Problem 3: Let (ABn) be a sequence of i.i.d. random variables with common distribution function F. This sequence mimics a Levy process in discrete time when we define recursively Bn := Bn-1+ Bn, Bo := 0. We let the filtration be generated by B; that is, Fn: = (Bo, B1, ..., Bn), n E {0, 1, 2, ...}. We define the processes recursively 50 := S+5, 50 := 1, s(1) := $2, + 512, Bn-1, S.") : = 1, S(2) :- 52 + sin(Sn21) + ABn, So := 0, S(3) : 53 + sin(Sn1) + Bn-1Bn, So := 0, S( 4) : = S( 4) Sn-1+ Sn-1 + Bn-1Bn, So := 0, (5) S(5) := S(5)+ Sm-1 + Sn 1+ Sn2 1Bn, So := 0 . (6) For all i E {0, 1, 2, 3, 4, 5} answer the following question: Is S() 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(4) such that ( S(2 ) , X (1), ..., X (k) ) becomes a Markov process