《《multivariate ito#39;s lemma》.pdf

Chapter 6 Multivariable Ito Calculus As is the case for ordinary Calculus, there exists a multi-variable version of Ito Calculus involving more than one Brownian. It is relevant for modelling situations in which there are several independent sources of uncertainty, for example stochastic volatility, credit risk with stochastic interest and hazard rates or multi-factor yield curve modelling. The extension of the one-variable Ito Calculus to several Brownian motion does not involve any really new ideas, once we have made clear what exactly is mean by a multi-dimensional Brownian motion. In fact, multi-dimensionality could conceivably be introduced in more than one way: either by looking at several stochastic processes, all parametrised by the same single time-parameter t, which is what will be done here, or by replacing the single t by a kind of ‘multi-dimensional time’ (t1 , . . . , tn ), which is what we will not do - the latter idea leads to the notion of Brownian sheets, which falls beyond the scope of this book (and which, at least to our knowledge, have to date not yet found applications in Finance). 6.1 n-dimensional Brownian motion We begin by defining multi-dimensional Brownian motion. A standard Brown- ian motion in Rn , or a standard n-dimensional Brownian motion, is stochastic process (Zt )t≥0 whose value at time t is simply a vector of n independent Brow- nian motions at t: Zt = (Z1,t, . . . , Zn,t ). (6.1) That is, each Zj,t is (the value at time t of) a one-dimensional Brownian motion, and the different components Zi,t , Zj,t (i = j ) are independent for all (possibly different) times t, t ≥ 0. We use the Z instead of W, since we want to reserve the latter for the more general case of correlated Brownian motion, which we intr