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Questions tagged [martingales]

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

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Optional sampling theorem with a.s. finite stopping time

I'm trying to prove the following generalized version of Doob's optional sampling theorem: Let $X$ be a square integrable martingale with respect to a filtration $\mathbb F = \left\{ \mathcal F_n \...
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min & sum of discrete stopping times using alternate definition

Let $A, B$ be stopping times of a process $X_0, ... X_n$ where $A$ is a stopping time iff $I_{A = k}$ is a function of $X_0, ... X_k$ (where $I$ denotes an indicator variable). We want to show that: ...
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probability with martingales 12.2 sum of zero-mean independent variables in L^2

I am struggling with the following theorem from David Williams, Probability with Martingales: THEOREM Suppose that $(X_{k}:k\in\mathbb{N})$ is a sequence of independent random variables such that, ...
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Super martingale and stopping time

I am a little confused about stopping times and martingales. Suppose I have a super-martingale $f(X_t)$ where $X_t$ is a martingale. Also, $f()$ is increasing in $X_t$. Consider a person looking to ...
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Probability problem (Durrett 4.7.4.) Exchangeable Sequence of Random Variables.

I am having trouble in solving the following problem; If $X_1, X_2, . . . \in \mathbb{R}$ are exchangeable with $EX_i^2 < \infty$ then $E(X_1X_2) ≥ 0.$ What I know is that the definition of ...
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Conditions for a process to be a martingale

Stumbled upon a problem which raised some questions in my mind, here's the deal : $ dS = (a + b.S)dt + (c + d.S) dW \\ X = \alpha(t).S + \beta(t) $ where $a,b,c,d$ are constants. No conditions were ...
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Martingale to estimate gain or loss of a gambler.

Let $Z_1, Z_2, Z_3, ...$ be a sequence of i.i.d random variables valued in $\{-1, 1\}$ and taking each value with equal probability. Define $X_n=\sum_{i=1}^n Z_i$ and $Y_n=X_n^2-n$. It can be easily ...
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Central limit theorem for square integrable martingales

Let $(M_t)_{t \in \mathbb{R}^*}$ be a square-integrable martingale. I am looking for a reference for the following convergence result : $$\frac{M_t}{\sqrt{\langle M_t \rangle}} \overset{d}{\to} \xi$$ ...
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Is $\{C^{X_n}\}_{n = 1}^{\infty}$ is a martingale?

Suppose $\{X_n\}_{n = 1}^{\infty}$ is a Galton-Watson branching process with ($P(X_1 = 1) = 1$ and $X_{n+1} = \Sigma_{i = 1}^{X_n} \theta_{i,n}$, where $\{\theta_{i,n}\}_{n = 1}^{\infty}$ are i.i.d. ...
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Join of random variables - relation to max

On Wikipedia in the article Snell envelope author uses 'join' operation on random variables. Following line is relevant: $$ U_n = X_n \lor \mathbb{E}^\mathbb{Q}[U_{n+1} | \mathcal F_n] \quad \text{...
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Find values of $a$ and $\lambda$ for which $Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale

Find values of $a$ and $\lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,b\in\mathbb{R}$ can be positive as well as negative, since $...
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Do non-negative submartingales with bounded second moments converge almost surely?

Suppose $\{X_n\}_{n = 1}^{\infty}$ is a discrete-time submartingale (a sequence of random variables, such that $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$), such, that $\exists C \in \mathbb{R} \forall n \...
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Do non-negative submartingales with finite second moment converge almost surely?

Suppose $\{X_n\}_{n = 1}^{\infty}$ is a discrete-time submartingale (a sequence of random variables, such that $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$), such, that $\forall n \in \mathbb{N} EX_n^2 <...
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Show that $X_t=e^{B(t)}-1-\frac{1}{2}\int_0^te^{B(s)}ds$ is a martingale

The problem tells me to show that $X_t$ is a martingale but I am getting that it is not. Here the assumed filtration is ${\scr F}_t=\{\sigma(X_s):s \le t\}$. Here is what I tried. $$E(X_t|{\scr F}_s) ...
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Example of a Non-negative Martingale Satisfying Certain Conditions

Question The question is to find an example of a non-negative martingale $(X_n)$ with $EX_n=1$ for all $n$ such that $X_n$ converges almost surely to a random variable $X$ where $EX\neq 1$ and $\text{...
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Martingale to power

Assume a Martingale $(X_i)$. My Question is what do I know about $(X_i^p)$ with $p\in \Bbb{N}_{\geq 2}$? I understand that I need $X_i\in\mathcal{L}^p$ to have any sort of (Sub-/Super-)Martingale.
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Conditional expectation of integral of Ornstein-Uhlenbeck process

Given that $X(t)$ is an Ornstein-Uhlenbeck process with $X(0) = x_0$, which is a Markov process, but not a Martingale, how could I go forward if I would like to calculate $E[\int_0^T X(s)ds | \...
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Martingale convergence theory question

I'm confused on what this proof would look like. I know that by Martingale Convergence Theorem: $$X= \lim Z_n / \mu^n$$ exists a.s. I also know that Kesten Stigum Theorem: Assume that $E[L] &...
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1answer
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Doob's Optional Stopping Theorem: $\xi_\tau$ vs $\xi_{\tau\land n}$

I have some troubles in understanding the Optional Stopping Theorem by Doobs. I have a bit of confusion about the following (Brzezniak, Zastawniak - Basic Stochastic Processes p. 58-59): Let $\...
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Finding out the sequence as Martingale.

Consider the sequence $\{X_n\}_{n\geq 1}$ of independent random variables with law $N(0, \sigma^2)$. Define the sequence $Y_n= \exp \bigg(a\sum_{i=1}^n X_i-n\sigma^2\bigg), n\geq 1,$ for $a$ a real ...
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Prove that $\{S_n^2-\sigma^2n\}$ is a martingale.

Let $S=\{S_n\}$ be a mean zero random walk with $EX_1^2=\sigma^2$. Prove that $\{S_n^2-\sigma^2n\}$ is a random walk. So we need that $\{S_n^2-\sigma^2n\}$ is integrable, (easy) $\{S_n^2-\sigma^2n\...
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Reference Request: Martingales - Convex Analysis (-Harmonic Analysis) relationship

During my undergraduate studies I've encountered martingales, convexity (and also partial differential equations). Fast forward a few years, as a PhD student in applied math, I often find myself ...
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Stopped random walk is not uniformly integrable

I know that in general Doob's Optional Stopping Theorem doesn't hold for unbounded stopping times, but that it does when the system up to the stopping time is uniformly integrable. One counter ...
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Interpreting Doob's Up-Crossing Inequality

Doob's upcrosssing inequality states that, for martingale $X_n$, if $U_n(a,b)$ counts the number of upcrossings through $(a,b)$ up to time $n$, then $E[U_n(a,b)]\leq \frac{1}{b-a}E[(a-X_n)_+]$. We ...
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1answer
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Continuous local martingale $M$ with $\langle M\rangle_t=t$ is a martingale

I am currently reading a proof of the Levy characterization of Brownian motion, and it seems it uses the following result. Let $(M_t,\mathcal F_t)_{t\geq 0}$ be a continuous local martingale with ...
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exponential martingale, expectation of stopping time [duplicate]

Let $\{B(t): t \ge 0 \}$ be a linear Brownian motion. Show that, for $\sigma >0$, the process $\{exp(\sigma B(t)-\sigma^2 t/2): t \ge 0 \}$ is a martingale. 2.Show, by taking derivatives $$\...
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1answer
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Doob's decomposition theorem relaxation for nonuniqueness

Doob's decomposition theorem states that, given any sequence of adapted, integrable functions $X_n$ (adapted to $\mathcal{F}_n$), we get an almost surely unique decomposition $X_n=A_n+Y_n$, where $Y_n$...
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Checking for martingale and Itô's formula

Assuming that $\{ W ( t ) | t \geq 0 \}$ is a Brownian motion, I'm trying to check whether the process $$X ( t ) = W ( t ) + 4 t$$ is martingale with respect to filter $\mathcal{F}_t$. For this we ...
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Square integrability of martingale?

So we had the following example in class: Assume $(\Omega, ℱ,(ℱ_{t}),ℙ)$ a filtered probability space and $X: \Omega \to \mathbb{R}$ a random variable such that $$ \mathbb{E}[|X|^{2}] < \infty \...
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Density of hitting time of absolute value of a Brownian motion

I am interested in the probability density of $$ \tau =\inf\{t\geq 0: \vert W(t)\vert = 1\} $$ where $W(t)$ is a standard Wiener process. I have two approaches in mind: First approach: I could ...
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$B_t^3 - 3t B_t$ is a $L^2$ martingale ($B_t$ being a standard Brownian motion)

By Itô's formula I get that \begin{align} d(B_t^3 - 3t B_t) &= (3 B_t^2 dB_t + 3 B_t dt) - 3(B_t dt + 3 t d B_t) \\ &= (3 B_t^2 + 3t) d B_t \end{align} which seems related to martingale ...
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Geometric Brownian Motion and Martingales?

Let Gt denote the canonical filtration of a Brownian motion Wt. Show that for any λ ∈ R, the S.P. Mt(λ) = exp(λWt − λ^{2}t/2), is a continuous time martingale with respect to Gt. Explain why its kth ...
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1answer
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Convergence of a cadlag martingale to a square-integrable limit

Let $(M_t)_{t\geq 0}$ be a cadlag uniformly integrable (UI) martingale with $M_t\rightarrow M_\infty$ a.s. as $t\rightarrow\infty$. If $\Bbb E[M_\infty^2]<\infty$, is it true that $\Bbb E[M_t^2]<...
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An example of a submartingale $X=\{X_n\}$ such that $\{X_n^2\}$ is a supermartingale.

A submartingale is a real-valued stochastic process $X=\{X_n\}$ adapted to a filtration $\{\mathcal{F}_n\}$ such that $$E[X_{n+1}\mid \mathcal{F}_n] \geq X_n.$$ For a supermartingale just reverse the ...
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Distribution of inter-arrival time of non-homogenous Poisson Process

What is the distribution of inter-arrival time of a non-homogenous Poisson Process? In other words, if $T_n$ is a non-homogeneous Poisson process with intensity $\lambda(s)$, and $S_{n+1} = T_{n+1} - ...
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Conditional law of Poisson process

1)What does the notation for $G_n$ and $H_n$ mean? 2)How would $G_n$ and $H_n$ look like for a Poisson process? 3)How to show that $v$ would be the compensator of a Poisson process? Thanks!
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Prove the infinum of supermartingales is still a supermartingale

In Tomas Bjork's Arbitrage Theory Continous Time (2009), Proposition 21.20 claimed that, in discrete time, the (pointwise) infinum of an arbitrary family of supermartingales is still a supermartingale....
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martingale $X(t)=e^{-\lambda t}f(B(t))- \int_{0}^{t} e^{-\lambda s} (\tfrac{1}{2}\Delta f(B(s))- \lambda f(B(s)))ds$

Let $f: \mathbb{R}^d \longrightarrow \mathbb{R}^d$ be twice continuously differentiable and $\{B(t): t \ge 0 \}$ be a d-dimensional Brownian motion. Further suppose that, for all $t>0$ and $x \in \...
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$V\in L^2$, $U$ is standard normal, $U,V$ independent$\implies$ $\lim_{\sigma \to \infty} E[V| V+\sigma U]=E[V]$ a.s.

How to show that \begin{align} \lim_{\sigma \to \infty} E[V| V+\sigma U]=E[V], a.s. \end{align} where $E[V^2]<\infty$ and $U$ is standard normal and $U$ and $V$ are independent. I was thinking ...
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Doob martingale concentration with approximate expectations

Consider a set of random variables: $$ \vec{X} = X_1,\ldots,X_n$$ and a function $f(\vec{X})$ used to construct the Doob martingale with elements $$ B_i = E_{X_{i+1},\ldots,X_{n}}[f(\vec{X}) | X_1,\...
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1answer
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Markov chain and martingale: Taking one step to the right or going back to zero

Consider the Markov chain on nonnegative integers that does the following: from any site $x \geq 0$ it jumps to $x + 1$ with probability $p$ and to 0 with probability $1 − p$. Is this chain ...
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1answer
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If a process is natural increasing, does that imply that the stopped process is natural increasing?

Let $A = \{A_t\}_{t\geq 0}$ be a right-continuous stochastic process on the filtration $\mathcal F = \{\mathcal{F}_t\}_{t\geq 0}$ and let $T$ be a stopping time. Suppose also that $A_t$ is natural ...
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Restriction martingale to a measurable set

If we have that $M_n$ is a martingale adapted to $\mathscr{F}_n\subset\mathscr{F}$ on a probability space $(\Omega,\mathscr{F},\mu)$ then if $A$ is an $\mathscr{F}_0$ measurable set we have $$M_n 1_A=...
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Compute the exit time of $M_t = W_t^2 - t$ from $[-R, R]$

I have a question in my textbook that states: Apply the optional stopping theorem for the martingale $ M_t = W_t^2 -t $ to show that $E[\tau] = R^2$. This exercise came right after the section on ...
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1answer
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Expected value- $E(1/3)^t$

$X_1, X_2, ... $ are independent random variables. And $P(X_n=1)=P(X_n=-1)=1/2$. $t=inf (n: X_1+X_2+...+X_n=1)$ Find $E(1/3)^{t}$. I tried to do it from the definition of expected value: $E(1/...
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1answer
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Submartingale characterization

is there a characterization of submartingales in terms of stopping times similar to the martingale case in A martingale characterization. Especially is the following true: If $X$ is an adapted ...
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Limit of an indicator martingale

Consider the Lebesgue measure on $[0,1]$. We define $I_k^n=[k2^{-n},(k+1)2^{-n})$, for $0\leq k\leq 2^n-1$, and $F_n=\sigma(\{I^n_k:k\})$. Finally $M_n=1_{I_0^n}2^n$. How do we show that $M_n$ is a ...
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Prove that $P(\exists n\in N:S_n=n-2008)=1$

Let $X_n$, $n=1,2,...$ be independent random variables. $P(X_n=1)=\frac{1}{2}=P(X_n=-1)$ $S_n=\sum_1^nX_k$, $T=inf[n:S_n=n-2008]$ a) Prove that $P(\exists n\in N:S_n=n-2008)=1$ b) Prove $ET\...
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1answer
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Varying definitions of a martingale

I frequently see that, over some filtration $\mathscr{F}_{n}, \{X_{n}\}$ is defined as a martingale if $E[X_{n+1}|\mathscr{F}_{n}]=X_{n}$. Sometimes, however, I see this extended to $E[X_{n+s}|\...
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In 'Probability: A Graduate Course' by Allan Gut, it is written '[…]in a sense life itself is a martingale.' Can someone explain how?

I hope this question doesn't infringe any rules. Original text (p. 480, 2nd edition): If we interpret a martingale as a game, part (ii) states that, on average, nothing happens, and part (i) ...