Paper 1, Section II, J

Stochastic Financial Models | Part II, 2017

(a) What does it mean to say that (Xn,Fn)n0\left(X_{n}, \mathcal{F}_{n}\right)_{n \geqslant 0} is a martingale?

(b) Let Δ0,Δ1,\Delta_{0}, \Delta_{1}, \ldots be independent random variables on (Ω,F,P)(\Omega, \mathcal{F}, \mathbb{P}) with E[Δi]<\mathbb{E}\left[\left|\Delta_{i}\right|\right]<\infty and E[Δi]=0,i0\mathbb{E}\left[\Delta_{i}\right]=0, i \geqslant 0. Further, let

X0=Δ0 and Xn+1=Xn+Δn+1fn(X0,,Xn),n0X_{0}=\Delta_{0} \quad \text { and } \quad X_{n+1}=X_{n}+\Delta_{n+1} f_{n}\left(X_{0}, \ldots, X_{n}\right), \quad n \geqslant 0

where

fn(x0,,xn)=1n+1i=0nxif_{n}\left(x_{0}, \ldots, x_{n}\right)=\frac{1}{n+1} \sum_{i=0}^{n} x_{i}

Show that (Xn)n0\left(X_{n}\right)_{n \geqslant 0} is a martingale with respect to the natural filtration Fn=\mathcal{F}_{n}= σ(X0,,Xn)\sigma\left(X_{0}, \ldots, X_{n}\right).

(c) State and prove the optional stopping theorem for a bounded stopping time τ\tau.

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