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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.

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If $S$ is a martingale, then alle processes $((HS)_t)_t$ are martingales

I have just started with financial mathematics in discrete times, and while reading about martingales I came across this statement without proof, which I am not sure how I would prove. I would appreciate any help! Statement: If $S$ is a martingale,…
variableXYZ
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Proof of $X^{(n)}_t$ converges to 0

I recently came across this question and reckon it should be a direct application of Doob's inequality (correct me if I am wrong). But I struggle to write formal proof. For each $n \in \mathbb{N}$, let $(X^{(n)}_{t})_{t\geq0}$ be a martingale on…
Cathy
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monotocity of a positive martingale

I am trying to understand the martingale convergence theorem, specifically the following Levy's upward theorem: Let $(\Omega, F, \mathbf{P})$ be a probability space and let $X$ be a random variable in $L^{1}$. Let $F_{*}=\left(F_{k}\right)_{k \in…
pineapple
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Will a nonnegative supermartingale converge to 0?

I want to know weather a non negative supermartingale converges to $0$. I have a hunch that it shall be so, but could not prove or disprove it. Is this correct? And if so, is there a way to prove it?
athos
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Local martingale convergence theorem

I really need help, suppose we have $$E[x(t)^2]\leq C\exp(Dt),$$ where $C$ and $D$ are positive constants, Is x^2 a martingale? Can we apply the martingale convergence theorem?
fidel
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When $X_t = {W_t}^n - k\int^{t}_{0} {W_s}^{n-2} \, ds$ is a martingale?

I know that $X_t = {W_t}^3 - 3\int^{t}_{0} W_s \, ds$ is a martingale, but my general question is: for what values of $k$, $X_t = {W_t}^n - k\int^{t}_{0} {W_s}^{n-2} \, ds$ is a martingale?
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Why $\mathbb P\left\{\sup_{s\in [0,t]}|M_s|\geq y\right\}\leq \frac{\mathbb E(|M_t|1_{\{\sup_{s\in [0,t]}|M_s|\geq y\}})}{y}$ holds?

Let $(M_s)$ a martingale s.t. $\mathbb E[M_t^2]<\infty $. Regarding the proof of $$\mathbb E[\sup_{0\leq s\leq t}M_s]\leq 4\mathbb E[M_t^2],$$ the person who answer the question in this post mention that $$\mathbb P\left\{\sup_{s\in [0,t]}|M_s|\geq…
Bruce
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Martingale with expectation

Let a probability space defined on $\Omega = [0 \ 1]$ and assume the probability of any interval be defined as the length of that interval. Let $\\$ $Y_i(w) = \begin{cases} 1 & \quad 0 \leq w \leq 1/i, \\ 0 & \quad otherwise \end{cases} $ and $X(w)…
eet
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super martingale which dominates

Could anyone tell me what it means by ''it dominates $\{E (Z|G_k), G_k\}, E(|Z|)<\infty$'' ? in the following statement? Thanks. Let $(Y_k, G_k)$ be a super-martingale and it dominates $\{E (Z|G_k), G_k\}, E(|Z|)<\infty$,
Myshkin
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About the martingale.

For martingale, we have $E[X_{n+1}-EX_n|X_1,\dots,X_n]=0$, can we say $X_{n+1}-X_n$ is independent of $X_1,\dots,X_n$?
Boommm
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Properties of Martingales

I found following problems about properties of martingales and I would like to know is my approach correct for first problem and how to exactly solve second one. Problem is following: Let $\{Y_n\}_{n\ge 0}$ be a martingale w.r.t. the filtration…
Chris125
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Does all $L^2(\Omega ,\mathcal F_T,\mathbb P)$ processes is a martingale?

In the Book of Schilling (Brownain motion), there is the following theorem I'm quite surprised by this theorem. It looks to mean that all $L^2(\Omega ,\mathcal F_T,\mathbb P)$ is a Martingale (or local martingale). Is this really true ?
Walace
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square-root of a martingale

A general result says that if $\{X_n\}$ is a martingale in $L^2$, then $\{X_n^2\}$ is also a martingale, and in general for a positive power $p>1$ this also holds. How about fractional powers, like $p=\frac{1}{2}$? I haven't been able to find a…
mj_indefinite
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Does a symmetric random walk converge almost surely?

Does a symmetric random walk in- one dimension- converge almost surely? Can we prove or disprove it by martingales?
peanut
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