More generally, for locally bounded integrands, stochastic integration preserves the local martingale property. I shall give some examples demonstrating this. which. 1.1 General theory Let (Ω,F,P) be a probability space. The key idea is contained in the definition of Itˆo integral, introduced later. It generalizes to integrals of the form R t 0 X(s)dB(s) for appropriate stochastic processes {X(t) : t ≥ 0}. 2 Examples The Itˆo isometry and the Itˆo formula are the backbone of the Itoˆ calculus which we now use to compute some stochastic integrals and solve some SDEs. agrees with the explicit expression for bounded elementary integrands . A Brief Introduction to Stochastic Calculus 3 2 Stochastic Integrals We now discuss the concept of a stochastic integral, ignoring the various technical conditions that are required to make our de nitions rigorous. The stochastic integral up to time with respect to , if it exists, is a map . For example, we can 1The convergence here, in general, is in probability or in L2 3 Stochastic integral Introduction Ito integral Basic process Moments Simple process Predictable process In summary Generalization References Appendices Ito integral II Let fX t: t 0g be a predictable stochastic process. Stochastic integrals u = fu t;t 0 is a simple process if u t = nX 1 j=0 ˚ j1 (t j;t j+1](t); where 0 t 0 t 1 t n and ˚ j are F t j-measurable random variables such that E(˚2 j) <1. In this section, we write X t(!) instead of the usual X tto emphasize that the quantities in question are stochastic. Definition. We know that an integral of a bounded elementary process with respect to a martingale is itself a martingale. satisfies bounded convergence in probability . precisely defining the set of functions for which the integral is well-defined. Example 1 Consider the experiment of ipping a coin once. We will use a set of time instants I. k−1), that is called the Ito integral. We define the stochastic integral of u as I(u) := Z 1 0 u tdB t = Xn 1 j=0 ˚ j B t j+1 B t j: Proposition The … As an example of stochastic integral, consider Z t 0 WsdWs. When this set is not specified, it will be [0,∞), occasionally [0,∞], or N. Let (E,E) another measurable space. Therefore Z t 0 WsdWs = 1 2W 2 t − 1 2t. The integral R 1 1 g(x)dP X(x) can be expressed in terms of the probability density or the probability function of X: Z 1 1 Proving the existence of the stochastic integral for an arbitrary integrator is, … This is an integral of a function (b[t]) with respect to a stochastic process, and when S is a function of Brownian motion (which it will be) this is called an Itô Integral. 1 Stochastic processes In this section we review some fundamental facts from the general theory of stochastic processes. In the following, W is the sample space associated with a probability space for an underlying stochastic process, and W t is a Brownian motion. 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