Problem 3: Let (Bn)n=1 be a sequence of i.i.d. random variables with common distribution function F. This sequence mimics a Lvy process in discrete time when we define recursively Bn:=Bn1+Bn,B0:=0! We let the filtration be generated by B; that is, Fn:=(B0,B1,,Bn),n{0,1,2,}. We define the processes recursively Sn(0):=Sn1(0)+Sn1(0),S0(0):=1,Sn(1):=Sn1(1)+Sn1(1)Bn1,S0(1):=1,Sn(2):=Sn1(2)+sin(Sn1(2))+Bn,S0(2):=0,Sn(3):=Sn1(3)+sin(Sn1(3))+Bn1Bn,S0(3):=0,Sn(4):=Sn1(4)+Sn1(0)+Bn1Bn,S0(4):=0,Sn(5):=Sn1(5)+Sn1(2)+Sn1(1)Bn,S0(5):=0. For all i{0,1,2,3,4,5} answer the following question: Is S(i) a Markov process? If your answer is yes, prove it. If you answer is no, justify your no and provide the smallest set of additional processes X(1),,X(k) such that (S(i),X(1),,X(k)) becomes a Markov process. Problem 3: Let (Bn)n=1 be a sequence of i.i.d. random variables with common distribution function F. This sequence mimics a Lvy process in discrete time when we define recursively Bn:=Bn1+Bn,B0:=0! We let the filtration be generated by B; that is, Fn:=(B0,B1,,Bn),n{0,1,2,}. We define the processes recursively Sn(0):=Sn1(0)+Sn1(0),S0(0):=1,Sn(1):=Sn1(1)+Sn1(1)Bn1,S0(1):=1,Sn(2):=Sn1(2)+sin(Sn1(2))+Bn,S0(2):=0,Sn(3):=Sn1(3)+sin(Sn1(3))+Bn1Bn,S0(3):=0,Sn(4):=Sn1(4)+Sn1(0)+Bn1Bn,S0(4):=0,Sn(5):=Sn1(5)+Sn1(2)+Sn1(1)Bn,S0(5):=0. For all i{0,1,2,3,4,5} answer the following question: Is S(i) a Markov process? If your answer is yes, prove it. If you answer is no, justify your no and provide the smallest set of additional processes X(1),,X(k) such that (S(i),X(1),,X(k)) becomes a Markov process