Submartingale Last updated
Nov 1, 2022
Table of Contents Properties Remarks # Definition 1Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) Probability Space
Probability Space
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. Let X : Ω → R T X: \Omega \to \mathbb{R}^{T} X : Ω → R T Adapted Process
Adapted Process
Definition
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) X : Ω → S T X: \Omega \to S^{T} X : Ω → S T ( S , Σ ) (S, \Sigma) ( S , Σ )
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wrt Filtration
Filtration
Definition
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) ( I , ≤ ) (I, \leq) ( I , ≤ ) B i ⊂ A \mathcal{B}{i} \subset \mathcal{A} B i ⊂ A
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$\mathcal{F}{*} := (\mathcal{B} {t})_{t \in T}o n on o n . . . Supermartingale
Supermartingale
Definition 1
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) X : Ω → R T X: \Omega \to \mathbb{R}^{T} X : Ω → R T
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if it satisfies the following properties:
X t ∈ L 1 ( Ω ) X_{t} \in L^{1}(\Omega) X t ∈ L 1 ( Ω ) ∀ t ∈ T \forall t \in T ∀ t ∈ T $\mathbb{E}(X_{t} | \mathcal{B}{r}) \geq X {r}$ Almost Surely for any r ≤ t r \leq t r ≤ t # PropertiesAn Adapted Process is a Supermartingale iff its negative is a Submartingale Expectations of a Submartingale are Non-Decreasing
Expectations of a Submartingale are Non-Decreasing
Statement
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) ( X t ) t ∈ T (X{t}){t \in T} ( X t ) t ∈ T
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# Definition 2Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) Probability Space
Probability Space
...
11/7/2022
. Let X : Ω → R N X: \Omega \to \mathbb{R}^{\mathbb{N}} X : Ω → R N Adapted Process
Adapted Process
Definition
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) X : Ω → S T X: \Omega \to S^{T} X : Ω → S T ( S , Σ ) (S, \Sigma) ( S , Σ )
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wrt Filtration
Filtration
Definition
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) ( I , ≤ ) (I, \leq) ( I , ≤ ) B i ⊂ A \mathcal{B}{i} \subset \mathcal{A} B i ⊂ A
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$\mathcal{F}{*} := (\mathcal{B} {n})_{n \in \mathbb{N}}o n on o n . . . Submartingale
Submartingale
Definition 1
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) X : Ω → R T X: \Omega \to \mathbb{R}^{T} X : Ω → R T
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if it satisfies the following properties:
X n ∈ L 1 ( Ω ) X_{n} \in L^{1}(\Omega) X n ∈ L 1 ( Ω ) ∀ n ∈ N \forall n \in \mathbb{N} ∀ n ∈ N $\mathbb{E}(X_{n+1} | \mathcal{B}{n}) \geq X {n}$ Almost Surely for any n ∈ N n \in \mathbb{N} n ∈ N Like in Martingale
Martingale
Definition 1
Let ( Ω , A , P ) (\Omega, \mathcal{A}, \mathbb{P}) ( Ω , A , P ) X : Ω → R T X: \Omega \to \mathbb{R}^{T} X : Ω → R T
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, Definition (2) and (1) are equivalent. The proof is very similar, except we also require Conditional Expectation is Non-Decreasing
Conditional Expectation is Non-Decreasing
Statement
Let ( Ω , G , P ) (\Omega, \mathcal{G}, \mathbb{P}) ( Ω , G , P ) B ⊂ G \mathcal{B} \subset \mathcal{G} B ⊂ G G \mathcal{G} G
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.