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.
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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a1c925e-790f-448e-9276-e5adcf0e8758%2F9eb11759-f171-4a0b-a65d-81744bca2af7%2F2lnif4_processed.png&w=3840&q=75)
Transcribed Image Text: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.
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