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17 Oct 2019: b) Hence show that, for all such claims C, the no-arbitrage time-0 price is given by erTE(C)where P is an equivalent probability measure on FT , to be ... a) Find the price at time 0 of such an option in the Black–Scholes model with interest rate r

16 Nov 2020: Let Xdenote the number of fixed points of σ, i.e. the number of 1 i n such that σ(i) = i. ... π(x)Ea[τ]. (b) Consider a finite, irreducible and aperiodic Markov chain.

7 Oct 2019: Use (b) to find random variables (Mn)n3 such that, forall n 0,. ... b) there exists an equivalent probability measure P̃ such that P̃ = P on F0 and, almost surely,.

18 Nov 2011: 2. Assuming Prohorov’s theorem for probability measures on Rd, show that if (µn,n 0) isa sequence of non-negative measures on Rd which is tight and such that. ... 3. Deduce that a.s. there exists no interval [a,b] with a < b such that B is Hölder

3 Oct 2021: Consider the followingconditions. (a) T n for some n 0,(b) there is a constant C < such that |Xn| C for all n T almost surely,(c) E(T) < and there ... is a constant C < such that |Xn1 Xn| C for all n < T.

16 Nov 2020: P 2tmix(x, y) 14π(y). and that there exists a transition matrix P̃ such that. ... PB(x, y) = P(x, y for x, y B)) is irreducible, in thesense that for all x, y B, there exists n 0 such that P nB(x, y) > 0.

20 Jan 2016: exist a,b,c,d R with ad bc = 1 such that. ... Show that thereexists λ (0,) such that σ(z) = λz for all z.

2 Nov 2011: E[eθY ] cosh θc exp(. 1. 2θ2c2. ). (b) Prove that if M is a martingale, with M0 = 0 and such that for some sequence (cn : n N)of positive constants, |Mn ... An is Fn1 measurable) such that A0 = 0, A is increasing and X = M A.2.

22 Feb 2016: X]t =. t0Asds. for a non-negative, previsible process (At)t0. Show that there exists a Brownian motion B (possiblydefined on a larger probability space) such that. ... Show thatthere exists a Brownian motion such that dXt = b(Xt)dt σ(Xt)dWt.

2 Oct 2009: algebra. 1.2 Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a, b) : a < b}, (b) {(a, b] : a < b}, (c) {(, b] : b ... Show that for every ε > 0, thereexists a finite union of disjoint intervals A = (a1, b1] (an, bn]