> <\body> <\hide-preamble> <\padded-normal|0.5fn|0.5fn> >||> > > \; \; <\center> > \; \; <\very-small> Version 1, 2020.05.22. Tutorial classes: Mon June 1st 16\U18 \ (Zoom) Min Liu \| Wed June 3rd 16\U18 (Zoom) Daria Frolova. Solutions in groups of 2 (at most). To be handled in or format via eCampus not later than 4pm Thursday May 28th. Use this sheet for your solutions and write them under the corresponding exercise. Fill out your names below. \; \; <\exercise> [Pts 2+2+2] Let be a one\Udimensional Brownian motion with =0> and let 0>, \>. <\enumerate-alpha> Let =inf0 :X=a+b*t|}>> denote the first passage time to the line . Show that\ <\equation> \\t|)>=\-bt/2>\\t>|]>, where =inf0:X=a|}>> is the first passage time to level . Recall that, by the reflection principle, the law of > is absolutely continuous with density <\equation*> f>=a*t\|)>\|)>>, where > is the standard normal density. Deduce that the law of > is absolutely continuous with density <\equation*> f>=a*t\/|)>\|)>>. take the conditional expectation w.r.t. >>]>. Show that, for 0>, <\equation*> \>*maxt>|)>|]>\t/2>|2b>,\>*maxt>|)>|]>\b*tet/2>,\>>. <\exercise> [Pts 2+2+3] Let be a \Udimensional Brownian motion with =0>.\ <\enumerate-alpha> Show that, for any \>, the process <\equation*> X=X-t*-y|)>t\ is independent of >. Let > denote the law of > on ;\|)>>. Show that \> is a regular version of the conditional distribution of given =y>. Compute the SDE satisfied by the canonical process under the probability measure > on the space ;\|)>>. (Hint: use Doob's -transform argument from the lectures) <\exercise> [Pts 3] Let be a positive continuous supermartingale such that |]>\\>. Let >=lim\>M>. Show that if >|]>=\|]>> then is a martingale and >\|\|]>=M>. >\|\|]>\M> and that |]>=\|]>> and conclude.]> <\exercise> [Pts 4] Prove directly that the -transform gives a transformation of martingale problems from the one with drift and diffusion > to another with same diffusion coefficient > but different drift >. \; <\initial> <\collection> <\references> <\collection> > > >