> <\body> <\hide-preamble> <\padded-normal|0.5fn|0.5fn> >||> > > \; \; <\center> > \; \; <\very-small> Version 1, 2020.06.10. Tutorial classes: Mon June 22nd 16\U18 \ (Zoom) Min Liu \| Wed June 24th 16\U18 (Zoom) Daria Frolova. Solutions in groups of 2 (at most). To be handled in or format via eCampus not later than 8pm Thursday June 18th. Use this sheet for your solutions and write them under the corresponding exercise. Fill out your names below. \; <\exercise> [Pts 2+2+2+2] Assume that =C0>;\|)>>, > is the \Udimensional Wiener measure and that is the canonical process on > and that the filtration >> is generated by . Consider a predictable \ >-valued drift given by a function 0>\\\\>. By tilting > via >b\X|)>> we obtain that, under the tilted measure > the process is a solution of the SDE <\equation*> \X=b+\W,t\0 where is a >\UBrownian motion. <\enumerate-alpha> Prove that if\ <\equation*> |\|>\C|\|>|)>,t\0,x\\, then Novikov's condition holds conditionally on > for intervals > such that > is small enough, i.e. <\equation*> \|\|>\u|)>\|\|]>\+\. Deduce that is a martingale. [Hint: prove that \|\|]>=Z> for small time intervals > and the conclude].\ Prove that <\equation*> \|X|\<\|\|\>>>\r|)>\2*d e/2*d t>t\0,r\0. where |X|\<\|\|\>>>> denotes the supremum wrt. the Euclidean norm of |)>>>. [Hint: use Doob's inequality for the submartingale X>> and optimize over \0>] Prove the same result as in (a) under the more general assumption that is a previsible drift such that <\equation*> |\|>\C|x|\<\|\|\>>,>|)>,t\0,x\\ where +\>. <\exercise> [Pts 2+2+2] Consider the one dimensional SDE <\equation*> \X=-X\t+\B,X=x, where is a standard Brownian motion.\ <\enumerate-alpha> Let =|)>> and =inf0:|\|>\L|}>>. Use Ito formula to show that there exists a constant > such that the process \eT|)>>fT>|)>> is a supermartingale.\ Deduce that \t|)>\0> as \>.\ Conclude that solutions of the SDE cannot explode (that is \sup T=\> a.s.). <\exercise> [Pts 2+2+2] If is a smooth curve in > with c(0) = 0,\ <\equation*> A=y-yx|)>\s describes the area that is covered by the secant from the origin to in the interval . Analogously, for a two-dimensional Brownian motion = (X, Y)> with = 0>, one defines the Lévy Area <\equation*> A=\Y-Y\X|)>. <\enumerate-alpha> Let (t)>, (t)> be >-functions, \>, and\ <\equation*> V=i*p*A-|2>+Y|)>+\. Use Itô formula to show that >> is a local martingale provided =\-p> and =\> Let \0>. Solutions to the equations for ,\> with |)>=\|)>=0> are\ <\equation*> \=p*tanh-t|)>|)>,\=-log cosh-t|)>|)>. Conclude that\ <\equation*> \>>|]>=|)>|)>. Show that the distribution of > is absolutely continuous with respect to the Lebesgue measure with density <\equation*> f>= x/2t|)>|)>,x\\. \; <\initial> <\collection> <\references> <\collection> >