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

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Proving that $W$ is Brownian motion, without stochastic calculus

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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8 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 $\partial D$. ...
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1answer
23 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 ...
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29 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 ...
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20 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, ...
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29 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
15 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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16 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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82 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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31 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
81 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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16 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?
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33 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
12 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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28 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
20 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
82 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
58 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
21 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 ...
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1answer
82 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
20 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 ...
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1answer
49 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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35 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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40 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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45 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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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 ...
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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
40 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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32 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. $$ X_0 =\mathbf{E}[X_\infty| \mathcal{F}_0] \; ?$$
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29 views

Riesz decomposition of a nonnegative supermartingale

$X=(X_n,\mathcal{F_n})$ is a nonnegative supermartingale, and moreover $EX_n\to 0$, i.e., it is a potential. If $X_n=M_n-A_n$ is the Doob decomposition, then $$EX_n=EM_n-EA_n=EX_0-EA_n,$$ so by the ...
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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 ...
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24 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. ...
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22 views

Martingale with bounded increments converges or diverges to $\pm \infty$ [duplicate]

Let $(M_n)$ be a martingale with $|M_n - M_{n-1}| \leq c$ for some fixed $c < \infty$. Check that the two disjoint events $$C:=\{M_n \text{ converges to a finite limit}\}, \; F:=\{\limsup M_n ...
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30 views

Proof of the optional sampling theorem

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a Filtration on $(\Omega,\mathcal{A})$ $X=(X_n)_{n\in\mathbb{N}_0}$ be a nonnegative $\mathbb{F}$-supermartingale ...
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17 views

Does Doob's optional sampling theorem hold for *backward* (sub)martingales?

Here is my setup: $X_1,X_2,\ldots\overset{\mbox{iid}}{\sim}\mathcal{N}(0,1)$. $N_k=\min\{n:|\bar{X}_n|\geq k/\sqrt{n}\}$. I want to show that $k_2>k_1$ implies ...
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51 views

$\mathcal{L}^2$-martingale with finite limit but infinite square variation

Find an $\mathcal{L}^2$-martingale $(M_n)$ with $M_n \rightarrow M_\infty$ almost surely for a finite real valued $M_\infty$ but $\langle M \rangle_n \rightarrow \infty$ almost surely. $\langle M ...
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27 views

How does expectation of martingales relate to “fair games”?

A martingale is defined as an $(\mathcal{F}_n)_{n\in\mathbb{N}}$-adapted stochastic process with $\mathbf{E}[|X_t|] < \infty$ for all times $t\in I\subset \Bbb{R}$ and ...
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1answer
55 views

Variance of the square variation process

Let $X = (X_n)_{n\in\mathbb{N}_0}$ a square integrable $(\mathcal{F_n})_{n\in\mathbb{N}_0}$-martingale. The predictable process $\langle X \rangle_n = \sum_{i=1}^n \Bigl(\mathbf{E}\bigl[X_i^2\vert ...
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1answer
53 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad ...
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1answer
42 views

Computation of a stochastic integral with respect to a local martingale

I am trying to compute the stochastic integral $$\int_{(0,t]}\mathbb{1}_{[a,b)}(s)dM_s$$ where $0 < a < b< \infty$ are constant and $M$ is a continuous local $L^2$-martingale. I am guessing ...
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1answer
47 views

Proof of stopping theorem for bounded stopping times

Let $\tau$ be a bounded stopping time and $X=X_n$ a martingale. Then $X_\tau$ is integrable and $E(X_\tau)=E(X_0)$. I need help with the proof at discrete time, at one step I am not sure I ...
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1answer
75 views

Quadratic Variation of a square-integrable Lévy process

I am having a problem with the following question. I have tried using the definition of square integrable martingales and quadratic variation, but just can't seem to get anywhere. Can anybody offer me ...
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1answer
22 views

Upper bound for martingale at a stopping time

This seems like a simple question, but I cannot figure out the following. Let $\{M_i\}_{i\geq 0}$ be a martingale adapted to a filtration $\mathcal{F}_i$, with the following conditions: ...
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1answer
34 views

Azuma's/Hoeffding's inequality for geometric series

Let $X_1,X_2,\dotsc$ be a sequence of a.s. bounded, zero-mean random variables. For $\alpha \in (0,1)$ define $Z_t$ as the geometric series with $Z_t = \sum_{i=1}^t\alpha^{t-i}X_i$ and $\mathcal{F}_k ...
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1answer
45 views

Is the geometric series of a set of $n$ RVs a martingale?

Let $X_1,\dotsc,X_n$ be independent, zero mean random variables and define $Y_k = \alpha^{n-k}X_k$. Is $\{Z_k\}$ with $Z_k = \sum_{i=1}^k Y_i = \sum_{i=1}^k \alpha^{n-i}X_i$ a martingale? I suppose ...
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0answers
25 views

How to show that stochastic exponent is integrable?

I need to prove that if $u: [0,T]\rightarrow \mathbb{R}$ is a deterministic square integrable function then stochastic exponential process defined : $M_{t} = exp(-\int_0^t \! u(s) \, \mathrm{d}W_{s} ...
2
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1answer
28 views

(Bt)^2 it is a martingale?

Well i think no, because the expected value is t E[(Bt)^2)=t, so it´s not constant, change with the time. Am I right? Propz
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1answer
37 views

Let $X_n$ be the $n$-th partial sum of i.i.d. centralized rv and $\mathcal{F}_m:=\sigma(X_n,n\le m)$, then $\text{E}[X_n\mid\mathcal{F}_m]=X_m$

Let $(\Omega,\mathcal{F},\text{P})$ be a probability space $\left(Y_i\right)_{i\in\mathbb{N}}$ be a sequence of i.i.d. random variables ...
1
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1answer
49 views

Prove $\sum E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converges to $X$ in $L^2$

Let $(X_n)_{n \in \mathbb{N}}$ be a martingale. Prove $\sum_n E((X_n-X_{n-1})^2)$ is finite iff $X_n$ converge to $X$ in $L^2$. It is not hard at first glance, but I cannot figure it out after many ...
2
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1answer
38 views

How to prove this Brownian motion convergence?

Let $W_t$ be a Brownian motion. How do I show the following? $$ \alpha > \frac{1}{2} \Rightarrow \lim_{t\rightarrow\infty} \frac{W_t}{t^{\alpha}} = 0 \text{ a.s.} $$ Showing convergence of this ...