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8#
发表于 2011-7-11 19:05
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Hmm.. I guess the usual way is that you look at the symmetric random walk at points 0, 1, 2,... and step sizes of +1 or -1 so for instance C(n) = sum of n coin flips where you get +1 if you have a head and -1 if you get a tail. Then you make the process continous by saying C(t) = the interpolated process (i.e., draw a line from all the C(n)'s to the next one). Then let W[n](t) = 1/sqrt(n)*C(t). Then let n -> infinity.
All those binomial problems can be stated as problems about C(n) and then W[n] pretty much inherits all the results you got on C(n). But some of those results are a pain in the butt because youre dealing with that n choose r stuff and sums and we just don't have very good machinary for sums (mostly) except to approximate them. The cool thing is that when the step sizes get smaller and smaller the approximations become better and better. Finally, they aren't approximations at all but results from calculus. And that opens up the whole world of chain rules, change of measure, etc that can be worked out for stochastic calc in much the way you do for any other calculus. And then problems that would be impossible for discrete cases become easy because you can use Ito's lemma and Girsanov's thm and all that other cool stuff to solve them.
Of course, the answers you get are wrong anyway in finance because all those stationary Markov kinds of assumptions are wrong, but you can still do really cool math (I especially like change of numeraire arguments that make impossible problems go away with a flick of the wrist). |
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