I want the answer of below question of financial mathematics

7:15 PM Sun 15 Nov Homework 2 (1 of 6) 3. Let p E (0, 1). Consider (Xi),-21 i.i.d., where each X,- = 1 with probability p, and = 1 with probability 1 p. Let Sn=X1+...+Xn (n2 1) and 30:0. Letfn=0(X,-:13in)=J(S,-:0in)(n>0). (i) Find all values of a E R such that (exp(aSn))n> 0 is a martingale with respect to (Fm-20. Consider now gambler '8 min with N 2 1 and p 6 (0,1): at each step, the gambler has a probability p to win 1, and a probability 1 p to lose 1. The fortune of the gambler at time n 2 0 is denoted by Y,1 (E {0, . . .,N}), starting from an initial fortune YO : yo 6 {1, . . . , N 1}. (ii) We rst assume that p 74 % Using (i) and the optional stopping theorem, compute the probability that the gambler eventually \"beats the casino\3. Let p 6 (0,1). Consider (X,),21 i.i.d., where each X,- = 1 with probability p, and = 1 with probability 1 3:). Let Sn=X1+...+Xn (n21) and 30:0. Letfn=J(X,- : lgign)=a(3,- : Ogign)(n20). (i) Find all values of o: 6 R such that (exp(os.'3',r,))>0 is a martingale with respect to ($0120. _ Consider now gambler's min with N 2 1 and p E (0, 1): at each step, the gambler has a probability p to win 1, and a probability 1 p to lose 1. The fortune of the gambler at time n 2 0 is denoted by Y", (E {0, . . .,N}), starting from an initial fortune Y0 = yo 6 {1, . . . , N 1}. (ii) We rst assume that p 7E %. Using (i) and the optional stopping theorem, compute the probability that the gambler eventually \"beats the casino\1. Let p E (0, 1). Consider (Xi)i>1 i.i.d., where Xi = 2 with probability p, and = -1 with probability 1 - p. Let Sn = X1+ ... + Xn (n > 1) and So = 0. Determine all values o E R such that (exp(Sn - an) )n20 is a martingale with respect to Fn = o(Xi : i1, where for each i 2 1, Xi = 1 with probability , Xi = 2 with probability , and Xi = -1 with probability ;. Let Fn := o(Xi : io as a martingale transform, and show that it is a martingale with respect to (Fn)n20. (iv) Denote o2 := Var(Xi). Show that ((Sn)2 - no?)n20 is a martingale with respect to (Fn)n20. (v) Is ((Sn)2)nyo a martingale with respect to (Fn)nzo?5. Let (Xi)izo be a discrete-time stochastic process, and consider the filtra- tion that it generates, i.e. Fn = o(Xo, ..., Xn), n 2 0. Consider two (Fi)izo-stopping times T and T'. For each of the following random times, say whether it is (always) an (Fi)>o-stopping time, or not. A detailed explanation should be given for each of them. (i) 71 = (T - 1) VO = max(T - 1, 0). (ii) 12 = T + 1. (iii) 13 = TAT = min(T, T'). (iv) TA = IT - T<><>