1 MTH 5420/ORP 5020 Stochastic Processes Name _____________________________________ Spring 2014 120 points FINAL EXAM (Thursday, May 1, 6:00 to 8:00 PM) PLEASE ATTACH THIS COVER PAGE TO YOUR PAPER Please note that longer solutions can reduce your score. 1. Give the TPM and the generator of the embedded Markov chain ???? generated by the a marked Poisson process, with marks ?? distributed as ??? ???? and the initial ? ? vector x??? ? ??? ?? ????. (20 pts) 2. Let ???? be a Markov process driven by the infinitesimal generator ? ? ? ? ? . ? ? ? ? ? ? ? ??? ? ? ? ? ? ? ? ? ? ? ? ? Given the initial vector x??? ? ??? ?? ?? ?s.t. ? ? ? ? ??, find the absorption time probability density function of the r.v. ? ? min?? ? ? ? ???? ? ??. Also, identify the r.v. ? for the special case of ?? ? ??. (20 pts) 3. Let ?? be a marked Poisson process of intensity ? (with initial vector x??? ? ??? ?? ?? ?? and with marks distributed type 1 geometrically with parameter ?. Let ???? ? ????? ? ???? ? ? ? ?, with ?? ? ? ? ? ? ?? ?? being a r.v. independent of ?? and with the distribution: ??? ? ? ?? ? ? ? ??? ? ?? ? ? ? ? ? ? . Find the autocovariance function of ? ? and investigate if ???? is WSS. (20 pts) 4. For ? ? ? and ? ? ?? calculate ?? ? ? . ? ?? ? ? ???? ? ? ? ? ? ???? Also give ?? for ? ? ?. (20 pts) 5. Consider the following stochastic integral equation ???? ? ?? ????? ? ? ??????????? ??? ? ? ?
2 where is a stochastic process. Denote ???? ???? ? ??????. Find the expression for Cov?????? ????? in terms of function ????. The latter can be calculated explicitly (see Bonus Problem 2). [Notice that you have to take into account two variants with respect to ? and ?.] (20 pts) 6. Let ???? be Brownian motion adapted to a filtration ???? in ???? and let ???? and ???? be adapted processes from ?. Let ???? be a stochastic process with ???? (nonrandom and positive) which satisfies the SDE ????? ? ?????????? ? ?????????????. Find the explicit solution to this SDE, namely show that ???? ?…