> <\body> <\hide-preamble> <\padded-normal|0.5fn|0.5fn> >||> > > \; \; <\center> > \; \; <\very-small> Version 2, 2020.05.07. Tutorial classes: Mon May 18th 16\U18 \ (Zoom) Min Liu \| Wed May 20th 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 14th. Use this sheet for your solutions and write them under the corresponding exercise. Fill out your names below. \; \; \; <\exercise> [Pts 3] Let be a continuous local martingale and T> two stopping times. Prove that =\\> a.s implies =M> for all > a.s. . =\>\M>>]. \; <\exercise> [Pts 3+3]\ <\enumerate-alpha> Consider the solution of the SDE in > <\equation*> \X=b|)>\t+\|)>\B,X=x, where is a -dimensional Brownian motion and \\>, :\\\d>> locally bounded continuous coefficients. Let > be the associated infinitesimal generator. Fix 0> and assume that :\\\> and \\\\0>> are continuous functions. Show that any bounded > solution \\\\> of the equation <\equation*> |||\s>u>||u-Vu,\\\,>>|>||,>>>>> has the stochastic representation <\equation*> u=\|)>expV|)>\s|)>|]>. In particular, there is at most only one solution of the PDE. =expV|)>\s|)>u|)>> is a local martingale]>. The price of a security is modeled by a geometric Brownian motion with parameters ,\\0>: <\equation*> \X=\ X \t+\ X \B,X=x\0. At price we have a running cost of > per unit time. The total cost up to time is then <\equation*> A=V|)>\s. Suppose that is a bounded solution to the PDE <\equation*> |||\s>u>||u-\Vu,\\\0>,>>|>||>>>> where > is the generator of . Show that the Laplace transform of > is given by\ <\equation*> \A>|]>=u. \; <\exercise> [Pts 3+3+3+2] Consider a family of diffusions |)>0,x\0>> satisfying the SDE <\equation*> \X=\X\t+X>\B,X=x, where \\,\\\0>>. Existence of strong solutions to this equation follows from the Yamada\UWatanabe theorem. Let ,|)>> be an independent copy of > and let =X+> for 0,x\0,y\0>. <\enumerate-alpha> Compute the SDE satisfied by and prove that |)>0>> has the same law of |)>0>>. Show that this implies that there exists a function 0>\\0>\\0>> such that <\equation> \X>|]>=e|)>>,x\\0> if we assume that the map \X>|]>> is continuous. Assume that 0>\\0>\\0>> is differentiable with respect to its first parameter. Apply Ito formula to G=e|)>X>> and determine which differential equation should satisfy in order for to be a local martingale. Prove that in this case eq. is satisfied (in particular, if a solution of the equation exists then it is unique). Find the explicit solution for the differential equation and using eq. prove that if =0> then <\equation*> \=0|)>=et|)>>,x,t\0. \; \; <\initial> <\collection> <\references> <\collection> >