Ito Calculus Research Articles

Local Solutions of Laplace, Heat, and Other Equations by Itô Processes

A method is presented for finding the solution of deterministic partial differential equations at an arbitrary point of the domain of definition of these equations referred to as the local solution. The method is based on the Ito calculus, properties of diffusion processes, and Monte Carlo simulation. The theoretical background of the proposed method is relatively difficult. However, the method has attractive features for applications. For example, the numerical algorithms based on the proposed method are simple, stable, accurate, local, and ideal for parallel computation. Numerical examples from mechanics are presented to demonstrate the use and the accuracy of the proposed method.

Stochastic Tools on Hilbert Manifolds: Interplay with Geometry and Physics

Projections via the action of a Hilbert Lie group of a class of semi-martingales (given by Ito fields) defined on Hilbert manifolds are investigated. Using Ito calculus, we show that the drift term arising in the projected process can be interpreted in terms of a regularised trace of the second fundamental form of the orbits. For group actions with finite dimensional orbit space, we introduce a notion of strongly harmonic functions resp. regularised Brownian motion, which project onto harmonic functions resp. onto Brownian motion, whenever the orbits are minimal (in a regularised sense). We relate this projection procedure of semi-martingales to the Faddeev-Popov procedure in gauge field theory.

Causal structure of quantum stochastic integrators

A class of concrete representations of a noncommutative Stratonovich calculus is defined and its relationship with the quantum Ito calculus of Hudson and Parthasarathy is made explicit. Given a quantum field interacting with a quantum mechanical system, it is possible to extract a quantum noise description for the field using a suitable scaling limit (here, the weak coupling limit). The motivation for our construction is to discuss the relationship between the micro-causality of a quantum field and the notion of macro-causality of the quantum noise which replaces it. We derive the Stratonovich quantum stochastic differential equation for the limit evolution operator and show that it agrees with the quantum stochastic limit theory of Accardi, Frigerio, and Lu when we convert to the Ito form. The Stratonovich approach, being inherently closer to the physical microscopic equations, leads to an overwhelmingly simplified derivation of the quantum stochastic limit equations of motion. The unification of the two quantum stochastic calculi is given and their physical origins explained.

A note on ergodic distributions in two-agent economies

This paper analytically characterizes the wealth distribution of two types of agents where each agent's wealth evolves according to an Ito process with time-varying drifts and diffusions. The paper proposes a simple method to derive a condition under which the wealth distribution is uniquely ergodic. An intuitive explanation of this condition is discussed.

An Approximation of the Ito and Stratonovich Stochastic Integrals by Elements of a Direct Product of Algebras of Generalized Random Processes

The paper investigates a problem of approximation of multidimensional Ito and Stratonovich stochastic integrals by finite sums of elements of a direct product of algebras of generalized random processes.

A Review of White Noise Analysis from a Probabilistic Standpoint

The main notions and tools from white noise analysis are set up on the basis of the calculus of Gaussian random variables and the S-transform. A new proof of the formula for the S-transform of Ito integrals is given. Moreover, measurability and the martingale property with respect to the Brownian filtration are characterized in terms of the S-transform. This allows to extend these notions to random variables and processes, respectively, in the space of Hida distributions.

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrodinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Ito integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrodinger equation.

Probability and Statistics Applied to the Practice of Financial Risk Management: The Case of J.P. Morgan's RiskMetrics™

This work describes applications of probability and statistics in RiskMetrics™, J.P. Morgan's methodology for quantifying market risk. The methodology implements an analytical approach to financial risk in trading, arbitrage, and investment based on the statistics of market moves in equities, bonds, currencies and commodities. The public unveiling of RiskMetrics™ in October of 1994 attracted widespread interest among regulators, competing financial institutions, investment managers, and corporate treasurers, while the available technical documentation offers us a unique opportunity for informed statistical research on the theory and practice of financial risk management. For the purpose of identifying problems for further research, this discussion focuses on five applications of statistics in RiskMetrics™, which range from data analysis of daily returns and locally Gaussian processes to stochastic volatility models and Ito processes for the term structure of interest rates. Another important theme of this discussion, however, is devoted to attracting statisticians to the study of financial risk management and developing the foundations for collaborative work with financial economists and practicing risk managers. For this reason, this is also an expository document that touches several areas of active statistical research with applications to problems of risk management.

On the martingale property for generalized stochastic processes

In the recent years, several groups have studied stochastic equations (e.g. SDE's, SPDE's, stochastic Volterra equations) outside the framework of the Ito calculus. Often, this led to solutions in spaces of generalized random processes or fields. It is therefore of interest to study the probabilistic properties of generalized stochastic processes, and the present paper makes some (rather naive) first steps into this direction. If we think of a generalized process as a mapping from the real line (or an interval) into aspace of generalized random variables (with some additional properties), then there is a wide range of choices for the latter: e.g., the space (S)* of Hida distributions (e.g. [HKPS]), the space (S)-1 of Kondratiev distributions (e.g. [KLS]), the Sobolev type space D* which is used within the Malliavin calculus and so on. Often, the generalized processes coming up in applications have a chaos decomposition with kernels which belong to L2(IRn), and in this paper we shall focus our attention on this situation. It will be convenient to work with a space G* which is larger than D*. This space has already been considered by several authors, cf. e.g. [PT] and the references quoted there. It turns out, that basic notions from the theory of stochastic processes like conditional expectation, martingales, sub- (super-) martingales etc. have a rather natural generalization to mappings from the realline into G* . The paper is organized as follows. In Section 2 we shall give a construction of the Ito integral (with respect to Brownian motion) of generalized stochastic processes, and compare it with the Hitsuda-Skorokhod integral (e.g. [HKPS]). In Section 3 we define generalized martingales and derive a number of properties. In particular, we prove that the generalized Ito integrals of Section 2 are indeed generalized martingales. Moreover, a representation of generalized martingales (in analogy with the Clark-Haussmann formula) will be shown, and we prove that the Wick product of two generalized martingales is again one. Finally, in Section 4 we define the notion of a generalized semimartingale and give a class of examples. In the remainder of this Introduction we provide the necessary background.

The pricing of Asian options under stochastic interest rates

The purpose of this paper is to analyse the effect of stochastic interest rates on the pricing of Asian options. It is shown that a stochastic, in contrast to a deterministic, development of the term structure of interest rates has a significant influence. The price of the underlying asset, e.g. a stock or oil, and the prices of bonds are assumed to follow correlated two-dimensional Ito processes. The averages considered in the Asian options are calculated on a discrete time grid, e.g. all closing prices on Wednesdays during the lifetime of the contract. The value of an Asian option will be obtained through the application of Monte Carlo simulation, and for this purpose the stochastic processes for the basic assets need not be severely restricted. However, to make comparison with published results originating from models with deterministic interest rates, we will stay within the setting of a Gaussian framework.

Nonlinear potentials of the cauchy-dirichlet problem for the integrodifferential Bellman equation

The Cauchy-Dirichlet problem for the integrodifferential Bellman equation, arising in the theory of contolled Ito processes, is investigated. Sufficient conditions are given, under which this problem has a viscosity solution Lipschitz continuous inx and Holder continuous int with the exponent 1/2. The proof is based on the method of nonlinear potentials.

Differential calculi on commutative algebras

A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in very much the same way we are used to from the geometrical arena underlying classical physical theories and models. In previous work, certain differential calculi on a commutative algebra exhibited relations with lattice structures, stochastics, and parametrized quantum theories. This motivated the present systematic investigation of differential calculi on commutative and associative algebras. Various results about their structure are obtained. In particular, it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of 1-forms. An example of such a product is provided by the Ito calculus of stochastic differentials. For the case where the algebra A is freely generated by `coordinates' x^i, i=1,...,n, we study calculi for which the differentials dx^i constitute a basis of the space of 1-forms (as a left A-module). These may be regarded as `deformations' of the ordinary differential calculus on R^n. For n < 4 a classification of all (orbits under the general linear group of) such calculi with `constant structure functions' is presented. We analyse whether these calculi are reducible (i.e., a skew tensor product of lower-dimensional calculi) or whether they are the extension (as defined in this article) of a one dimension lower calculus. Furthermore, generalizations to arbitrary n are obtained for all these calculi.

A stochastic physical system approach to modeling river water quality

In this paper, concepts of network thermodynamics are applied to a river water quality model, which is based on Streeter-Phelps equations, to identify the corresponding physical components and their topology. Then, the randomness in the parameters, input coefficients and initial conditions are modeled by Gaussian white noises. From the stochastic components of the physical system description of problem and concepts of physical system theory, a set of stochastic differential equations can be automatically generated in a computer and the recent developments on the automatic formulation of the moment equations based on Ito calculus can be used. This procedure is illustrated through the solution of an example of stochastic river water quality problem and it is also shown how other related problems with different configurations can be automatically solved in a computer using just one software.

A basis for iterated stochastic integrals

We explore the algebra of the sets of iterated Stratonovich or Ito integrals that appear in the stochastic Taylor series expansion of the solution to a stochastic differential equation. The algebra of iterated Stratonovich integrals with pointwise multiplication is a shuffle algebra, allowing us to apply results from work on shuffle algebras. The pointwise product of Ito integrals is a modified shuffle product. Lyndon words provide an algebraic basis for both sets of iterated integrals. This basis is similar to, but simpler than, that obtained by Sussmann using Hall words.

An alternative approach to stochastic calculus for economic and financial models

Abstract Application of the Ito stochastic calculus to problems in economics and finance raises several modeling issues. McShane's canonical model and alternative stochastic calculus for handling these models resolves these issues in a satisfactory manner. This paper explores the application of McShane's approach to four areas: empirical estimation and testing of stochastic models, Fischer's model of the demand for index bonds, option pricing, and optimal investment under price level uncertainty.

Stochastic Integration of Processes with Finite Generalized Variations. I

In this paper the $L^1$-stochastic integral and the mixed stochastic integral of a process $Y$ with respect to a process $X$ is defined in a way that extends Riemann-Stieltjes integration of deterministic functions with respect to $X$. The $L^1$-integral will include the classical Ito integral. However, the concepts of "filtration" and adaptability do not play any role; instead, the $p$-variation of Dolean functions of the processes $X$ and $Y$ is the determining factor.

Spectral representation of fractional Brownian motion in n dimensions and its properties

Fractional Brownian motion (fBm) provides a useful model for processes with strong long-term dependence, such as 1/f/sup /spl beta// spectral behavior. However, fBm's are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation. To facilitate the study of this problem, another model is provided for the construction of fBm from a white-noise-like process by means of a stochastic or Ito integral in frequency of a stationary uncorrelated random process. Also a generalized power spectrum of the nonstationary fBm process is defined. This new approach to fBm can be used to compute all of the correlations, power spectra, and other properties of fBm. In this paper, a number of these fBm properties are developed from this model such as the T/sup H/ law of scaling, the power law of fractional order, the correlation of two arbitrary fBm's, and the evaluation of the fractal dimension under various transformations. This new treatment of fBm using a spectral representation is extended also, for the first time, to two or more topological dimensions in order to analyze the features of isotropic n-dimensional fBm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The algebra of iterated stochastic integrals

Using results from work on shuffle algebras, we show that Lyndon words provide an algebraic basis for the sets of iterated Stratonovich or Ito integrals that appear in the stochastic Taylor series expansion of the solution to a stochastic differential equation and give a method for rewriting these stochastic integrals in terms of the basis. This basis is similar to, but simpler than, that obtained by Sussmann ([19]) using Hall words. We also show how the shuffle product can be used to obtain moments of stochastic integrals and hence reprove a result of Kloeden and Platen ([9]) using a purely combinatorial approach.

An infinite expansion for non-linear filtering

An abstract filtering equation is derived for a partially observed system. An infinite series expansion is obtained for the best mean square estimate in terms of the iterated Ito integrals with respect to the innovation process.

Non-linear oscillators under parametric and external poisson pulses

The extended Ito calculus for non-normal excitations is applied in order to study the response behaviour of some non-linear oscillators subjected to Poisson pulses. The results obtained show that the non-normality of the input can strongly affect the response, so that, in general, it can not be neglected.