Two-parameter martingales and their quadratic variation
Since the publication of the pioneering paper of \textit{R. Cairoli} and \textit{J. B. Walsh} [Acta Math. 134, 111-183 (1975; Zbl 0334.60026)] the theory of two-parameter processes has undergone an active development, especially during the second part of the last decade. The basic survey of \textit{P. A. Meyer} [Théorie élémentaire des processus aléatoires à deux indices, Colloq. E.N.S.T.-C.N.E.T., Paris 1980, Lect. Notes Math. 863, 1-39 (1981; Zbl 0461.60072)] provides an account of the stage of this theory at the beginning of the eighties and may be considered as an introduction to a general theory of two-parameter processes. Some fundamental results such as the regularity properties for square integrable martingales were already presented in this paper of Meyer, but the theory was just at an initial stage. In the last five years stochastic calculus for two-parameter martingales has been developed by several authors, in particular L. Chevalier, P. Imkeller, N. Frangos, D. Nualart, and M. Sanz. These authors have proved different types of Ito formulas for two-parameter processes, but they have considered mainly the case of martingales with continuous paths. One of the basic ingredients in proving these two-parameter versions of the Ito formula is the notion of quadratic variation. In the non-continuous case the situation is much more complicated and, except in some special examples (like the Poisson case), stochastic calculus has not yet been developed. The main objective of this monography is to show the existence of the quadratic variation for a (not necessarily continuous) two-parameter martingale bounded in L \(log^+ L\). To accomplish this goal the author introduces in a clear and detailed way all the necessary notions and results on two-parameter processes which are relevant to the problem. In that sense this work can be regarded as an updated presentation of the fundamental notions of the general theory of two-parameter processes. Chapter I is devoted to prove what can be called ``one-directional'' projection and decomposition theorems. In this chapter the two-parameter processes are regarded as one-parameter processes depending in a measurable way on the other coordinate. In this situation it can be shown that this measurability is preserved when taking optional and predictable (dual) projections. Also, decomposition theorems for submartingales depending reasonably on a parameter, and Garsia-Neveu type inequalities are deduced. In Chapter II, and changing the point of view, the author introduces the techniques which are properly of two-parameter nature, like the notion of a stopping line. The main result of this chapter is the regularity theorem (existence of a right continuous version possessing limits along the remaining three quadrants) for L \(log^+ L\)-integrable martingales. Here the basic assumption of conditional independence for the underlying two-parameter filtration, which was not relevant in Chapter I, plays a fundamental role. As a consequence two-directional projection and decomposition theorems can be derived. The regularity result for two- parameter martingales and some other material presented in this chapter are due to \textit{D. Bakry} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, 149-157 (1979; Zbl 0419.60051), and ibid. 55, 55-71 (1981; Zbl 0446.60039)]. The results on the quadratic variation of two-parameter martingales are based on a deep investigation of the structure of these martingales, and in particular, on the form of their discontinuities. In the one-parameter case the notion of stopping time is essential to study these subjects, but in the two-parameter case the corresponding notions of stopping point and stopping line are of less utility. In that sense the author has been forced to find some methods that do not use the notion of stopping. The first step is to study the discontinuities of a regular two-parameter martingale. Here an additional complication, in comparison with the one- parameter case, appears. Given a regular function f on \([0,1]^ 2\), it may have three different kinds of discontinuities at a given point \(t=(t_ 1,t_ 2):\) 0-jumps, 1-jumps and 2-jumps, defined respectively by the following properties: \[ \lim_{h\downarrow 0}(f(t)-f(t_ 1,t_ 2-h)-f(t_ 1-h,t_ 2)+f(t_ 1-h,t_ 2-h))\neq 0, \] \[ \lim_{h\downarrow 0}(f(t)-f(t_ 1-h,t_ 2))\neq 0,\quad \lim_{h\downarrow 0}(f(t)-f(t_ 1,t_ 2-h))\neq 0. \] Then the main result on the structure of two-parameter square integrable martingales says that such a martingale can be decomposed into four orthogonal parts: a continuous part plus three i-jump parts \((i=0,1,2)\). The idea in obtaining this result consists in gradually extracting the jump parts of M and compensating them using the projection and decomposition theorems proved in Chapters I and II. This process is executed in several steps, until the desired decomposition is deduced. This decomposition theorem allows to deduce the main result of the paper: the existence of the quadratic variation for a two-parameter square integrable martingale M. In fact, the quadratic variation of M is simply obtained as the orthogonal sum of the quadratic variations of the corresponding elementary components. An approximation argument allows to extend the result to martingales bounded in L \(log^+ L.\) The main results on the structure and quadratic variation are proved in Chapter IV, and Chapter III contains some preliminary facts about the jumps of a martingale and their compensators. Finally, a counterexample of an \(L^ 1\)-bounded martingale with no quadratic variation is given.
general theory of two-parameter processes
two-parameter versions of the Ito formula
scientific article; zbMATH DE number 192999