# Questions tagged [martingales]

For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.

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### Does the Martingale Representation theorem hold both ways?

Can the Martingale Representation theorem be used to assume that the integral with respect to Brownian motion, $B(t,\omega)$, $$X=\int^{T}_{0}B^{4}(t,\omega)dB(t,\omega)$$ is a square integrable ...
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### Counterexample for continuous time sub-martingale convergence

We know that if $X_t$ is a right continuous sub-martingale with $\sup_t \mathbb{E}[X_t^+] < \infty$ then $\lim_{t \rightarrow \infty} X_t$ exists almost surely, but I haven't been able to find a ...
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### Show that a L2 martingale with stationary, independent increments has a certain quadratic variation

I need help solving the following exercise out of chapter 1 of Karatzas, Shreve: 5.20 Exercise. Suppose $X\in\mathscr{M}_2$ has stationary, independent increments, and $(\mathscr{F}_t)_t$ is the ...
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### Quadratic variation of true martingale

We know that for a continuous local martingale $M$ the quadratic variation $\left<M \right>$ is such that $M^2- \left< M \right>$ is a continuous local martingale. Is it true that if $M$ ...
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### $L^p$ Boundedness of a martingale

So I recently read a paper where the authors claim that if for some martingale $(M_t)_{t\geq 0}$ we have $$\mathbb E[M_{t+s}^p]-\mathbb E[M_s^p]\leq \exp(-cs) (\mathbb E[M_{t}^p ]-\mathbb E[M_t]^p)$$ ...
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### The conditional expectation of iid random variables

Let $Y_1,Y_2,\dots,Y_n$ be a sequence of iid random variables. Each of them is integrable. Let $X_1=(Y_1+Y_2+\cdots+Y_n)/n,X_2=(Y_1+Y_2+\cdots+Y_{n-1})/(n-1),\dots,X_{n-1}=(Y_1+Y_2)/2,X_n=Y_1$ ...
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### Given a martingale $Y=\mathbb{E}\{M_n|\mathcal{F}_n\}$, why does the below red inequality hold true?

Given a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$, let $Y$ be a random variable defined on it such that $Y\in \mathcal{L}^1$ and $M_n=\mathbb{E}\{Y|\mathcal{F}_n\}$ be a martingale. ...
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### Show martingality property for r.v. $S_{\infty}$, given some assumptions. Could you please detail your answer to my two points?

Given a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$, let $(S_n)_{n\geq1}$ be a martingale and a sequence of uniformly integrable random variables. Also assume $S_n\rightarrow S_{\infty}$ ...
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### $X=M+A$ bounded semimartingale, then $M$ and $A$ are bounded? Counterexample?

Let be $X=M+A$ a semimartingale with $M$ being a local martingale and $A$ an adapted process a finite variation. If $M$ and $A$ are bounded, then of course $X$ is bounded as well. Is the converse true?...
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### Optional sampling theorem St. Petersburg paradox

I want to show that the optional sampling theorem does not hold for unbounded stopping times using the example of the St. Petersburg paradox/St. Petersburg game. We have a consecutive (fair) coin toss ...
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### Show this statement holds for a right continuous martingale.

Let $T > 0$ and consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ on which there is a filtration $$\{\mathcal{F}_t: 0 \leq t \leq T\}$$ Let $M:=\{M_t: 0 \leq t \le T\}$ be a ...
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### A doubt on the proof of Martingale Convergence Theorem on Jacod-Protter

Theorem: Let $(X_n)_{n\geq1}$ be a submartingale such that $\sup\limits_{n}\mathbb{E}\{X_n^{+}\}<\infty$. Then, $\lim\limits_{n\rightarrow\infty} X_n = X$ exists a.s. (and is finite a.s.). Moreover,...
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$\rho,\tau$ - stopping times such that $0\le \rho\le\tau\le T$ and $M(t),F_t)$ square integrable martingale. Prove that $\mathbb{E}((M(\tau)-M(\rho))^2|F_{\rho})=\mathbb{E}(M^2(\tau)-M^2(\rho)|F_{\rho}... 1answer 19 views ### Why the process$n \mapsto e^{-aT_n}f(X_{T_n})$is a super martingale. Let$M := (e^{-aT_n}f(X_{T_n}); \mathcal{F}_n)$, where$a$is constant,$\mathcal{F}_n$is appropriately defined filtration and$T_n$is$n$th jump of an independent (independent of$X_t$) Poisson ... 0answers 39 views ###$\text{lim inf}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}}\leq \text{lim sup}_{t \downarrow 0}\frac{\int_{0}^{t}E[X_{s}^{2}]ds}{t^{2}}$Let$g(t) = E[X_{t}^{2}]$, the starting point is the following set of inequalities which hold for all$t \geq 0: \begin{align} - \int _{0}^{t} 2g(s)ds + t \leq g(t) \leq \int _{0}^{t} 2g(s)ds + t \... 0answers 13 views ### I am trying to construct a processY$in the martingale representation of the following two terms (i)$M_t = \mathbb{E}[B_t^2|F_t]$(ii)$M_t = \mathbb{E}[\text{max}\{B_t,0\}|F_t]$Where$B$is a one-dim. Brownian Motion and$F$its$P$-completed canonical filtration. I think in (ii) we can ... 1answer 26 views ### Prove that process is a local martingale.$ \{X_{t}\}_{t\geq 0}, \{Y_{t}\}_{t\geq 0} $- Ito processes. From Ito's formula we have got:$$X_{t}Y_{t} = X_{0}Y_{0} + \int_{0}^{t}X_{s}dY_{s} + \int_{0}^{t}Y_{s}dX_{s} + \int_{0}^{t}dX_{s}dY_{s} ... 1answer 24 views ### Why is the fact that the sequence$(M_n)_{n\geq0}$is increasing shown in the following way? My alternative I quote Jacod-Protter Given a probability space$\left(\Omega,\mathcal{F},\mathbb{P}\right)$and an increasing sequence of$\sigma$-algebras$\left(\mathcal{F}_n\right)_{n\geq0}$, let$M=\left(M_n\...
Establish the optional sampling theorem for a right-continuous submartingale $\{X_{t},\mathcal{F}_{t}, t \in [0,\infty)\}$ under either of the following conditions: $a)$ $T$ is a bounded optional ...