# Questions tagged [martingales]

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

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### prove that $(Y_n)_n$ is a submartingale

If $(X_n)_{n \in \mathbb{N}}$ is a nonnegative submartingale for the filtration $(\mathcal{F}_n)_n$ and F is an increasing and convexe function. We suppose that F has a differentiable function f. (F ...
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### If $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for $X_t$ a local martingale, is $X_t$ a proper martingale?

Let $X_t$ be a continuous local martingale. Suppose $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for any $t > 0$. Is it true that $X_t$ is a proper martingale? We ...
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### Find the compenstor of the standard ito integral

Let $B_t$ be a Brownian motion and let $\{\mathcal F_t : a<t<b\}$ be a filtration such that for each $t$ we have that $B_t$ is $\mathcal F_t$ measurable and for and $s<t$, the random variable ...
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### Put-Call Parity for a stock with dividends in Black-Scholes

I want to find a put-call parity relation for European options in the Black-Scholes model where the underlying stock pays a continuous dividend. I know that the relation will be S_0e^{-\delta T} - ...
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### A stopped process is adapted

I am trying to understand the proof of Theorem 2.2.2(Optional Stopping Theorem) in Fleming and Harrington's Counting Processes and Survival Analysis. Let $\{X(t):0\leq t<\infty \}$ be a right-...
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### Càdlàg Feller process is quasi-left-continuous

I've been working in Chung's "Lectures from Markov Processes to Brownian Motion", and I got stuck at Exercise 1 from 2.4. The objective of the problem is to give a short proof of the quasi-left-...
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### If $\sum_{n=1}^\infty E(X_n-X_{n-1})^2/b_n^2 < \infty$, then $X_n/b_n \rightarrow 0$ a.s.

This is Durrett Exercise 5.4.9. I'm trying to show that if $X_n$ is a martingale, and $b_m \uparrow \infty$, $\sum_{n=1}^\infty E(X_n-X_{n-1})^2/b_n^2 < \infty$, then $X_n/b_n \rightarrow 0$ a.s....
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### Definition of Martingale

I'm a little bit confused by the definition of a martingale. We say that a sequence of RV's is a martingale if: $E[X_n|X_{n-1}]=X_{n-1}$. I'm confused because it seems like the LHS should be just a ...
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### Stopping theorem counterexample

I have been thinking about the conventional counterexample to stopping theorems where the stopping time is not bounded. For example, flip a fair coin infinitely many times, representing heads with 1 ...