Problem 1. Discrete stochastic integrals for general martingales. n=0 Let (Mn) be a martingale with respect to a filtration (Fn)-0. Consider a previsible process (Zn) such that EZ2 < ∞ for all n. We define a stochastic process (In) by Io = 0, and n In = Zk (Mk - Mk−1), k=1 for k = 1,..., N. (a) Prove that E[In] = : 0. (b) Find an expression for E[12]. (Note that it will depend on M.) (c) Prove that (In) is a martingale.

Problem 1. Discrete stochastic integrals for general martingales. n=0 Let (Mn) be a martingale with respect to a filtration (Fn)-0. Consider a previsible process (Zn) such that EZ2 < ∞ for all n. We define a stochastic process (In) by Io = 0, and n In = Zk (Mk - Mk−1), k=1 for k = 1,..., N. (a) Prove that E[In] = : 0. (b) Find an expression for E[12]. (Note that it will depend on M.) (c) Prove that (In) is a martingale.

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.4: Hyperbolas

Problem 5ECP: Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.

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