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A Probability Path.pdf

# 50 two genetics models let xn n 0 be a markov chain

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50. Two genetics models. Let {Xn, n ::: 0} be a Markov chain on the state space {0, 1, ... , M} with transition matrix {p;j. 0 :::; i, j :::; M}. For the following two models, compute 1/J; := P[ absorbtion at MJXo = i] and show 1/J; is the same for either model. (a) Wright model: ( M) ( · )j ( · )M-j Pij = j 1- , fori= 0, ... , M; j = 0, ... , M. (b) Moran model: Define and set i(M- i) p;= M2 i =O , ... ,M Pij=p;, j=i-1,i+1,i:f=OorM, POj =8oj, PMj = 8Mj, Pij =0, otherwise. 51. (a) Suppose {(Xn, Bn), n ::: 0} is a positive supermartingale and v is a stopping time. Show (b) Suppose Xn represents an insurance company's assets at the start of year n and suppose we have the recursion X n+l = X n + b - Yn, where b is a positive constant representing influx of premiums per year and Yn,

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440 10. Martingales the claims in yearn, has N(JL, a 2 ) distribution where f.L <b. Assume that {Yn, n 1} are iid and that Xo = 1. Define the event 00 [Ruin]= Urxn < 0]. n=l Show P[Ruin] e-Z(b-J.L)/u 2 (Hint: Check that {exp{ -2(b - JL)a- 2 Xn, n 0} is a supermartingale. Save come computation by noting what is the moment generating function of a normal density.) 52. Suppose Bn t !3 00 and fYn, n E N} is a sequence of random variables such that Yn -+ Y 00 (a) If IYn I Z E L 1, then show almost surely that (b) IfYn Y 00 , then in L1 (c) Backwards analogue: Suppose now that Bn .j.. !3_ 00 and IYnl =::: Z E L1 and Yn -+ Y -oo almost surely. Show almost surely. 53. A potential is a non-negative supermartingale {(Xn. Bn). n 0} such that E(Xn) -+ 0. Suppose the Doob decomposition (see Theorem 10.6.1) is Xn = Mn -An. Show 54. Suppose f is a bounded continuous function on lR and X is a random vari- able with distribution F. Assume for all x E lR f(x) = k. f(x + y)F(dy) = E(f(x +X)). Show using martingale theory that f (x + s) = f (x) for each s in the support of F. In particular, if F has a density bounded away from 0, then f is constant. (Hint: Let {X n} be iid with common distribuion F and define an appropriate martingale.)
10 . 17 Exercises 441 55. Suppose {X j , j 1} are iid with common distribution F and let Fn be the empirical distribution based on Xt. ... , Xn . Show Yn :=sup IFn(x)- F(x)l xe!R is a reversed submartingale. Hint: Consider first {Fn(x)- F(x), n 1}, then take absolute values, and then take the supremum over a countable set.) 56. Refer to Subsection 10.16.5. A price system is a mapping n from the set of all contingent claims X to [0, oo) such that n(X)=OiffX=O, VXeX, n(aX + bX') = an(X) + bn(X'), for all a 0, b 0, X, X' E X . The price system n is consistent with the market model if n(VN(¢)) = n(Vo(¢)), for all admissible strategies ¢ . (i) If P* is an equivalent martingale measure, show n(X) := E*(X/S<:\ "'X EX, defines a price system that is consistent. (ii) If n is a consistent price system, show that P* defined by is an equivalent martingale measure. (iii) If the market is complete, there is a unique initial price for a contingent claim.

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S.I. Resnick, A Probability Path , Modern Birkhäuser Classics, ss Media New York 2014 DOI 10.1007/978-0-8176-8409-9, © Springer Science+Busine 443 References [BD91] P. J. Brockwell and R. A. Davis.
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