# Questions tagged [martingales]

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

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### Brownian motion: how is $\mathbb E [\sup_{t \ge 0} |\langle M\rangle_{t \wedge \tau_n}|^2] < \infty$ satisfied?

I'm reading a theorem at page $43$ of these notes, i.e., Proposition 7.12. Let $M$ be a continuous local martingale with respect to a filtration $\left(\mathcal{F}_t, t \in \mathbb{R}_{+}\right)$ ...
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### A proof of Lévy's theorem: how to obtain the independence $X_t-X_s \perp \mathcal{F}_s$?

I'm reading about Lévy's theorem at page $42$ of these notes, i.e., Let $X$ be a continuous local martingale such that $X_0=0$ a.s. and $\langle X\rangle_t=t$ a.s., $\forall t \in \mathbb{R}_{+}$. ...
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### 20-sided dice 100 rounds game

In a game there is a 20-sided die. At the start of the game it is on the table and the 1 face is facing upright. In each of the 100 rounds you have 2 options: you can either roll the dice or you can ...
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### Matrix Azuma Inequality with non-zero mean

Based on theorem 7.1 of the paper User-friendly tail bounds for sums of random matrices, we have: Consider a finite adapted $\{ \mathbf{X}_k\}_{k=0}^{\infty}$ of self-adjoint matrices in dimension $d$,...
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### Can this integrability assumption be weakened in a way that the stochastic integration by parts still holds?

$\newcommand{\Ex}{\mathbb E} \newcommand{\diff}{~\mathrm d}$Recently, I have read the integration by parts formula for a continuous semi-martingale in these notes. Theorem Let $X$ and $Y$ be ...
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### Let X be a semimartingale, prove that $[X,[X,X]^c] = 0$

Let $X$ be a semimartingale and $[X,X]^c$ the continuous part of the quadratic variation of $X$. I need to prove that $[X,[X,X]^c] = 0$. I thought it might be useful to divide the SM into its ...
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We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn. $(X_n)_{n \in \mathbb N}... 0 votes 0 answers 29 views ### How to show that Ornstein-Uhlenbeck is not a martingale with$E[X(t)]=E[X(s)]$for all$s,t\in [0,\infty)$? If we are given that$X=\{X(t)\}_{t\in[0,\infty)}$is an O-U-Process, which is defined as the continuous centered Gaussian process with covariance function given by $$\Gamma(s,t)=\frac{\sigma^2}{2\... 1 vote 0 answers 33 views ### Show that the following sequence is a martingale. Exercise: Let X_i be a sequence of independent random variables with \mathbb{E}[X_i] = 0 and \text{Var}(X_i) = \sigma_i^2. Show that the sequence$$ S_n = \sum_{i=1}^{n} (X_i^2 - \sigma_i^2) $$... 2 votes 0 answers 42 views ### Proving that a Stochastic integral is a Martingale I am trying to prove the following:$$\lim_{n\rightarrow \infty} E\left(\int_{0}^{t \wedge \tau_{n}} e^{-\delta s} v(X_{s})X_{s} dW_{s}\right)=0$$where \tau_{n} is a sequence of stopping times with ... 2 votes 1 answer 79 views ### Meaning of: A continuous martingale of finite varaition is almost surely constant Theorem: If \{X(t)\}_{t \in [0,\infty)} is a continuous martingale of finite variation then almost surely it holds that X(t)=X(0) for all t \in [0,\infty). Does this mean that almost every path ... 0 votes 1 answer 33 views ### How the requirement of Itô's lemma is satisfied in this theorem about integration by parts? I'm reading a theorem (about integration by parts) from page 9 of these notes, i.e., Theorem Let X and Y be continuous semi-martingales such that$$ \mathbb E \bigg [ \int_0^t X_s^2 \mathrm d \... 3 votes 1 answer 497 views ### What's the precise statement of the continuous-time optional stopping theorem? I searched high and low in a number of probability / financial mathematics textbooks and surprisingly cannot find any precise statement of the continuous time optional stopping theorem. In particular, ... 0 votes 0 answers 23 views ### Under which condition is a stochastic integral a Martingale I know that for a predictable process$H(s)$with$\mathbb{E}\left[\int_0^T H(s)d \langle M\rangle(s) \right]<\infty$the stochastic integral w.r.t to a continuous Martingale$M$is a Martingale. ... 0 votes 1 answer 46 views ### Why is the completeness of the filtration needed here? I am studying how the usual conditions of the filtration are used. Answers in this post In the definition of a Stopping Time, how important are the conditions on the Filtration being complete and ... 1 vote 2 answers 265 views ### Show that a martingale$\{X_n\}$is bounded in$L^2$if and only if$EX_n^2<\infty$for each$n$and$\sum_{n\ge1}E(X_{n+1}-X_n)^2<\infty$A martingale$\{X_n\}$is bounded in$L^2$by definition if$\sup\limits_nEX_n^2<\infty$. Show that a martingale$\{X_n\}$is bounded in$L^2$if and only if$EX_n^2<\infty$for each$n$and \... 1 vote 1 answer 50 views ### Quadratic Variation of a Semimartingale and jumps Consider the process$X(t) = A(t) + M(t)$, where$\{A\}$is a finite variation predictable process and$\{M\}$is a local martingale. Suppose that$\{A\}$does not have any jump, hence it is a ... 2 votes 1 answer 65 views ### How$\int_1^{\infty} P[Y>t] dt \leq \frac 1 {\alpha -1}$in this proof of martingale maximal inequality? I have just encountered this question Suppose$({X_n}, n \in \mathbb N)$is a martingale. Let$n \ge 1$and$\alpha > 1$such that$E\left[|X_n|^\alpha\right]<\infty$. Then$$E\left[\max_{0\... 0 votes 0 answers 22 views ### Does$X_t = M_t+V_t$have to hold everywhere in the definition of a semi-martingale? I'm reading about continuous semi-martingale from page 8 of these notes. Let$(\Omega, \mathcal F, \mathbb P)$be a probability space. Definition. A continuous semi-martingale w.r.t. a filtration$(\...
I'm reading a proof of $L^p$ maximal inequality from these notes. In the proof, we have $\left\|X_n^*\right\|_p^p \le C_p^p\left\|X_n\right\|_p \left\|X_n^*\right\|_p^{p / q}$ after Hölder's ...