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 … 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. Therefore Z t 0 WsdWs = 1 2W 2 t − 1 2t. Taking f(x) = x2 in Itˆo formula gives 1 2dW 2 t= W dW + 1 2dt. We will use a set of time instants I. For example, we can 1The convergence here, in general, is in probability or in L2 3 satisfies bounded convergence in probability . It generalizes to integrals of the form R t 0 X(s)dB(s) for appropriate stochastic processes {X(t) : t ≥ 0}. We know that an integral of a bounded elementary process with respect to a martingale is itself a martingale. 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. 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. More generally, for locally bounded integrands, stochastic integration preserves the local martingale property. The key idea is contained in the definition of Itˆo integral, introduced later. Definition. I shall give some examples demonstrating this. 1 Stochastic processes In this section we review some fundamental facts from the general theory of stochastic processes. precisely defining the set of functions for which the integral is well-defined. 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 . Example 1 Consider the experiment of ipping a coin once. 1.1 General theory Let (Ω,F,P) be a probability space. In this section, we write X t(!) 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 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. which. 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