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

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Equivalent Stopping Times for Brownian Motions

For standard Brownian motion $B$, define stopping time $T_1:=\inf\{t>0: B_t = 3\}$ and $T_2:=\inf\{t>0: B_t = -3\}$ and $T_3 := \min\{T_1, T_2\}$. Can I say that $T_3 = \inf\{t>0, B_t \in ...
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0answers
34 views

Let us assume that m balls are thrown independently at random into n bins.

Let us assume that $m$ balls are thrown independently at random into $n$ bins. Let $X$ denote the number of bins that contain afterwards exactly one ball. I want calculate $Pr(X=0) $ In literature I ...
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2answers
23 views

Sub-Martingale and Martingale

An integrable sub-martingale $S_t$ with $\mathbb E(S_t)$ being a constant is a martingale. Is this statement true, please? I think so.
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1answer
34 views

Is this Stochastic integral a martingale ?

Let $(B_t)$ be a Brownian motion and set $X_t = \int_0^t B_t^2 dB_s$. Is $X_t$ martingale? My idea is to rewrite $X_t$ in terms of Ito's Formula $(f(x) = \frac{1}{3}x^3)$ $X_t = \int_0^t B_t^2 dB_s ...
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1answer
45 views

How to show the following definition gives Wiener measure

On the first page of Ustunel's lecture notes, he defines the Wiener measure in the following way: Let $W = C_0([0,1]), \omega \in W, t\in [0,1]$, define $W_t(\omega) = \omega(t)$. If we denote by ...
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0answers
18 views

Limit of sum of stochastic increments [closed]

Let $(X_t)$ be a continuous local martingale with respect to a filtration $\mathcal{F}_t$. Show that the limit $ \lim_{n\to \infty} \sum_{i=1}^{2^n} (X_{i/2^n}^4 - X_{(i-1)/2^n}^4)^2$ exist in ...
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1answer
25 views

Prove X is a martingale

Prove $X = (X_n)_{n \geq 0}$ is a martingale w/rt $\mathscr{F}$ where X is given by: $X_0 = 1$ and for $n \geq 1$ $X_{n+1} = 2X_n$ w/ prob 1/2 $X_{n+1} = 0$ w/ prob 1/2 and $\mathscr{F} = ...
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2answers
28 views

Brownian and Brackets

A continuous martingale with deterministic bracket must be a Brownian motion. Is this statement ture or not, please? If true, how to show it? If not, what is a counter example?
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1answer
19 views

A question on semi-martingale and its variations

With probability one, paths of semimartingales have unbounded variation. What I know is that a martingale is also a semi-martingale, for example, Brownian motion. Hence, Brownian is an example of ...
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1answer
32 views

Zero mean but not a martingale

I am looking for a simple stochastic process which has zero mean for all $t\geq0$ but it is not a martingale. I been looking in to local martingales but having trouble keeping the mean zero for all ...
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1answer
9 views

Martingale property of negative Brownian motion

Let $B_t$ be Brownian motion, with $B_0=0$. Next define $M_t=-B_t$. Have I understood it correctly that $M_t$ is not a Martingale? $E[M_t]=0$ $E[M_{t+1}|M_t]=-M_t$ and therefore not a Martingale? ...
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0answers
57 views
+50

Dose “optional stopping theorem” imply “optional sampling theorem”?

Suppose $X$ is a martingale,$\tau$ and $\sigma$ are two stopping times which satisfy (a)$\sigma\le\tau$ and (b)the "optional stopping theorem" holds,that is to say: $$\mathbb E[X_\sigma]=\mathbb ...
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1answer
14 views

measurability in backwards martingales

$X$ is a backwards martingale with $X_0\in L^1 $ According to the convergence theorem:$X_{-n}\to X_{-\infty} $ a.s. But how to get the conclusion that $X_{-\infty}$ is $\mathcal F_{-\infty} $ ...
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1answer
31 views

Seeking for counterexample in closed martingale.

In the martingale convergent theorem: $X_n$ is a martingale with $\sup_n\mathbb E[|X_n|]<\infty$. Then there exist a $X_\infty\in L^1$ such that $X_n\to X_\infty\text{ a.s.}$ I want to find a ...
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0answers
17 views

Problems about the upcrossing lemma.

The following pictures comes from "Probability with Martingales" which denotes a stochastic integral(discrete): $$Y:=H\cdot X$$ Here $H$ is previsible.According to the gambling strategy ,$H=0$ in the ...
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1answer
48 views

Show independence of stochastic integral and stochastic process

Let $ M_t $ and $ N_t$ be two continuous local martingales with respect to a filtration $ \mathcal{F}_t $. Suppose that $ M_t $ and $ N_t$ are independent and set $X_t = \int_0^t M_s^4 \mathrm{d} M_s ...
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1answer
31 views

Why can $\int_0^t f''(X_s) \, d\langle X \rangle_s$ not be a local martingale?

We know from Itos formula, if $X$ is a continuous local martingale and $f$ has two continuous derivatives, we can write $f(X_t)$ as $$ f(X_t) = \int_0^t f'(X_s) dX_s + \frac{1}{2} \int_0^t ...
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1answer
12 views

Remove drift from exponential Weiner process

I have the following problem: let $X_t$ solve $$ dX_t = b X_t \, dt + \sigma X_t \, dW_t$$ where $W_t$ is a Weiner process. Find $s(\cdot)$ such that $Y_t = s(X_t)$ is a martingale. We can see by ...
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1answer
22 views

If $\{X_n\}$ is a martingale, then $E[X_n-X_{n-1}]=0$

Apparently this should be quite simple, but I have been trying for a while and can't seem to get this. Let $\{X_n\}$ be a martingale, then we have: $$E[X_n-X_{n-1}]=0$$ According to some notes I found ...
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1answer
25 views

Showing a sequence of Random Variables (Submartingale) Converges.

If $X_n$ and $Y_n$ are non-negative, integrable, and measurable with respect to $\mathscr{F_n}$, with $\mathscr{F_n} \subset \mathscr{F_{n+1}}$, and suppose that $E[X_{n+1}|\mathscr{F_n}] \leq X_n + ...
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1answer
20 views

A simple question about the definition of martingales

The definition of Martingale denotes that $E(M_{n+1}|\mathcal{F}_n)=M_n$. This implies $E(E(M_{n+1}|\mathcal{F}_n))=E(M_n)$. Then does it mean that $E(M_{n+1})=E(M_n)$ using the tower property? If so, ...
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1answer
43 views

Prove $A_t := W_t^3-3t W_t$ a martingale

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let $(A_t)_{t \geq 0}$ where $A_t = W_t^3 - 3tW_t$. ...
2
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1answer
37 views

What is the difference between $\mathbb E[Z|\mathcal G]=Y$ and $\mathbb E[Z|\mathcal G]\stackrel{\text{a.s.}}{=}Y$?

I'm somewhat confused by the definition of martingale: Let $(\Omega, \mathcal F, \mathcal F_n, \mathbb P)$ be a filtered probability space. We call $(X_n)_{n\in\mathbb N}$ martingale if for all ...
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55 views

Martingale Can't be Strictly Increasing

If the sample paths of a martingale are almost surely continuous and not constant on any interval, is it true that they are almost surely not increasing on any interval? Edit for clarity: Let ...
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0answers
38 views

Is $X_t = t \xi$ a martingale?

Consider a random value $\xi(\omega), \ \omega \in \Omega$ defined in $(\Omega,\mathbb{P},\mathcal{F})$. Prove or disprove that process $X_t = t \xi$ is a martingale. In my opinion it is not a ...
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0answers
38 views

Why this is a martingale?

Setup: $W$ probability space $Z_i : W \to L_i $ random variables ($L_i$ finite, for example $\{0,1\}$) $f: Z_1 \times \ldots \times Z_n \to \mathbb{R}$ $X_i := \mathbb{E}[f \mid Z_1,..,Z_i]$ Why ...
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1answer
32 views

For $X_n$ iid, $S_n=\sum X_n$, $\mathscr{G}_n=\sigma(S_n,S_{n+1}, \dots)$, show $E(X_j|\mathscr{G}_n)=E(X_1|S_n)$

If $(X_n)$ is iid in $L^1$, and $S_n = \sum_{i=1}^{n} X_i$ and $\mathscr{G_n} = \sigma(S_n, S_{n+1},...)$, then show that $E[X_1|\mathscr{G_n}]=E[X_1|S_n]$, and that ...
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1answer
16 views

Showing $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ for Martingale $M_n$.

Let $(M_n)$ be a martingale with $M_n \in L^2$. $S,T$ are bounded stopping times w $S\leq T$. Show $M_T, M_T$ are both in $L^2$ and that $E[(M_T-M_S)^2|\mathscr{F}_S]=E[M_T^2-M_S^2|\mathscr{F}_S]$ ...
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2answers
98 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
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2answers
57 views

Doob's decomposition and submartingale with bounded increments

Let $(X_n)_{n \geq 0}$ be a submartingale defined on some filtered probability space $(\Omega, \mathcal{F}, ({\mathcal{F}}_n)_{n \geq 0}, \mathbb{P})$. It is a standard fact that $X_n = X_0 + M_n + ...
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1answer
42 views

Application of martingale convergence theorem

I am struggling with this question: Let $(X_n : n \geq 1)$ be a zero mean martingale in $L^2$. Show that, for $\lambda >0$, \begin{equation} \mathbb{P} \bigg( \max_{1 \leq k \leq n} X_k \geq ...
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0answers
38 views

A problem on super/sub martingale

Let $(X_n, \mathscr{F_n}), n \geq 0$ be a super martingale and $T$ an $\{F_n\}$-stopping time a.s. bounded by $N \lt \infty$. Show that $$ E[|X_T|] \leq 3 \max_{n \leq N} E[|X_n|]$$ I can prove that ...
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1answer
22 views

Is random walk on half-line a martingale?

Let $X_n$ denote a random walk on $\mathbb Z^+$ starting at $0$. Is it a martingale? In Probability with Martingales by David Williams on page 99 it is claimed that it is, but I cannot understand ...
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1answer
33 views

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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1answer
29 views

how to find ALL $\sigma$-algebras of a given sample space?

I have a sample space $\Omega$={$\omega_{1}$,$\omega_{2}$,$\omega_{3}$,$\omega_{4}$} and I need to find ALL $\sigma$-algebra on $\Omega$. I know how to construct some $\sigma$-algebra like ...
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1answer
47 views

Proof of extinction probability in Galton-Watson-process using a Martingale

this problem is somewhat similar to the thread The extinction probability of Galton-Watson process from a Martingale perspective. I want to show, that for a Galton-Watson-process $Z_0,Z_1,\ldots$ with ...
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2answers
99 views

Proving integrability of a random variable involving stopping times

Let $X_1, X_2,...$ be i.i.d integrable random variables in $\mathbb{R}$ with $\mathbb{E}[X_i] =0$ and $\mathbb{P} (X_i >0) >0$. Let $x>0$, $S_0 = x$, and $S_n= x + \sum_{i=1}^{n} X_i $. For ...
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3answers
59 views

Prove $X_n = \frac{1}{a_n} 1_{(0,a_n]}$ a martingale

With the following premises, I want to prove, that the series of random variables $(X_n)_{n \in\mathbb{N}}$ is a Martingale: Let $\Omega := (0, 1] \subset \mathbb{R}, \mathfrak{F}$ the ...
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2answers
48 views

One Question on Law of Total Probability

Let $(X_n)$ with $n \in \mathbb N_0$ be a discrete martingale. Then I read the following identity which is said to be derived from the law of total probability. $$ \mathbb EX_m = \left( \sum_{n=0}^m ...
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2answers
33 views

Prove Kolmogorov's SLLN by martingale.

Suppose $\xi_i$ are i.i.d. and $\mathbb E(|\xi_1|)\lt\infty$ Let $X_n=\sum_{i=1}^n\xi_i$ Then we have $\frac{X_n}{n}\to \mathbb E(|\xi_1|) $a.s. In the proof of this theorem: ...
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1answer
29 views

Intuition underlying stopped martingales

Let $X$ be a martingale and $T$ a stopping time. Define the stopped martingale $X_{\min\{T,n\}}$. What is the intuition underlying this process? It is quite confusing here. $X$ is random and $T$ is ...
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1answer
51 views

An inequality in martingale

Suppose $X_n$ is a supermartingale,for $\lambda>0$ ,we have the following inequality: $$\lambda\mathbb{P}(\inf_{n\leq k}X_n\leq-\lambda)\leq\int_{[\inf_{n\leq k }X_n\leq -\lambda]}(-X_k) ...
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1answer
83 views

Expectation of Square of Stopping Time

Let $B_t$ be standard Brownian motion and $a < 0 < b$. Define stopping time $T$ as follows. $$T = \min \{t \geq 0: B_t \in \{a, b\} \}.$$ The expectation of $T$ is $\mathbb ET = |a|b$ and can be ...
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0answers
23 views

Control of the expected false discovery rate

I've been looking at why exactly the Benjamini and Hochberg procedure controls the expected false discovery proportion. More specifically, assuming $N$ hypotheses with corresponding $p$-values; for a ...
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1answer
98 views

Problem about Random walk and Stopping time.

Here is an example in "Probability with Martingales" My questions are: (1)Does equation (a) hold for $T=\infty$? (2)The equation:$$\mathbb{E}M_T^\theta=1=\mathbb{E}[(sech \theta)^Te ...
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1answer
23 views

Square Integrable local martingale or locally square integrable martingale?

I have a question about martingales. What is the difference between "locally square integrable martingale" and "square integrable local martingale"? In particular, which set does $M_{loc}^2$ ...
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1answer
25 views

A problem about martingale with stopping time .

In Durrett's "Probability,theory and examples": Suppose $X_n$ is supermartingale and $H_n$ is predictable. define: $$(H\cdot X)_n\triangleq\sum^n_{m=1}H_m(X_m-X_{m-1}) $$ $N$ is stopping time and ...
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0answers
62 views
+100

How to get closed form solutions to stopped martingale problems?

Way back when, I took a course in stochastic processes in college. I remember being frustrated by the plethora of abstract proofs without much in the way of how to use them to get actual results. It ...
0
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1answer
42 views

For a Poisson process prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be $V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $ Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a ...
2
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2answers
40 views

Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$ [closed]

Consider an experiment of rolling two dice. Let $X$ be the value of the first die and $Y$ the sum of the two dice. Find $E(X / Y)$, ie, obtain the value of $E(x/y) (y)$ for all $y$ Good evening, I ...