# Tagged Questions

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

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### Prove that a process with independent increments and a constant mean is a martingale

how to prove that a process with independent increments and a constant mean is a martingale? in a solution to this problem i found this : $X_t - X_s$ is independent from $F_s$ hence : ...
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### What is the difference between a martingale and doob's martingale?

Every sequence that was termed as a doob's martingale, I was able to deduce that it was also a martingale. So here are few of my questions: 1) Is it correct to say that every doob martingale is also ...
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### Sufficient unconditional moment condition for the convergence of $\sum_n (X_n - E[X_n])$

Let $\mathcal F_n$ be a filtration and $X_n$ be $\mathcal F_n$ measurable. Then $M_n = \sum_{k=1}^{n} (X_k -E[X_k])$ is a $\mathcal F_n$ measurable martingale. Let's assume that it is a square ...
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### Absolute expectation of stopped martingale

Let $M_0,M_1,\dots$ be a martingale with respect to $X_0,X_1,\dots$ and $T$ be a stopping time with respect to $X_0,X_1,\dots$ Define $T_n=\min\{n,T\}$ and let $M_{T_n}$ be the stopped martingale. By ...
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### Martingale Transform counterexample

I am studying discrete time martingale theory and came across the classical "You can't beat the system" theorem: given a martingale $M$ and a previsible process $C=(C_n)_{ n \ge 1}$ such that $C_n$ is ...
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### Modelling the ballot theorem as a martingale.

The page 19 in the link http://www.imada.sdu.dk/~jbj/DM839/FL15.pdf provides the explanation of what a ballot theorem is and how we can prove that it is a martingale. It takes a random variable $S_k$ ...
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### Stopping time and the Martingale stopping theorem.

According to the book that I am reading, A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of ...
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### A clarification on $L_{loc}^2$ process and stochastic exponential

In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process. Later (pp. 329-330) for a process ...
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### Conditional expectation and Radon Nikodym derivative.

Assume $(\Omega,\mathcal{F}, \mathbb{P})$ is a probability space, $\{F_t\}_{t\leq T}$ is an adapted filteration and $M_t$ is a martingale with respect to that with $M_0=1$. We can define another ...
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### Existence of compensator process under the assumption of local integrability and finite variation

I am reading a proof regarding existence of compensators under the assumption of local integrability in which I don't quite understand: Definition: The compensator of a cadlag adapted process $X$ ...
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### Theorem 4.14 Brownian Motion and Stochastic Calculus

I have been reading the proof of Theorem 4.14 of Karatzas' book. I wonder whether there is a typo in the description of the process $\eta^{(n)}_{t}$ as $\xi^{(n)}_{t+}-\min({\lambda,A_{t} })$ ...
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### Why does this “wlog” make sense: $L^p$-norms of random variables

Let $$\overline{X_n}:=\max_{0 \leq m \leq n} X_m^+$$ for a sequence of random variables $X_i, i \geq 1$ (in fact, it is a submartingale), where $X_m^+:=\max(X_m,0)$. We want to show that ...
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### A simple symmetric random walk is adapted

$\newcommand{\ee}{\mathbb{E}}$The fact that for all $n$ we have $\ee[S_n \mid \mathcal{F_{n-1}}]=S_{n-1} ~\text{a.s.}$ and $\ee[ |S_n|]<\infty$ is usually shown explicitly when showing something ...
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### A stochastic process $X$ with values in a separable Banach space $E$ is a martingale iff $f(X)$ is a martingale for all $f\in E^\ast$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space and ...
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### How to generalize a fact (convex function of a mtg is submtg) about martingales to multivalued martingales?

It's known that a convex function of a martingale is a submartingale. What about martingales with values in $\mathbb{R}^{n}$? Is is true that a subharmonic function of such a martingale is a ...
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### Is this process a martingale?

Given $X_t=\int_0^t s W_s dW_s$ and the process $M_t=X_t^3-\int_0^t X_sY_s ds$. Find $Y_t$ such that $M_t$ is a martingale. I started thinking that $X_t$ can be seen as: $dX_t=tW_tdW_t$ then by ...
Above is my question. I am, unfortunately, stuck on part (a)! Below are my workings. I've just spotted a typo -- at one point, an "$\exp$" is missing, but it's fairly obviously supposed to be there. ...
### Sequence of Martingales convergent in $L^1$-norm
Suppose $X^n_t$ is a sequence of martingales on a filtered probability space $\left(\Omega,\mathcal{F},\mathbb{P},\left(\mathcal{F_t}\right)_{t\in\left[0,T\right]}\right)$, that is for $\Delta>0$ ...