Stochastic Integration

The main purpose of this article is to propose a reasonable definition for the stochastic integration (S.I.) of progressive processes w.r.t. semimartingales. This S.I. generalizes that of predictable processes w.r.t. semimartingales as well as the stochastic Stieltjes integration. This S.I. is proposed in §1. We give also in §1 an exponential formula for semimartingales using this S.I.. The rest of this paper consists of several remarks on the theory of stochastic integration which are mostly of pedagogical interest. In §2 we propose a new construction of the S.I. of predictable processes w.r.t. local martingales. A simple proof of the integration by parts formula is given in §3. Finally, we propose in §4 a short proof of Meyer’s theorem on compensated stochastic integrals of local martingales.AMS Subject Classification60H05

Consider A Closed E of the Real Line R and a (Measureble) Function Defined on R and Strictly Concave on Every Component of R/E. The Present Paper Deals with the Following Marticgale Problem: Find a Continuous Local Martingale (X,F), Given on a Family () of Probability Spaces With for every such that x is stopped at the debut of e, and the process is a local martingale up to We show that this martingale problem possesses a unique solution. This solution is a strong Markov continuous local martingale. Furthermore, it is a pure continuous local martingale and therefore satisfies the previsible representation property. Conversely, for every strong Markov continuous local martingale (X,F) we can find a closed subset E of R and a function f strictly concave on every component of R\E such that (X, F) is the unique solution of the above martingale problem. In the particular case that E and f(X) we recover the theorem of P. Levy on the martingale characterization of the Brownian motion.

Recently, van Neerven, Weis and the author, constructed a theory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integrator is an arbitrary real-valued continuous local martingale. We give several characterizations of integrability and prove a version of the Itô isometry, the Burkholder–Davis–Gundy inequality, the Itô formula and the martingale representation theorem.

In this paper we define a new type of quadratic variation for cylindrical continuous local martingales on an infinite dimensional spaces. It is shown that a large class of cylindrical continuous local martingales has such a quadratic variation. For this new class of cylindrical continuous local martingales we develop a stochastic integration theory for operator valued processes under the condition that the range space is a UMD Banach space. We obtain two-sided estimates for the stochastic integral in terms of the $\gamma $-norm. In the scalar or Hilbert case this reduces to the Burkholder-Davis-Gundy inequalities. An application to a class of stochastic evolution equations is given at the end of the paper.

We present a new test for the “continuous martingale hypothesis”. That is, a test for the hypothesis that observed data are from a process which is a continuous local martingale. The basis of the test is an embedded random walk at first passage times, obtained from the well-known representation of a continuous local martingale as a continuous time-change of Brownian motion. With a variety of simulated diffusion processes our new test shows higher power than existing tests using either the crossing tree or the quadratic variation, including the situation where non-negligible drift is present. The power of the test in the presence of jumps is also explored with a variety of simulated jump diffusion processes. The test is also applied to two sequences of high-frequency foreign exchange trade-by-trade data. In both cases the continuous martingale hypothesis is rejected at times less than hourly and we identify significant dependence in price movements at these small scales.

We abstract and extend the stochastic integration with martingale integrators to more general processes for which the dominated convergence theorem is still valid. The motivation here is to obtain a unified treatment of several different stochastic integrals, available in the literature, by means of a generalized boundedness principle based on a fundamental idea formulated by S. Bochner. After presenting the semi-martingale integrals in the next section, to serve as a key example, the desired boundedness principle is treated in detail in Section 2. It is also shown there, and in Section 3, that the earlier integrals fit in this frame work; and several applications are worked out to exhibit the universality of the principle, including some vector and multiparameter cases. The rest of the chapter is devoted to the existence (and unicity) of solutions of both linear and nonlinear higher order stochastic differential equations and its progression to stochastic flows for the L 2,2-bounded case. This work takes up Sections 4 and 5 below, and most of Section 4 appears in book form for the first time. Several other results are included in the Complements section.

In this chapter, we will briefly introduce the martingale theory and Ito’s stochastic analysis. First we introduce some basic concepts of continuous time stochastic processes and the definitions of four basic types of process: Markov process, martingale, Poisson process, and Brownian motion, as well as their basic properties, including Doob-Meyer’s decompositions of continuous local submartingales and quadratic variation processes of continuous local martingales and continuous semimartingales. Then we introduce the stochastic integrals of measurable adapted processes w.r.t. the Brownian motion. Finally, we introduce some useful tools for Ito calculus, such as Ito’s formula, Girsanov’s theorem, and the martingale representation theorem. Ito stochastic differential equations and Feynman-Kac formula are also presented. This chapter is self-contained. For a small number of results, we omit their proofs and refer to Karatzas and Shreve (1991) or Revuz and Yor (1999).

A new proof for the conditions of Novikov and Kazamaki

Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition

K. Ito invented his famous stochastic calculus on Brownian motion in the 1940s. In the 1960s and 1970s, the “Strasbourg school,” headed by P.A. Meyer, developed a modern theory of martingales, the general theory of stochastic processes, and stochastic calculus on semimartingales. It turned out soon that semimartingales constitute the largest class of right continuous adapted integrators with respect to which stochastic integrals of simple predictable integrands satisfy the theorem of dominated convergence in probability. Stochastic calculus on semimartingales not only became an important tool for modern probability theory and stochastic processes but also has broad applications to many branches of mathematics, physics, engineering, and mathematical finance.

In this dissertation, we focus mainly on the further study of the new stochastic integral introduced by Ayed and Kuo in 2008. Several properties of this new stochastic integral are obtained. We first introduce the concept of near-martingale for non-adapted stochastic processes. This concept is a generalization of the martingale property for adapted stochastic processes in the It\^o theory. We prove a special case of It\^o isometry for the stochastic integral of certain instantly independent processes. We obtain some formulas for expressing a new stochastic integral in terms of It\^o integrals and Riemann integrals. Several generalized versions of It\^o's formula for the new stochastic integral obtained by Ayed and Kuo are given. We also provide some examples to illustrate the ideas.

Stochastic integration on the Cuntz algebra O∞

Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.

This paper is concerned with a version of the Lebesgue dominated convergence theorem (DCT) which has been stated for the Kurzweil–Stieltjes integral of real functions. Our objective in this work is to analyze the extension of this result to include vector functions with values in Banach spaces. We establish that the mentioned convergence theorem for the Kurzweil–Stieltjes integral can be formulated in weaker versions for reflexive and separable Banach spaces, and spaces having the Schur property, nonetheless it is not verified in the general case.

In this chapter we give a brief overview of the basic mathematical concepts used for pricing financial products. Modern pricing theory is mainly based on describing the ups and downs of market prices via stochastic processes. Therefore, we start with the definition and characteristics of stochastic processes in Section 2.1. Usually the price-changes of the most basic financial instruments are modelled in terms of a so-called stochastic differential equation (SDE). The price of a financial derivative is considered to be a function of the basic financial instruments’ prices. One of the central tools in stochastic calculus which can be applied to determine the stochastic differential equation for the prices of these derivatives, Itô’s lemma, is described in Section 2.4. This concept is closely related to the stochastic integral, which is defined in Section 2.3 to be the mean square limit of some random sums, carefully put together so that the resulting limit is a (local) martingale. And in fact, martingales are one of the most important elements needed for the evaluation of financial instruments. They are defined in Section 2.5 and are used to describe the prices of financial derivatives in terms of conditional expectations. The Feynman-Kac formula provides a partial differential equation (PDE) that corresponds to such conditional expectations as described in Section 2.6. The advantage of dealing with partial differential equations is that they can be solved numerically. Also, PDE methods come naturally to applied mathematicians and physicists. This approach led Black and Scholes [BS73] to their famous equation for pricing European options. Their model will be discussed in Section 3.5. By contrast, martingales and Itô (stochastic) calculus come naturally to probabilists, and this approach led Merton [Mer74] to his option pricing formula.KeywordsStochastic ProcessStochastic Differential EquationConditional ExpectationWiener ProcessLocal MartingaleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.