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Categories

Data Science & Statistics

Answered: Can properties of stochastic processes, such as the martingale property, be generalized to a corresponding local property?
Posted on Tuesday September 21, 2021

Many properties of stochastic processes, such as the martingale property, can be generalized to a corresponding local property. The local properties can be more useful than the original property...

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Answered: What is meant by filtration of \(\sigma\)-algebras?
Posted on Tuesday September 21, 2021

For an ordered set \(T\), a filtration of  \(\left(\mathcal{F}_{t}\right)_{t \in T}\) is a collection of \(\sigma\)-algebras on an underlying set \(\Omega\), satisfying \(\mathcal{F}_{s}...

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Answered: What is the filtered probability space?
Posted on Tuesday September 21, 2021

A filtered probability space, or stochastic basis, \(\left(\Omega, \mathcal{F},\left(\mathcal{F}_{t}\right)_{t \in T}, \mathbb{P}\right)\) consists of a probability space \((\Omega, \mathcal{F},...

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Answered: What is the strong law of large numbers?
Posted on Tuesday September 21, 2021

A sequence of random variables \(X_{1}, X_{2}, \ldots\) with finite expectations in a probability space is said to satisfiy the strong law of large numbers if
\[
\frac{1}{n}...

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Answered: Prove the martingale convergence theorem
Posted on Tuesday September 21, 2021

Let \(\left(X_{n}\right)_{n \in \mathbb{N}}\) be a supermartingale such that \(\mathbb{E}\left|X_{n}\right| \leq M\), and let \(a<b\). We define a random variable counting how many times the...

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