1 Markov Processes and Martingales ( 20 points) Consider our usual double coin-flip example: FP({})={HH,HT,TH,TT}=D()=41 and let's consider times I={0,1,2} and the filtration F=(Ft) given by F0={,}F1={,{HH,HT},{TH,TT},}F2=Fe=() This models the information you would learn as you flip the coins. F0 is the trivial -algebra, and means you are not allowed to know anything yet. t=0 therefore represents before you flip the coins. F1 means you are allowed to know the first coin, but not the second, so t=1 represents after you flip the first coin but before you flip the second. F2 is the discrete -algebra meaning you are allowed to know everything. For each of the following four adapted stochastic processes, tell me whether they are Markov and/or a martingale. That is, tell me whether they are i) Markov only, ii) a martingale only, iii) both Markov and a martingale, or iv) neither. Hint: there are four processes and there are four possible answers. Maybe I gave you one of each? (a) The process X(0,)X(1,HH)X(1,TH)X(2,HH)X(2,HT)X(2,TH)X(2,TT)=0=X(1,HT)=1=X(1,TT)=0=2=1=1=0 (Note: this is the process that counts the number of heads.) (b) The process X(0,)X(1,HH)X(1,TH)X(2,HH)X(2,HT)X(2,TH)X(2,TT)=0=X(1,HT)=0=X(1,TT)=0=1=0=0=0 (Note: this is the process that indicates whether two heads have been observed.) (c) The process X(0,)X(1,HH)X(1,TH)X(2,HH)X(2,HT)X(2,TH)X(2,TT)=1=X(1,HT)=1.5=X(1,TT)=0.5=2=1=1=0 (Note: this is the process tracking the expected number of heads.) (d) The process X(0,)X(1,HH)X(1,TH)X(2,HH)X(2,HT)X(2,TH)X(2,TT)=0=X(1,HT)=0=X(1,TT)=0=1=1=0=0 (Note: this is the payoff from a gamble where you win $1 if both coins come up heads, lose $1 if the first coin comes up heads and the second comes up tails, and otherwise win or lose nothing.)