satisfies bounded convergence in probability . k−1), that is called the Ito integral. 1 Stochastic processes In this section we review some fundamental facts from the general theory of stochastic processes. For example, we can 1The convergence here, in general, is in probability or in L2 3 A stochastic process f(t;w): [0;¥) W!R is adapted if, 8t 0, f(t;w) depends only on the values of W agrees with the explicit expression for bounded elementary integrands . The stochastic integral up to time with respect to , if it exists, is a map . 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. 1.1 General theory Let (Ω,F,P) be a probability space. 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 … The key idea is contained in the definition of Itˆo integral, introduced later. More generally, for locally bounded integrands, stochastic integration preserves the local martingale property. 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. I shall give some examples demonstrating this. Taking f(x) = x2 in Itˆo formula gives 1 2dW 2 t= W dW + 1 2dt. 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 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. We will use a set of time instants I. 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. Definition. When this set is not specified, it will be [0,∞), occasionally [0,∞], or N. Let (E,E) another measurable space. It generalizes to integrals of the form R t 0 X(s)dB(s) for appropriate stochastic processes {X(t) : t ≥ 0}. Example 1 Consider the experiment of ipping a coin once. Therefore Z t 0 WsdWs = 1 2W 2 t − 1 2t. 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. We know that an integral of a bounded elementary process with respect to a martingale is itself a martingale. which. Proving the existence of the stochastic integral for an arbitrary integrator is, … 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. In this section, we write X t(!) 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