# Questions tagged [martingales]

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

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### Sum and product of Martingale processes

Given two Martingale processes $(X_t)$ and $(Y_t)$, are their sum $(X_t+Y_t)$ and their product $(X_t \times Y_t)$ also Martingale? If not, will the two $(X_t)$ and $(Y_t)$ being independent grant ...
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### Why is stopping time defined as a random variable?

I've been given a crash course in stochastic processes and martingales for the purposes of a semester project on them. The guy I'm working with has been, I feel, a little vague in the definition of ...
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### Bounded (from below) continuous local martingale is a supermartingale

Suppose $M(t)$ is a continuous local martingale. That is, there exists a sequence of stopping times $T_n$ which almost surely increase to $\infty$, and such that $M(t\wedge T_n)$ is a martingale for ...
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### Can you make money on coin tosses when the odds are against you?

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\left\lfloor\frac{n}{2^b-1}\right\rfloor$ dollars. Wager $x$ dollars on random variable $C$ ...
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### Supermartingale with constant Expectation is a martingale

In my lecture notes they use the fact, that every supermartingale $(M_t)$ for which the map $t\mapsto E[M_t]$ is constant is already a martingale. Unfortunately I can't prove it. Some help would be ...
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### Monkey typing ABRACADABRA and gamblers

Problem: A monkey is sitting at a typewriter, typing a letter (A-Z) independently and with uniform distribution each minute. What is the expected amount of time that passes before ABRACADABRA is ...
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### Does law of large numbers converge in $L^1$?

I've seen the law of large numbers stated mainly in two (or three) forms: $S_n/n$ converges in probability (weak law) and converges almost surely (strong law). Also, there is convergence in the $L^2$-...
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### Martingale not uniformly integrable

I've come across a statement that implies that non-negative martingales for which $\{M_{\tau}\mid \tau \ \rm{stopping} \ \rm{time}\}$ is not uniformly integrable exist. I personally can't think of an ...
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### Martingale oscillating between three values

I'm self-studying Martingales. I came accross the following exercise (exercise 4.3.1.) in Durrett's Probability Theory and Examples (5th Edition). Exercise. Give an example of a martingale $X_n$ ...
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### Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
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### Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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### $L^p$ submartingale convergence theorem

In class today, we learned about the familiar $L^p$ martingale convergence theorem: For $p >1,$ if $X_n$ is a martingale with $\sup \mathbb{E}|X_n|^p <\infty$, then $X_n \rightarrow X$ a.s. and ...
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### Distribution at First Time a Sum Reaches a Threshold

Consider the following problem. Roll a die many times, and stop when the total exceeds $M$, for some prescribed threshold $M$. Call this time $\tau$, and call the running score after $n$ rolls $X_n$. ...
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### NON-martingale approach to ABRACADABRA problem

The well-known ABRACADABRA problem states (see D. Williams, "Probability with martingales", for example): a monkey is typing letters A-Z randomly and independently of each other, each letter ...
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### Probability on first hitting time of Brownian motion with drift

I am struggling with the following problem: Let $B$ be a one dimensional Brownian motion and $a,b>0$. Show that $$P[B_t=a + bt \text{ for some } t\geq 0] = e^{-2ab}.$$ The following hint is ...
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### What are some martingales for asymmetric random walks?

Here are some examples for symmetric ones: https://mathoverflow.net/questions/55092/martingales-in-both-discrete-and-continuous-setting/55101#55101 Is there a similar list for asymmmetric random ...
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### Prove this is a Martingale

Prove that $$Z_t:=\frac{e^{W_t^2/(1+2t)}}{\sqrt{1+2t}}$$ is a $\mathscr{F}_t$-martingale. I have tried all the usual manipulations without any success. The only useful fact I think should be used is ...
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### On "for all" in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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