Questions tagged [martingales]

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

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time in the realized sequence, the expectation of the next value in the sequence is equal to the present observed value even given knowledge of all prior observed values. Expressing the following mathematically:

A sequence of random variables $X_0, X_1, \dots$ with finite means such that the conditional expectation of $X_{n+1}$ given $X_0, X_1, X_2, \dots, X_n$ is equal to $X_n$, i.e., $$\mathbb{E}[X_{n+1}\mid X_0, X_1, \dots, X_n] = X_n.$$

A one-dimensional random walk with steps equally likely in either direction $(p=q=\frac12)$ is an example of a martingale.

See these lecture notes for more information.

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Why did my friend lose all his money?

Not sure if this is a question for math.se or stats.se, but here we go: Our MUD (Multi-User-Dungeon, a sort of textbased world of warcraft) has a casino where players can play a simple roulette. My friend has devised this algorithm, which he himself…
Konerak
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Martingale theory: Collection of examples and counterexamples

The aim of this question is to collect interesting examples and counterexamples in martingale theory. There is a huge variety of such (counter)examples available here on StackExchange but I always have a hard time when I try to locate a specific…
saz
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Criteria for being a true martingale

Could you kindly list here all the criteria you know which guarantee that a continuous local martingale is in fact a true martingale? Which of these are valid for a general local martingale (non necessarily continuous)? Possible references to the…
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Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the first time after time $s$ that the process hits $a$…
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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 their sum and product Martingale? Thanks!
Tim
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The Laplace transform of the first hitting time of Brownian motion

Let $B_t$ be the standard Brownian motion process, $a > 0$, and let $H_a = \inf \{ t : B_t > a \}$ be a stopping time. I want to show that the Laplace transform of $H_a$ is $$\mathbb{E}[\exp(-\lambda H_a)] = \exp (-\sqrt{2\lambda} H_a)$$ by…
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Example of filtration in probability theory

I'm studying Martingales and before them filtrations. Given a probability space $(\Omega, F, P)$ I define a filter $(F_n)$ as a increasing sequence of $\sigma$-algebras of $F$, such that $F_t \subset F$ and $t_1 \leq t_2 \Longrightarrow F_{t_1}…
J. Goles
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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 stopping times, and I can't seem to find anything…
Jack M
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Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le i \le n}$ a family of measurable functions. My goal…
vanna
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Azuma's inequality to McDiarmid's inequality?

I was going through some notes on concentration inequalities when I noticed that there are two commonly-cited forms of McDiarmid's inequality. Long story short: I know how to prove the weaker one from Azuma's inequality. I also know how to prove the…
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What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should cover continuous-time martingale, stochastic integrals…
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Filtration and measure change

I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand the concept of "filtration". Yes, the definition of filtration is straight forward, it's set of $\sigma$-algebra. However, when it comes to…
athos
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How to show the following process is a local martingale but not a martingale?

I am confronted with the same problem in this thread, which hasn't had a complete answer yet. In the sequel, we denote by $Y^T$ the stopped process $Y^T_t = Y_{T \wedge t}$. Consider the following process: $$ X_t = \begin{cases} W_{t/(1-t)}^T…
Dormire
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Is there a discrete-time analogue of Doléans-Dade exponential?

For a continuous martingale $X$, we have the Doléans-Dade exponential: $$\epsilon(X)_t=\exp\left(X_t-\frac{1}{2}[X]_t\right)$$ What is the "correct" analogue, if one exists, for some discrete-time martingale $M$? (Unfortunately in discrete time this…
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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 appreciated. By definition of a supermartingale we…
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