(4.1) Prove that if a process X, is both a supermartingale and submartingale with respect to...

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(4.1) Prove that if a process X₁, is both a supermartingale and submartingale with respect to {Y} then it is a martingale with respect to {Y}. (4.2) Consider a Markov chain S₁, which at each transition either goes up 1 step with probability p or down 1 step with probability g = 1 - p. Prove that (q/p) is a martingale. (4.3) If X, is a supermartingale and Ta Markov time with respect to {Y}, then the stopped process XTA is a supermartingale. (4.1) Prove that if a process X₁, is both a supermartingale and submartingale with respect to {Y} then it is a martingale with respect to {Y}. (4.2) Consider a Markov chain S₁, which at each transition either goes up 1 step with probability p or down 1 step with probability g = 1 - p. Prove that (q/p) is a martingale. (4.3) If X, is a supermartingale and Ta Markov time with respect to {Y}, then the stopped process XTA is a supermartingale.

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