A Differentiation Theory for Itô's Calculus

Abstract

A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic variation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications, some of which we address in an upcoming article.     

Two-Parameter Lévy Processes Along Decreasing Paths

Let {X_t1,t2:t₁,t₂ 3 0}\{X_{t_{1},t_{2}}:t_{1},t_{2}\geq0\} be a two-parameter Lévy process on ℝ ^d . We study basic properties of the one-parameter process {X _x(t),y(t):t∈T} where x and y are, respectively, nondecreasing and nonincreasing nonnegative continuous functions on the interval T. We focus on and characterize the case where the process has stationary increments.     

Paths and the geometry of l'Hôpital's Rule

In this short note is presented an easy and instructive proof of l'Hôpital's Rule in both the 0/0 and ∞/ ∞ cases, obtained by translating the theorem into a simple geometric statement about paths in the plane.     

Some Results of Backward Itô Formula

Abstract

We use the notion of backward integration, with respect to a general Lévy process, to treat, in a simpler and unifying way, various classical topics as: Girsanov theorem, first order partial differential equations, the Liouville (or Lyapunov) equations and the stochastic characteristic method.     

The Peano phenomenon for Itô equations

The weak convergence of the measures generated by the solutions of stochastic Itô equations with low diffusion is studied, as the diffusion tends to zero. It is proved that the limiting measure in the presence of the Peano phenomenon for a relevant ordinary differential equation is concentrated on its extreme solutions with definite weights. The formulas for their calculation are given.     

Semimartingale: Itô or not ?

Itô semimartingales are the semimartingales whose characteristics are absolutely continuous with respect to Lebesgue measure. We study the importance of this assumption for statistical inference on a discretely sampled semimartingale in terms of the identifiability of its characteristics, their estimation, and propose tests of the Itô property against the non-Itô alternative when the observed semimartingale is continuous, or discontinuous with finite activity jumps, and under a number of technical assumptions.     

Itô's formula for linear fractional PDEs

In this paper, we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itô-type formula for the process X.     

Itô maps and analysis on path spaces

We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross–Sobolev spaces of differentiable functions and proving their intertwining with solution maps, , of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given , is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.     

Diffusion maps tailored to arbitrary non-degenerate Itô processes

We present two generalizations of the popular diffusion maps algorithm. The first generalization replaces the drift term in diffusion maps, which is the gradient of the sampling density, with the gradient of an arbitrary density of interest which is known up to a normalization constant. The second generalization allows for a diffusion map type approximation of the forward and backward generators of general Itô diffusions with given drift and diffusion coefficients. We use the local kernels introduced by Berry and Sauer, but allow for arbitrary sampling densities. We provide numerical illustrations to demonstrate that this opens up many new applications for diffusion maps as a tool to organize point cloud data, including biased or corrupted samples, dimension reduction for dynamical systems, detection of almost invariant regions in flow fields, and importance sampling.     

Linear information for approximation of the Itô integrals

We study optimal approximation of stochastic integrals in the Itô sense when linear information, consisting of certain integrals of trajectories of Brownian motion, is available. Upper bounds on the nth minimal error, where n is the fixed cardinality of information, are obtained by the Wagner–Platen algorithm and are O(n ^???3/2) or O(n ^???2), depending on considered class of integrands. We also show that Ω(n ^???2) is a lower bound which holds even for very smooth integrands.     

On Itô s formula for multidimensional Brownian motion

Consider a d-dimensional Brownian motion X = (X ¹,…,X ^d ) and a function F which belongs locally to the Sobolev space W ^1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f _k (X), X ^k ] involving the weak first partial derivatives f _k of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X ^k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Received: 16 March 1998 / Revised version: 4 April 1999     

Change-point inference on volatility in noisy Itô semimartingales

This work is concerned with tests on structural breaks in the spot volatility process of a general Itô semimartingale based on discrete observations contaminated with i.i.d. microstructure noise. We construct a consistent test building up on infill asymptotic results for certain functionals of spectral spot volatility estimates. A weak limit theorem is established under the null hypothesis relying on extreme value theory. We prove consistency of the test and of an associated estimator for the change point. A simulation study illustrates the finite-sample performance of the method and efficiency gains compared to a skip-sampling approach.     

Estimates for first exit times of non-Markovian Itô processes

First exit times and their path-wise dependence on trajectories are studied for non-Markovian Itô processes. Estimates of distances between two exit times are obtained. In particular, it follows that first exit times of two Itô processes are close if their trajectories are close.     

Lyapunov exponents of nilpotent Itô systems with random coefficients

The paper considers the top Lyapunov exponent of a two-dimensional linear stochastic differential equation. The matrix coefficients are assumed to be functions of an independent recurrent Markov process, and the system is a small perturbation of a nilpotent system. The main result gives the asymptotic behavior of the top Lyapunov exponent as the perturbation parameter tends to zero. This generalizes a result of Pinsky and Wihstutz for the constant coefficient case.     

Nonlinear Lebesgue and Itô Integration Problems of High Complexity

We analyze the complexity of nonlinear Lebesgue integration problems in the average case setting for continuous functions with the Wiener measure and the complexity of approximating the Itô stochastic integral. G. W. Wasilkowski and H. Woźniakowski (2001, Math. Comp., 685–698) studied these problems, observed that their complexities are closely related, and showed that for certain classes of smooth functions with boundedness conditions on derivatives the complexity is proportional to ε⁻¹. Here ε>0 is the desired precision with which the integral is to be approximated. They showed also that for certain natural function classes with weaker smoothness conditions the complexity is at most of order ε⁻² and conjectured that this bound is sharp. We show that this conjecture is true.     

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an It5 type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the It5 type set-valued stochastic differential equation.