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In order to use the theory of stochastic integration, much like in classical integration, certain rules are of fundamental importance. The most famous of these, ‘Ito’s Differential Rule’, generalizes the chain rule from classical calculus. Deriving this rule and exploring its consequences are the aims of this chapter.

In this chapter, we show that the definition of the stochastic integral can be extended to a larger class of integrands than the predictable ones, when either a mild condition on the Doleans measure \( \mu_{M} \) is satisfied or M is continuous.

This chapter provides well-known results concerning the properties of two-parametric martingales and stochastic integration on the plane. We begin with the auxiliary chapter, which also contains some facts which are of independent interest. Our standard references for the results below are [20–24, 40, 42, 44, 47, 48, 65, 71].

Let (Ω, F, P) be a complete probability space and (F t )t ≥ 0 a nondecreasing right-continuous family of sub-σ-fields of F where F 0 contains all A ∈ F with P(A) = 0. Suppose that M = (M t )t ≥ 0 is a real martingale adapted to (F t )t ≥ 0 such that almost all of the paths of M are right-continuous on [0, ∞) and have left limits on (0, ∞). Let V = (V t )t ≥ 0 be a predictable process with values in [−1, 1] and denote by N= V∙ M the stochastic integral of V with respect to M: N is an adapted right-continuous process with left limits on (0, ∞) such that $${N_t} = \int_{\left[ {0,\,t} \right]} {{V_s}\,d{M_s}} \,a.s.$$

Much of this paper will be devoted to answering a question about stochastic integrals, Question 1 below. Question 2 is even more basic. The primary goal is to introduce a method that can be used to study these and other questions about martingale maximal functions.

How does the size of a stochastic integral vary with the choice of the predictable integrand and the semimartingale integrator? One of our goals here is to throw new light on this question, especially in the case that the integrator is not necessarily a martingale but belongs to some other class of semimartingales.

Let W(t, ω) be a standard Wiener process, W t ≡ W(t) ≡ W(t, ∙). A function φ(t, ω) is called nonanticipating iff for all t ^ 0, φ(t, ∙) is measurable with respect to {W s : 0 % s % t}. The Ito stochastic integral $$f(\omega )\, \equiv \,\int {_0^1 \varphi (t,\,\omega )d_t \,W(t,\,\omega )}$$ is defined for any jointly measurable, nonanticipating φ such that for almost all \(\Omega, \int_0^1 \Phi^2(t, \Omega)dt < \infty\) (Gikhman and skorokhod (1968), Chapter 1, Section 2). It is known that it is defined for any jointly measurable, nonanticipating φ such that for almost all \(E \int_0^1 \Phi^2(t, \Phi) dt < \infty, {\rm then} Ef = 0 {\rm and} Ef^2 < \infty.\). Representation of an arbitrary measurable f as a stochastic integral was stated, but later retracted, by J.M.C. Clark (1970, 1971).

Let \(\{W(t),t\in \mathbb{R}^+\}\) be a Wiener process, and \(\{g(t),{\mathcal{F}}_t^W,t\in \mathbb{R}^+\}\) a stochastic process (recall that the previous notation means that g is adapted to a natural filtration \(\{{\mathcal{F}}_t^W\}\) of the Wiener process). Let \({\mathcal{F}}^W=\sigma\{W_t, t\geq 0\}.\) A process g is said to belong to the class \(\hat{\mathcal{L}}_2([a,b])\) if it is measurable and \({{\mathsf{E}}}\int_a^b g^2(s) d s

The purpose of the stochastic anticipating calculus is to develop a differential and integral calculus involving stochastic processes which are not necessarily adapter to the Brownian motion {W t ,t ≥0}. This stochastic calculus is mainly used to formulate and solve stochastic differential equations of the form. $$\left\{ \begin{gathered} {\text{ }}d{X_t} = \sigma (t,{X_t})d{W_t} + b(t,{X_t})dt, \hfill \\ {\text{ }}{X_0} \in {\mathbb{R}^m}, \hfill \\ \end{gathered} \right.$$ where the coefficients σ(t,x), b(t,x) or the initial condition X 0 depend on the whole trajectory of the Brownian motion W.

Odd Olaf Aalen was born in 1947 in Oslo, Norway, and he grew up there. He completed a Masters degree thesis at the University of Oslo in 1972 under the guidance of Jan Hoem. This thesis developed some methods for estimating the efficacy and risks involved in the use of intrauterine contraceptive devices (IUDs). Hoem had suggested that time-continuous Markov chains might be useful for modeling the progression of events experienced by a woman following the insertion of an IUD, as this was a model that Hoem had found useful in other contexts. The need to fit such models non-parametrically started Aalen thinking about the approach that he was to develop fully in his ground-breaking 1978 paper that is reprinted here.

Let {w(•), ℱ(•)} be a Brownian motion in ℝ, defined on some probability space (Ω, ℱ, P). It is supposed that ℱ(•) is right continuous and that ℱ(0) contains the null sets.

Survey of preceding results. We start by recalling and making more precise the definitions and previously obtained results pertaining to martingales and semimartingales (cf. Volume I, Chapter II, Section 2 and Chapter III, Section 4).