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