is more accurate and its time-step size is larger than that for the SCDM. It is relatively easy to implement in a computer compared with the SCDM. It has potential to be applied to MDOF nonlinear systems subject to random excitations.

Contingent claims valuation when the security price is a combination of an Ito process and a random point process

This paper develops several results in the modern theory of contingent claims valuation in a frictionless security market with continuous trading. The price model is a semi-martingale with a certain structure, making the return of the security a sum of an Ito-process and a random, marked point process. Dynamic equilibrium prices are known to be of this form in an ArrowDebreu economy, so there is no real limitation in our approach. This class of models is also advantageous from an applied point of view. Within this framework we investigate how the model behaves under the equivalent martingale measure in the P ∗- equilibrium economy, where discounted security prices are marginales. Here we present some new results showing how the marked point process affects prices of contingent claims in equilibrium. We derive a new class of option pricing formulas when the Ito process is a general Gaussian process, one formula for each positive L 2[O, T]-function. We show that our general model is complete, although the set of equivalent martingale measures is not a singleton. We also demonstrate how to price contingent claims when the underlying process has after-effects in all of its parameters.

Extension of the ito calculus via the malliavin calculus

The Ito formula is extended to the tempered distributions "evaluated" on the trajectories of a nondegenerate Ito process in the sense of P. Malliavin. To do this the Ito integral is extended to vector-valued adapted distributions on Wiener space. Also a Galerkin type approximation using the Skorohod integral or the divergence operator is given for the diffusion processes. At the final section we give a sufficient condition for the existence of a smooth density for the filtering of nonlinear diffusions with the help of the techniques of the Malliavin calculus and the theory of nuclear spaces.

Direct integration operators and their stability for random response of multi-degree-of-freedom systems

The stochastic central difference and stochastic Houbolt methods are introduced in this paper for the computation of the response of complex structures of nonstationary random excitations. The structures are approximated by the finite element method such that they can be treated as multi-degree-of-freedom systems. These methods can be regarded as equivalents to their deterministic counterparts. They have the potential to deal with highly nonlinear structures and mechanical systems. Stability of algorithms are also reviewed and discussed. It is concluded that within the domain of Newtonian dynamics any attempt to address quantitatively the issue of stability in the nonlinear regime would be futile. Results computed by the stochastic central difference method for two linear multi-degree-of-freedom systems are compared with those obtained by Ito's calculus approach. The implication of the numerical results is discussed.

A Malliavin-Type Anticipative Stochastic Calculus

Two extensions of the Ito integral are developed, and put in the perspective of derivative operators in the Malliavin calculus. The divergence operator, $\delta$, is constructed, and its properties and action on these two extended integrals are described. Discussion of iterated stochastic integrals and the extended stochastic integrals as functions of their upper limits is also included.

The random eigenvalue problem for a differential equation containing small white noise

By means of the extended Ito integral and the shooting method, this paper concerns the random problem containing white noise B ( t , ω ) B\left ( {t, \omega } \right ) : \[ [ − p ( t ) u ′ ( t ) ] ′ + [ q ( t ) + ε B ( t , ω ) ] u = λ u , u ( 0 ) = 0 , u ( 1 ) = 0. {\left [ { - p\left ( t \right )u’\left ( t \right )} \right ]’} + \left [ {q\left ( t \right ) + \varepsilon B\left ( {t, \omega } \right )} \right ] u = \lambda u, \qquad u\left ( 0 \right ) = 0, \qquad u\left ( 1 \right ) = 0. \] When ε \varepsilon is small, the existence and asymptotic expansions for the solutions can be obtained, and the normal properties for the first correction terms can be proved. Formulas for evaluation are derived and one example of the Schrödinger equation is given to illustrate the whole procedure.

Green measures of Ito processes

Green measures of Ito processes

Fluctuation-dissipation relations in the presence of multiplicative noise

Abstract This paper analyses the modifications that are warranted in the fluctuation-dissipation theorems by the addition of a multiplicative noise in the Langevin equation modeling the dynamics of a system. The concepts relating the Stratonovich and Ito integrals are employed to derive new relations for the first and second fluctuation-dissipation relations and the virial theorem. For the three dimensional case, the results for the mean and mean square of the angular momentum are obtained for the corresponding Langevin equation with multiplicative noise.

Optimal control in wide-sense stationary continuous-time stochastic models

This paper provides a rigorous treatment of the optimal control problem for a continuous-time stochastic model, with an infinite-horizon quadratic cost function, under weaker assumptions concerning the white-noise innovations than have been made in the literature on this subject. Since it is not assumed that the innovations are generated by Brownian motion, or even that the sample paths of the state variables are integrable, the Ito calculus is inapplicable. The solution depends on a more recently developed theory of stochastic differential equations, based on random measure theory, and on the ergodic theorem for a wide-sense stationary process.

A two-sided stochastic integral and its calculus

Let X be a forward diffusion and Y a backward diffusion, both defined on [0,1], X t and Y t being respectively adapted to the past of a Wiener process W (·), and to its future increments. We construct a “two-sided” stochastic integral of the form. $$\mathop \smallint \limits_0^t \Phi (u,X_u ,Y^u )dW(u)$$ which generalizes the backward and forward Ito integrals simultaneously. Our construction is quite intuitive, and leads to a generalized stochastic calculus. It is also shown that for each fixed t, our integral coincides with that defined by Skorohod in [18].

Wigner-Weisskopf atom and quantum Stochastic dilations

Quantum stochastic processes are operator processes in Fock space adapted in a natural way with respect to the tensor product factorization of the space. One can describe the usual classical stochastic processes (e.g., Brownian, Poisson) as well as new variety of non-commuting processes in this language. In this framework, quantum stochastic calculus (like Ito calculus in the classical case) has been developed by Hudson and Parthasarathy. Using this calculus, one can rewrite the simple Wigner-Weisskopf model of a decaying atom. In this model, the system (atom) is coupled linearly to the environment (a bath of infinite degrees of freedom represented by a reducible bosonic quantum stochastic process at non-zero temperature). The quantum stochastic differential equation of evolution is solved, and the bath variables are averaged out by taking vacuum expectation values leading to a law of relaxation of the state of the system.

Torsional Response of Long Span Bridges in Presence of Turbulence

The effects of spanwise wind correlation and turbulence intensity on the stability of the torsional response of long span bridges are studied. An aerodynamic coefficient model is used for the self‐excited forces. Wind turbulence is assumed to be a wide‐band, weakly stationary in time, random field whose correlation time is much shorter than the relaxation time of the bridge. The equation for the uncoupled torsional response is transformed into an Ito stochastic differential equation. The first‐ and second‐order moments of the response are obtained using standard methods of Ito calculus. An application is presented for the case of a particular long span bridge. Different levels of turbulence and spanwise wind correlation are used. For a constant turbulence level, reducing the spanwise correlation increases the critical wind velocity. For a constant spanwise correlation, increasing the turbulence level reduces the critical wind velocity.

A systematic elimination procedure for Ito stochastic differential equations and the adiabatic approximation

We study a class of nonlinear Ito stochastic differential equations (with possibly state dependent diffusion coefficients), in which the variables can be divided into linearly damped (slaved) variables s and linearly undamped variablesu (order parameters). We devise a systematic and constructive procedure to eliminate the slaved variables. We take explicit time and chance dependence of the slaved variables into account, the latter via a family of diffusion processesZ t (v) . These act as fluctuating coefficients of the Center Manifoldst=s(ut, t,Z t (v) (v=2, 3, ...)) and appear explicitly in the elimination procedure. We show how in the Ito calculus fluctuating and deterministic coefficients of the Center Manifold are more completely separated than in the previously treated Stratonovich case [1]. The adiabatic approximation is defined as a partial summation of the elimination expansion and the stochastic generalization ofs=0 is derived. We show how thus ambiguity of stochastic calculi is removed. Closed form summations are given in two examples. We briefly indicate the potential use of perturbation theory techniques in the systematic elimination procedure.

Stochastic response of nonlinear structures with parameter random fluctuations

The random response of a nonlinear structural system is examined when its parameters are experiencing random fluctuations with time. The treatment is based on the recent developments in the mathematical theory of stochastic differential equations. These include the Ito stochastic calculus and the Fokker-Planck equation approach to derive a general differential equation that describes the evolution of the statistical moments of the response coordinates. The differential equation is found to constitute an infinite coupled set of differential equations that are closed via two different closure schemes. The system response is determined in the neighborhood of internal resonance condition and for various random intensities of the system parameters. It is found that the random modal interaction is governed mainly by the internal resonance ratio and the stiffness fluctuation intensity. The effect of the random damping fluctuation on the system response is found to be very small compared to the stiffness fluctuation effect.

Mathematical problems in the multi-compartmental model with temporal fluctuations

The stochastic differential equation of a multi-compartmental model with temporal fluctuations in its transfer coefficients can be solved by using either the Stratonovich or the Ito integral. Although the mathematical expressions for the solutions are different from each other, computer simulation for a typical example gave practically the same results as long as the fluctuation in transfer coefficients remained sufficiently small. However, when this fluctuation exceeded a certain limit and the structure of system could no longer be maintained, these two integral methods yielded results that were different from each other. The mathematical reason for the different results can be understood by examining the two methods on the basis of the Fokker-Planck equation.

Nonlinear smoothing algorithms using white noise model

Nonlinear smoothing algorithms are studied where the white noise disturbance in the observation is treated directly, instead of the usual modelling through its integrated version of Brownian motion. This framework yields results already in the robust form. Furthermore, the Ito integrals involving the observations do not appear in this approach, thereby eliminating complications arising out of considering backward Ito integrals in the smoothing problems.

Stochastic quantization of Einstein gravity.

We determine a one-parameter family of covariant Langevin equations for the metric tensor of general relativity corresponding to DeWitt's one-parameter family of supermetrics. The stochastic source term in these equations can be expressed in terms of a Gaussian white noise upon the introduction of a stochastic tetrad field. The only physically acceptable resolution of a mathematical ambiguity in the ansatz for the source term is the adoption of Ito's calculus. By taking the formal equilibrium limit of the stochastic metric a one-parameter family of covariant path-integral measures for general relativity is obtained. There is a unique parameter value, distinguished by any one of the following three properties: (i) the metric is harmonic with respect to the supermetric, (ii) the path-integral measure is that of DeWitt, (iii) the supermetric governs the linearized Einstein dynamics. Moreover the Feynman propagator corresponding to this parameter is causal. Finally we show that a consistent stochastic perturbation theory gives rise to a new type of diagram containing ``stochastic vertices.''

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.