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# Class Note for MATH 456 at UMass(5)

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Massachusetts taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15
Math 456 Spring 2006 Dr Alexandros Sopasakis page 52 3121 Multi dimensional Ito formula Similarly the higher dimensional Ito formula is necessary in the case of a multi dimensional Ito process The following process is an example of a multi dimensional Ito process Xm Uldt UlldBl den Undt UmdBl where each of the Bis are one dimensional Brownian motions and the above is essentially an n dimensional Ito process We can write the above in matrix notation as follows dXt 2 MM tdBa Based on this process the multi dimensional Ito formula is Theorem 19 Suppose that at gnaw gpt is twice cantimwus Then the pracess Wt W933 is again an to pracess for which the fallawing farmula halds for each campanent k n 1 n 629k dYkt t Xtdt Z 2 t XtdXt 7 i1 where ijdt and Example Brownian motion on the unit sphere Use Ito s formula to write the following stochas tic process Xt cos B t sin B in standard form dXt utdt vtdBt where B t is considered to be 1 dimensional Brownian motion As usual one of our main task will be to de ne the function gt Here gt will in fact be a vector instead of a single function We therefore de ne in vector notation Wt W J 8 cos 3 sin 3 or written out in each component as M gt m case Ygt ggt sina where we choose a B t and B t denotes the usual one dimensional Brownian motion Using the multi dimensional Ito formula from Theorem 19 above we obtain rst for gl 1 le sin BtdBt 5 cos Btdt 339 Math 456 Spring 2006 Dr Alexandros Sopasakis page 53 Similarly for y we have using Ito formula 1 dYg cos BtdBt 5 sin Btdt 340 Using vector notation and the fact that Xt 2 Ya 2 Y1 t Y2t cos Btsin B we could write both of the above formulations 339a 340 as follows 1 dXt KXtdBt Xtdt Where K is the matrix K i 01

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