world:martingale
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world:martingale [2021/03/20 08:46] rdouc [Proof] |
world:martingale [2022/03/16 07:40] (current) |
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| - | ====== Statement: Martingale convergence results ====== | + | ===== Supermartingale convergence results ===== |
| <WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
| **__Theorem__**. | **__Theorem__**. | ||
| - | Let $\mcf=(\mcf_n)_{n\in\nset}$ be a filtration and let $\seq{X_n}{n\in\nset}$ be an $\mcf$-adapted sequence such that | + | Let $\mcf=(\mcf_n)_{n\in\nset}$ be a filtration and let $\seq{X_n}{n\in\nset}$ be an $\mcf$-adapted sequence of $\lone$-random variables such that |
| * $M=\sup_{n\in\nset} \PE[(X_n)^-]<\infty$ | * $M=\sup_{n\in\nset} \PE[(X_n)^-]<\infty$ | ||
| * for all $n\geq 1$, we have $\PE[X_{n}|\mcf_{n-1}]\leq X_{n-1}$. | * for all $n\geq 1$, we have $\PE[X_{n}|\mcf_{n-1}]\leq X_{n-1}$. | ||
| that is, $\seq{X_n}{n\in\nset}$ is a $(\mcf_n)_{n\in\nset}$-supermartingale, with negative part bounded in $\lone$. | that is, $\seq{X_n}{n\in\nset}$ is a $(\mcf_n)_{n\in\nset}$-supermartingale, with negative part bounded in $\lone$. | ||
| - | Then, almost surely, $X_\infty=\lim_{n\to\infty} X_n$ exists and is finite. | + | Then, almost surely, $X_\infty=\lim_{n\to\infty} X_n$ exists and is in $\lone$. |
| </WRAP> | </WRAP> | ||
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| - | ===== Proof ===== | + | ==== Proof ==== |
| Let $a<b$ and define $C_1=\indiacc{X_0<a}$ and for $n\geq 2$, | Let $a<b$ and define $C_1=\indiacc{X_0<a}$ and for $n\geq 2$, | ||
| $$ | $$ | ||
| - | C_n=\indiacc{C_{n-1}=1,X_{n-1} \leq b}+\indiacc{C_{n-1}=0,X_{n-1} \leq a} | + | C_n=\indiacc{C_{n-1}=1,X_{n-1} \leq b}+\indiacc{C_{n-1}=0,X_{n-1} < a} |
| $$ | $$ | ||
| In words, the first time $C_n $ flags $1$ is when $X_{n-1}<a$. Then it flags $1$ until $X_{n-1}$ goes above $b$. Then it flags $0$ until $X_n$ goes below $a$. So consecutive sequences of $C_n=1$ are linked with upcrossings of $[a,b] $ for $(X_n) $. Now, define | In words, the first time $C_n $ flags $1$ is when $X_{n-1}<a$. Then it flags $1$ until $X_{n-1}$ goes above $b$. Then it flags $0$ until $X_n$ goes below $a$. So consecutive sequences of $C_n=1$ are linked with upcrossings of $[a,b] $ for $(X_n) $. Now, define | ||
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| & \leq \sum_{a,b\in \mathbb{Q}, a<b} \PP(U_\infty[a,b]=\infty)=0 | & \leq \sum_{a,b\in \mathbb{Q}, a<b} \PP(U_\infty[a,b]=\infty)=0 | ||
| \end{align*} | \end{align*} | ||
| - | which shows that $X_\infty=\lim_{n \to \infty} X_n$ exits almost surely. Morevoer, $\PE[X_0] \geq \PE[X_n]=\PE[X^+_n]-\PE[X^-_n]$ so that | + | which shows that $X_\infty=\lim_{n \to \infty} X_n$ exits almost surely. |
| + | |||
| + | Moreover, $\PE[X_0] \geq \PE[X_n]=\PE[X^+_n]-\PE[X^-_n]$ so that | ||
| $$ | $$ | ||
| - | \sup_n \PE[|X_n|]=\sup_n \lr{\PE[X^+_n]+\PE[X^-_n]} \geq \PE[X_0] +2 \PE[X^-_n] \leq \PE[X_0] +2 M<\infty | + | \sup_n \PE[|X_n|]=\sup_n \lr{\PE[X^+_n]+\PE[X^-_n]} \leq \PE[X_0] +2 \sup_n \PE[X^-_n] \leq \PE[X_0] +2 M<\infty |
| $$ | $$ | ||
| - | which implies by Fatou's lemma that $\PE[|X_\infty|]=\PE[\liminf_{n}|X_\infty|] \leq \liminf_{n} \PE[|X_\infty| \leq \sup_n \PE[|X_n|]<\infty$ | + | which implies by Fatou's lemma that $\PE[|X_\infty|]=\PE[\liminf_{n}|X_n|] \leq \liminf_{n} \PE[|X_n|] \leq \sup_n \PE[|X_n|]<\infty$. The proof is completed. |
| - | ===== Corollary ===== | + | ===== Corollary: Submartingale convergence results ===== |
| {{anchor:submartingale:}} | {{anchor:submartingale:}} | ||
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| <WRAP center round tip 80%> | <WRAP center round tip 80%> | ||
| - | **__Corollary__** Assume that $\mcf=(\mcf_n)_{n\in\nset}$ is a filtration and let $\seq{X_n}{n\in\nset}$ be an $\mcf$-adapted sequence such that | + | **__Corollary__** Assume that $\mcf=(\mcf_n)_{n\in\nset}$ is a filtration and let $\seq{X_n}{n\in\nset}$ be an $\mcf$-adapted sequence of $\lone$ random variables such that |
| * $M=\sup_{n\in\nset} \PE[(X_n)^+]<\infty$ | * $M=\sup_{n\in\nset} \PE[(X_n)^+]<\infty$ | ||
| * for all $n\geq 1$, we have $\PE[X_{n}|\mcf_{n-1}]\geq X_{n-1}$. | * for all $n\geq 1$, we have $\PE[X_{n}|\mcf_{n-1}]\geq X_{n-1}$. | ||
| that is, $\seq{X_n}{n\in\nset}$ is a $\mcf$-submartingale, with positive part bounded in $\lone$. | that is, $\seq{X_n}{n\in\nset}$ is a $\mcf$-submartingale, with positive part bounded in $\lone$. | ||
| - | Then, almost surely, $X_\infty=\lim_{n\to\infty} X_n$ exists and is finite. | + | Then, almost surely, $X_\infty=\lim_{n\to\infty} X_n$ exists and is in $\lone$. |
| </WRAP> | </WRAP> | ||
world/martingale.1616226371.txt.gz · Last modified: 2022/03/16 01:37 (external edit)
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