> <\body> <\hide-preamble> <\padded-normal|0.5fn|0.5fn> >||> > > \; \; <\center> > \; \; <\very-small> Version 2, 2020.06.10. Tutorial classes: Mon June 15th 16\U18 \ (Zoom) Min Liu \| Wed June 17th 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 3+3+3+4] Let a solution of the SDE in > <\equation> \X=b|)>\t+\B, with a vectorfield \\> measurable and with linear growth. <\enumerate-alpha> Prove that for all 0>, almost surely <\equation*> A=|)>|\|>\s\\, and therefore the process is unique in law. Find a (deterministic) increasing function \\> such that, almost surely <\equation*> sup0>|f>\\. [Hint: find a constant such that 0>|f>\0>|f>\\> a.s.] Use Girsanov's transform to prove that the process is Markov when is a bounded vectorfield. (Bonus) Try to extend the proof of the Markov property for of linear growth. <\exercise> [Pts 5] Let =C,\|)>> with the Borel >-field and > the law of the Brownian motion starting at . Let the unique solution of the SDE with V> and a positive > function such that <\equation*> V|\|>-\V\-Lx\\. Use the path-integral formula\ <\equation*> \|)>|)>=>f|)>exp|)>-V|)>-V|)>|\|>-\V|)>|)>\s|)>\\|)> to show that for any two bounded functions and under appropriate conditions on : <\equation*> f|)>ge>\x=fg|)>e>\x which shows that > is symmetric wrt. the measure >\x> and taking show that >\x> properly normalized is an invariant measure for the SDE <\equation*> \X=-\V|)>\t+\B, meaning that if > is taken with probability distribution e>\x> then\ <\equation*> \|)>|]>=\|)>|]>, for all 0>.\ [Hint: let > the conditional law of the Brownian motion > to have =y>, i.e. the Brownian bridge. Prove that the under > the process |~>=\> has law > and use the path integral] <\exercise> [Pts 3+3] Prove a Fubini theorem for stochastic integrals. Let ,\|)>> be a measure space and ,\,\>,\|)>> a filtered probability space. <\enumerate-alpha> Let |)>> a sequence of functions :\\\\\> which are \\> measurable (product >-field) and such that ,\|)>|)>> converges in probability for any fixed \\>. Prove that there exists an \\> measurable function \\\\> for which ,\|)>||\>X,\|)>> for any \\>. [Hint: here the difficulty is the measurability of the limit , consider the sequence |)>> defined by |)>=1> and <\equation*> n|)>=infn|)>:supm>\,\|)>-X,\|)>|\|>\2|]>\2|}> and prove that X|)>>,\|)>> exists a.s. and conclude] Let \\0>\\\R> be a bounded function which is measurable w.r.t. \\> where > is the predictable >-field on 0>\\>. \ Let be a continuous martingale on ,\,\>,\|)>>. Prove that there exists a function \\\\> measurable for \\> which is a version of the stochastic process \J|)>\H,s|)>\M> and for which it holds <\equation*> >J|)>m\|)>=>H,s,\|)> m\|)> |]>\M,a.s. for any bounded measure on ,\|)>>. Hint: prove that <\equation*> \>H,s,\|)> m\|)> |]>\M->J|)>m\|)>|)>|]>=0. \; \; <\initial> <\collection> <\references> <\collection> >