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

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Martingale: Whay $\mathbb E[S_n]=\mathbb E[S_1]$.

I've got a theorem (without proof) that say: If $(S_n)$ is a martingale refer to $(X_n)$, then $\mathbb E[S_n]=\mathbb E[S_1]$. I don't really understand why. Is there an intuitive why to see ...
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2answers
15 views

Martingale: why $\mathbb E[S_{n+m}\mid X_1,…,X_n]= S_n$.

Let $(S_n)$ a martingale by ratio to $(X_n)$ (I'm not sure if the terme "by ratio" is correct, I hope you'll understand). A lemma of my lecture say: $$\mathbb E[S_{n+m}\mid X_1,...,X_n]= S_n,\quad ...
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1answer
35 views

Amazing property of martingales

let $Y_1,Y_2,..$ be a sequence of equally distributed, independent and positive random variables. Consider $X_n = Y_1…Y_n$. Under which condition is $X_n$ a (super)-martingale? Show that neglecting ...
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27 views

Chromatic number $\chi(G)=600$, $P(\chi(G|_S)\leq 200) \leq 2^{-10}$

I am learning martingale and Hoeffding-Azuma inequality recently but do not how to apply the those inequality or theorem here. Let $G=(V,E)$ be a graph with chromatic number 600,i.e. $\chi(G)=600$. ...
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24 views

Does the random variable $f(\tau)M_\tau$, where $M$ is a martingale and $\tau$ is a stopping time, have zero expectation?

Suppose that $M:=\{M_t\}_{t\geq0}$ is a martingale adapted to some filtration $\mathcal{F}:=\{\mathcal{F}_t\}_{t\geq0}$ with $M_0\equiv0$ and that $\tau$ is an $\mathcal{F}_t$-stopping time. Suppose ...
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1answer
13 views

Bound Involving Submartingales

Let $(X_{j})_{j \geq 1}$ be a sequence of random variables with $X_{j}$ having mean zero and a finite moment generating function $\phi_{j}(\xi) = E(e^{\xi X_{j}})$ for all $\xi$ in a neighborhood $J$ ...
2
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1answer
18 views

First hit of a martingale

I came across this result somewhere and I don't grasp its proof in its entirety. Let $M$ be a continuous martingale such that $M_0 = 0$. Define $\tau_x = \inf\{t\geq 0: M_t =x \}$. Then, $$P\{\tau_a ...
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1answer
44 views

Local martingale but not martingale

On wikipedia there is an example of a local martingale which is not a martingale, but I do not understand why it is a local martingale. We have the process $ X_t = \begin{cases} ...
8
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1answer
50 views

Doob-style second moment martingale inequality

Let $\{X_k\}_{k=0}^{\infty}$ be a martingale, supposing $X_0 = 0$ and $E[{X_n}^2] <\infty$. Prove that $$P\left(\max_{1\le k \le n} X_k \ge r \right) \le \frac{E[{X_n}^2]}{E[{X_n}^2] + r^2}$$ ...
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33 views

F measurable and conditional expectation.

(a):I found it easily cause sum of measurable sets are measurable. (b),(c): I know limsup(Sn/n) is also measurable but I can't prove that just sup(Sn/n) is measurable. (d): I solved it by using the ...
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1answer
15 views

Uncorrelated successive differences of martingale

I read somewhere that given a martingale ${X_n}$, the successive differences of the martingale series are uncorrelated, namely $X_i −X_{i−1}$ is uncorrelated with $X_j −X_{j−1}$ for $i \neq j$. I ...
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1answer
31 views

Proving a.s. convergence for martingales

Let $ε_n, n > 1$, and $V_n, n > 0$, be independent random variables, with $P(ε_n = 1) = P(ε_n = −1) = 1/2$, $P(V_n = 1) = p_n, P(V_n = 0) = 1 − p_n$, for all n. Define $X_n$ inductively by $X_0 ...
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1answer
22 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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1answer
24 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
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1answer
27 views

Show that $((N_t-t)^2-t)_{t \geq 0}$ is a martingale for a Poisson process $(N_t)_{t \geq 0}$

I am asked to show that if $N$ is a poisson process of intensity $1$, then: $X_t=N_t-t$ is a martingale. $X_t^2-t$ is a martingale. I have done the first part easily, using independence of ...
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1answer
12 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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0answers
16 views

Application of Doob's Optional Stopping Time Theorem on new stopping time

Consider a random walk on a line starting at 0. On each step the probability of moving in either direction (right or left) is 1/2. There are two particular points on the line -a, and b. If $\tau$ is ...
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1answer
56 views

Exponential of a uniform integrable martingale is a submartingale

For reference I want to prove this Lemma: Let $M$ be a uniformly integrable martingale with the additional property that $\mathbb{E}[ \exp(M_\infty)] < 1$. Then $\exp(M)$ is a uniformly ...
2
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1answer
22 views

Density of running supremum of Brownian motion until a stopping time

I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as ...
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2answers
52 views

A variation of Lévy's characterization of Brownian motion

It is shown here, without using stochastic calculus, that if $W_t$ is a standard Brownian motion, then $$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$ is a martingale, where $f\in C^2$ and compactly ...
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0answers
13 views

Markov chain converges to boundary

I am learning martingale and related concepts recently and come across the following problem. Suppose $D$ is a bounded, connected, open subset of $\mathbb{R}^2$ with boundary $\partial D$. ...
2
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1answer
30 views

Examples of Wiener Martingales

$(X_t,\mathcal{F}_t)$ is called a Weiner martignale if i) $X_t$ is a Wiener Process ii) $(X_t,\mathcal{F}_t)$ is a martingale. (Here $\mathcal{F}_t$ is an increasing $\sigma$-field family). Let ...
2
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1answer
32 views

Checking if $B_t^3 $ and $3tB_t$ are martingales?

$$\mathbb{E}[ B_t^3 - 3tB_t + 3B_t | \mathcal{F}_s]$$ $$\mathbb{E}[B_t^3 | \mathcal{F}_s] - 3\mathbb{E}[t B_t | \mathcal{F}_s\}$$ $$\mathbb{E}[(B_t^3 - B_s^3 + B_s^3) | \mathcal{F}_s] + [ not \space ...
2
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1answer
23 views

Finding $b$ such that $e^{5B_t - bt}$ is a martingale

I have $X_t = e^{5B_t}$ and Where $B_t$ is brownian motion at time $t$. $M_t = X_t \cdot e^{-bt}$ I need to find a value for $b$ such that $M_t$ is a martingale. I am encountering difficulty, ...
0
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1answer
54 views

Proving a sequence forms a martingale

Let $\Omega = \mathbb N = \{1,2,3,\cdots\}$ and $\mathscr F_n$ be the $\sigma$-field generated by the sets $\{1\},\{2\},\cdots,\{[n+1,\infty)\}$ Define a probability on $\mathbb N$ by setting ...
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1answer
16 views

Martingale roulette system

I'm making a roulette system simulator, specifically right now the Martingale roulette system. So what I do know about the system that there is an Anti-Martingale too, which is the same, but you have ...
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17 views

Stochastic Integral martingale if no $dt$ term? [duplicate]

There is a proposition in my book that For a process $M_t$ to be a martingale, it is necessary that its stochastic differential $dM_t$ has no $dt$ term. Why is this exactly? My guess is that it ...
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1answer
97 views

Show that $f(W_t)-\frac{1}{2} \int_0^t f''(W_s) \, ds$ is a martingale without using Itô's formula

I'm learning the basics about Brownian motion (I know nothing about stochastic calculus), and I've shown that if $W(t)$ is a standard Brownian motion, then $W(t)^2-t$ is a martingale. Now I'm trying ...
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1answer
35 views

Exponential Martingales - Properties

This question relates to the exponential martingale, \begin{align} Y(t) = \exp\left(-\int_{0}^{t} \lambda(s)\,dW(s) - \tfrac{1}{2} \int_{0}^{t} \lambda^2(s)\,ds \right) \end{align} and specifically ...
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2answers
82 views

Martingale representation theorem application

Let $X = \exp(W_{T/2}+W_T)$. I try to figure the adapted process $g(s)$ such that according to the MRT we have $$X = \mathbb{E}[X]+\int^T_0 g_s dW_s.$$ I can figure out $X = \exp(2W_{T/2}+W_{T-T/2})$ ...
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0answers
19 views

Is the product of exponents of normal iid variables a martingale?

I am told that $X_1,\:X_2,\:,\dots$ is a sequence of i.i.d random variables, where $X_i\sim N(\mu,\sigma^2)$ for $i=1,2,\dots$ and that $Y_N=e^{X_1}e^{X_2}\dots e^{X_N}$. Is $Y_N$ a martingale?
2
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1answer
38 views

Exponential Martingales

This is a two-part question concerning exponential martingales. It is stated that an application of Ito's lemma to \begin{align} \rho_t = \exp\left[-\int_{0}^{t} \lambda_s\,dW_s - ...
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1answer
13 views

Why is right-continuity important in the martingale convergence theorem

Let $(X_t)$ be a right-continuous super-martingale such that $\sup_t E[X_t^-] < \infty$. Then $\lim_{t \to \infty} X_t = X_\infty$ a.s. where $X_\infty$ is integrable. I am trying to prove this. I ...
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1answer
29 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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1answer
22 views

How to Prove Unboundedness

Suppose I have a submartingale $X_k$, what results/theorems can be useful if I want to show that $X_k$ is unbounded in the limit. There are results (basically bounding $\mathbb{E}X_k$) for convergence ...
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1answer
85 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
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1answer
61 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
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1answer
24 views

Local martingale being true martingale

I am doing a question Let $X$ be a continuous local martingale and suppose $\mathbb{E}\left[\sup\limits_{0\leq s\leq t} |X_s|\right]<\infty$ for each $t\geq 0$. Then $X$ is a true martingale. In ...
2
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1answer
97 views

Continuous time Uniform Integrability and convergence in L1 for (super)martingale

I need to prove the following theorem by reasoning as in discrete-time, since we have proven this theorem in discrete time. Theorem Let $M$ be a right-continuous supermartingale that is bounded in ...
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1answer
23 views

Can this attempt to prove Ito Isometry for Elementary Processes be fixed?

So I have found this link which I will try after writing this post, but I would like to see if my original attempt (which is his/her attempt there) can be made to work. The reason I want this to work ...
1
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1answer
52 views

Justifying a step in proving $M_{S\wedge T} = \mathbb{E}[M_T | \mathcal{F}_S ]$

$S,T$ are stopping times and $M$ is a (right) continuous martingale. My lecturer set this as an exercise and I am given a solution(essentially split $M_T = M_T \mathbf{1}_{S≤T} + M_T ...
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1answer
50 views

Martingales and stopping times question

Let $X_n$ be iid r.v.s such that $P(X_n=1)=P(X_n=-1)=1/2$, and $S_n=\sum_{k=0}^{n}X_k$. Define $S_0=0$ a.s. . Prove that for all $k,n \in \mathbb{N}$, $\mathbb{E}[S^2_{n \wedge T_k}]=\mathbb{E}[{n ...
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1answer
44 views

Show that a certain functional of Brownian motion is a martingale

Question: Show that $(W^2_{t}-t)^2 - 4 \int_{0}^{t} W^2_{u} du$ is a martingale. I understand how to show that $(W^2_{t}-t)$ is a martingale, and I know that $4 \int_{0}^{t} W^2_{u} du$ is the ...
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0answers
46 views

2d random walk on the nonnegative quadrant using martingale techniques

I know the basics of (discrete time) martingales, and I'd appreciate any help and suggestions on how to prove the following using martingale techniques. Let $Z_n$, $n\ge 0$ be a random walk on the ...
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29 views

The square of an Ito integral is not a martingale

I just had a lecture on martingales and my teacher said something which I thought was interesting but he said wasn't important to the course. I was wondering if you guys could help me on this. We ...
0
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1answer
29 views

Not all martingales $Y$ can be represented $Y = H\bullet X$ for a given $X$

This is given as a counterexample that not all martingales $Y$ with $Y_0 = 0$ can be represented as $H\bullet X$ (= "discrete stochastic integral" ... wherever this term comes from??) for a given ...
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1answer
41 views

Exercise about martingale convergence

Let $p \in [0, 1]$, consider a stochastic process $(X_n)_{n\in\mathbb{N}_0}$ with $X_0 = x_0 \in [0, 1]$ and the following dynamics: For $n\in \Bbb{N}_0$, conditional on $X_0, X_1, \ldots, X_n$, we ...
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0answers
33 views

Convergence theorem for uniformly integrable martingales

This is a theorem in my textbook: Why "for all $n\in\mathbb{N}$" and not "for all $n\in\mathbb{N}_0$"? What's wrong with setting $n=0$, e.g. $$ X_0 =\mathbf{E}[X_\infty| \mathcal{F}_0] \; ?$$
2
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0answers
36 views

A characterization of quadratic variation for $L^2$ martingales

I am trying to prove the following statement but I am totally at a loss. Let $(A_t)$, $t \in \mathbb{R}^+$ be an adapted (with respect to the filtration $(\mathcal{F}_t)$) continuous integrable ...
2
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0answers
25 views

Conditional Borel-Cantelli lemma

Let $A_1, A_2, \ldots$ be events with $A_n\in\mathcal{F}_n$. Show that $$\biggl\{\sum_{n=1}^\infty \mathbf{P}[A_n|\mathcal{F}_{n-1}]=\infty\biggr\} = \limsup_{n\rightarrow\infty} A_n \text{ a. ...