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Question 5 (a) Consider a Poisson process {N}+20 with rate > 0. Show that Xt = Nt - Xt is both a Markov process and a martingale with respect to the natural filtration {F}t, where F is the o-algebra generated by {N,: 0

Question 5 (a) Consider a Poisson process {N}+20 with rate > 0. Show that Xt = Nt - Xt is both a Markov process and a martingale with respect to the natural filtration {F}t, where F is the o-algebra generated by {N,: 0

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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Question 5 (a) Consider a Poisson process {N}+20 with rate > 0. Show that Xt = Nt - Xt is both a Markov process and a martingale with respect to the natural filtration {F}t, where F is the o-algebra generated by {N,: 0<s≤t}. (b) Let {X}to be a martingale with respect to a filtration F≥0. Suppose T is a stopping time with respect to {F}t (i.e. {T ≤ t} € Ft) for all t≥ 0. Prove that if T is a bounded stopping time, then {XT^t}+20 is also a martingale, where T^t= min (T, t).
Transcribed Image Text:Question 5 (a) Consider a Poisson process {N}+20 with rate > 0. Show that Xt = Nt - Xt is both a Markov process and a martingale with respect to the natural filtration {F}t, where F is the o-algebra generated by {N,: 0<s≤t}. (b) Let {X}to be a martingale with respect to a filtration F≥0. Suppose T is a stopping time with respect to {F}t (i.e. {T ≤ t} € Ft) for all t≥ 0. Prove that if T is a bounded stopping time, then {XT^t}+20 is also a martingale, where T^t= min (T, t).
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