> <\body> <\hide-preamble> <\padded-normal|0.5fn|0.5fn> >||> > > \; \; <\center> > \; \; <\very-small> Version 1, 2020.06.1. Tutorial classes: Mon June 8th 16\U18 \ (Zoom) Min Liu \| Wed June 10th 16\U18 (Zoom) Daria Frolova. Solutions in groups of 2 (at most). To be handled in or format via eCampus not later than . Use this sheet for your solutions and write them under the corresponding exercise. Fill out your names below. \; \; <\exercise> [Pts 2+2+2+2] Prove that a Brownian motion in > will write your name (in cursive script, without dotted 'i's or crossed 't's). Let be a two dimensional Brownian motion on > and observe that >=*t>-B|)>> \ for > has the same law as . Let \\> a smooth parametrization of your name. Let us agree that the Brownian motion >> spells your name (to precision \0>) if <\equation> sup>>-g|\|>\\. <\enumerate-alpha> For \> let > be the event that holds for > and >. Check that the events |)>\>> are independent and |)>=\|)>> for all 0>. Conclude that if |)>\0> then \ infinitely many of the >s will occur almost surely. \ Show that <\equation> \>|\|>\\|]>\0. Using and Girsanov's transform to show that |)>\0> (Hint: construct a measure > so that -g> is a Brownian motion) Prove that a similar result holds for only continuous. <\exercise> [Pts 3] Let |)>> be a solution of the martingale problem with drift and diffusion >. Generalise appropriately the Girsanov transform to construct a measure > under which the process solves a martingale problem with a different drift. For simplicity, assume that all the necessary integrability conditions are satisfied. (What takes the place of the Brownian motion?) <\exercise> [Pts 3+3+3] Given smooth, bounded functions \\>, \\>. Consider the operator > on |)>> given by <\equation*> H=--i A|\|>+V We will assume that this operator is self\Uadjoint (with suitable domain), bounded from below and with discrete spectrum. We will denote > its smaller eigenvalue which we will assume simple (i.e. of multiplicity one). Let > the complex valued solution to <\equation*> \\=-H \,\=\, which we will assume to exist, to be once differentiable in and twice in and be bounded with bounded derivatives.\ <\enumerate-alpha> Find a suitable functions \\> with which we can give the following Feynman\UKac representation for >: <\equation*> \=\|)>expB|)>\X+C|)>\s|]>|}> where under > the process is a -dimensional Brownian motion starting at \>. Prove that the lowest eigenvector of > is strictly positive everywhere. Use the above representation to prove the <\equation*> E\E. =1> and argue that \c et>\+o> where =E\> and conclude]>> \; <\initial> <\collection> <\references> <\collection> > > >