First of all what this has to do with RN theorem is if they re equivalent then there s a density.-"Is Girsanov the only way to get to an EMM?" The probability triplet is sometimes useful in structuring these ideas. If you take the natural algebra of W then yes every equivalent measure can be got through an exponential. If you take the canonical space, every equivalent measure comes from a Cameron-Martin transform; then the likelihood ratio can be defined without reference to W by using adequate projections, though technical care is required if the lambdas are unbounded-cf Dellacherie & Meyer 6.43. You could very well have a richer probability space, and define equivalent measures on that, that have nothing to do with W The only necessary and sufficient condition is that:!conditional on W the density has expectation 1! . -Alan: I'm a big fan offer me a faculty job please. Your first reaction is tied to the space of continuous paths: if the covs are different the laws are singular, eg using a.s. short time asymptotics. (By the way, for financial pricing and hedging it's the law of S that counts, and as such:!you only need to respect the projection of Vt onto S's past!) If you carry out your program i,ii,iii there is simply no chance of the two measures being equivalent. More precisely, we have defined (S,V) using Doob regularisation, and this means a.s. there is no discontinuity of type 1. You can write some jump dynamics realised by an SDE seen in the Skorokhod space, but still "has jump size * at time *" is a cylinder set so its law won't be equivalent to that of our (S,V).-frenchX: I am looking for tenure track,how is the job market in physics/chemistry for those with qfin interests? A BM under P won't always/exactly be a BM under M.. In certain cases you can "prove Girsanov" ie mass 1, but "Novikov" is not verified, like many cases around Heston, so in this case it's not such a great suggestion.. Well the RN derivative is martingale, and therefore due to right-continuity of the filtration, we can chose a continuous version (this is a very fascinating result of Doob).I'm hoping someone can chime in to add stuff about what the jump times of semimartingales can look like, and for example what is the relationship between the laws of VGs with different parameters.
Last edited by croot
on September 29th, 2011, 10:00 pm, edited 1 time in total.