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

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1answer
46 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 ...
0
votes
1answer
53 views

How do I prove that a martingale has a constant expected value?

I can´t prove that a martingale has constant expected value. $$ \mathbf{E}[M_t]=\mathbf{E}[M_0] $$ Thanks people.
2
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1answer
34 views

How to show the sum is a martingale

In the hypothesis of the martingale central limit theorem, my book says that given a sequence of random variables $X_n$ with the condition that $E(X_n \mid \mathcal F_{n-1}) = 0$, then $S_n = ...
2
votes
1answer
36 views

Show that a function of a symmetric random walk is a martingale

Suppose $S_n = (X_n,Y_n)$ is a symmetric random walk on $\mathbb{Z}^2$. Show that $G_n = X_n^2 + Y_n^2 - n$ is a martingale. What is true about $E_{(x_0,y_0)}[|S_n|]$? Find an upper bound for ...
2
votes
1answer
37 views

Supermartingale of product of random variables

Let $Y_1, Y_2, \ldots$ be independent and identically distributed non-negative random variables. Define $X_n := \prod_{i=1}^n Y_i$ for $n\geq 1$. I have to show that if $(X_n)_n$ is a supermartingale ...
-1
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1answer
26 views

Reference book on Backward Martingale theory

I am looking for an introduction to the topic of BACKWARD MARTINGALES possibly with good intuition (it can be either notes or a book) AND a reference book on the topic.
5
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2answers
124 views

Example: satisfying $E(X_{n+1}\mid X_n)=X_n$ but not a martingale

I am wondering if there is such a sequence of random variables $(X_n)_{n=0}^\infty$ such that $\mathbb{E}(X_{n+1}\mid X_n)=X_n$ for all $n\geq0$ but which is not a martingale with respect to the ...
2
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0answers
28 views

Supermartingales and optimal strategies for a game

Your winnings per unit stake on game $n$ are given by independent random variables $\epsilon_n$ such that $P(\epsilon_n=1)=p$, $P(\epsilon_n=-1)=q$ with $1/2<p=1-q<1$. Let $C_n$ be your stake on ...
1
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2answers
83 views

Is this stochastic process a martingale?

I have the following process: $X_t=tB_t-\int^{t}_{0}B_s \ ds$ where $B_t $ is a Brownian motion. Is this a Gauß-process and/or a martingale? Can someone help me with this? And how can I calculate ...
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0answers
24 views

Good reference on stopping times and continuous time change

I've been trying to look at stopping times and continuous time change in martingales but have trouble understanding without some concrete examples. Anyone knows of any good references that might be ...
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0answers
26 views

Question on exponential martingale

I was reading the first proof here on exponential martingale, https://fabricebaudoin.wordpress.com/2012/09/27/lecture-23-time-changed-martingales-and-planar-brownian-motion/ It says that "Let $ ...
4
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1answer
75 views

Does this stopping time always have infinite first moment?

Let $X_1, X_2, X_3, \ldots$ be i.i.d. random variables with zero mean and let $S_n := X_1 + \ldots + X_n$. Does $T := \inf\{n: S_n > 0\}$ always have infinite first moment? In the trivial case, ...
1
vote
1answer
74 views

Counterexample in optional stopping martingale

Problem: Give an example of submartingale $\{X_n\}$ with $\sup_nE |X_{n-1}-X_n|<\infty$ and stopping time $N$ with $E[N]<\infty$ such that $\{X_{n\wedge N}\}$ is not uniformly integrable. ...
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0answers
39 views

Size-Biased Galton Watson Tree.

First of all, i am not sure whether this question belongs here or to stack overflow. Let me write here, first, i will give a definition of size-biased distribution. then i will give definition of ...
2
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0answers
22 views

Martingale energy inequality

I am reading a book on BMO martingales which uses a so-called energy inequality. I have not been able to find a solid reference for this. Can someone please give a reference to these inequalities. ...
1
vote
1answer
37 views

Positive submartingales

Let $\{X_n\}$, $n>0$ be a positive submartingale with $X_{0} = 0.$ Let $V_n$ be random variables such that $V_n \in\mathcal F_{n−1}$ for all $n \geq 1$. $B > V_1 > V_2 > \dots > 0$ ...
2
votes
4answers
68 views

Show that $M_n = X_n^2 - n$ is a martingale

Suppose $X_n$ is a symmetric random walk on $\mathbb{Z}$. To show that it is a martingale I need to show $$ \mathbb{E}[M_{n+1}|X_{0:n}] = M_n $$ $$ \begin{align} \mathbb{E}[M_{n+1}|X_{0:n}] &= ...
2
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0answers
31 views

Stochastic control with stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ and smooth functions $u,F: [0, +\infty) \rightarrow \mathbb{R}$, how can we optimise the ...
2
votes
1answer
27 views

4th-moment bound on continuous local martingale

I am struggling with this question: Let $X$ be a continuous local martingale with $X_0=0$, and such that $\mathbb{E} (\langle X \rangle^{p/2}_t) < \infty$, for all $t \geq 0$ and $p \geq 2$. ...
0
votes
1answer
28 views

Expectation of Itö integral related to martingale property

A reference stated : "Because $I(t)$ is a martingale and $I(0) = 0$, we have $E[I(t)]=0$ for all $t\geq0$" What are the role of the martingale and initial value to determine that $E[I(t)] = 0$? ...
0
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1answer
27 views

$p$-variation of a continuous local martingale

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| ...
4
votes
2answers
70 views

Localisation in the proof of Ito's formula

I am reading Karatza's and Schreve's book "Stochastic Calculus and Brownian Motion" and I don't understand a strange thing as follows: Let $X=X_0 + A +M $ be a semimartingale, where $A$ is a ...
1
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1answer
36 views

Conditional independence of random variables

Let $(X_t),$ $(Y_t)$ be independent bounded martingales (for filtration $ \{ \mathcal{F}_t \}$ )which converge to $X_\infty$ and $Y_\infty$ respectively, by the martingale convergence theorem. Let $\{ ...
0
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1answer
54 views

Existence of localizing stopping times that reduce a local martingale to a square integrable martingale

Something is weird from a proof that I am reading: The well-known theorem of characterization of quadratic variation states that: Suppose $X$ is a continuous local martingale and $A$ is a continuous ...
2
votes
1answer
64 views

Prove Kolmogorov's zero one law using martingales

I am supposed to provide a martingale proof of Kolmogorov's zero-one law. Hint Let $X_n$ be independent random variables and let $\mathcal C_\infty$ be the corresponding tail $\sigma$-algebra. Let ...
1
vote
1answer
34 views

Discrete stochastic process is predictable iff it is natural

So this question was one we were asked to proof during class. The formulation is as follows: Now the question is about c). I was able to proof that $\mathrm{E} M_nA_n =\mathrm{E}M_{n-1}A_n$ via ...
0
votes
1answer
33 views

showing it's a martingale, should be simple but I'm missing a constant?

I'm working through example 1.4 on page 3 of http://people.maths.ox.ac.uk/xu/Martingale_convergence.pdf I'm having trouble showing that $X_n = 2^n 1_{[0,1/2^n)}$ is a martingale. I'm not sure why my ...
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0answers
35 views

Inverse Bessel Process as strict local martingale without Ito's formula?

Is there a way to prove that the inverse Bessel process $|B_t|^{-1}$ is a local martingale without using Ito's formula, considering that the Green's function $$g(x)=\int_0^\infty ...
3
votes
1answer
32 views

Why doesn't $E[E[X|\mathcal{F_n}]]=E[X]$ work for $X=(Z_{n+1}-Z_n)^2$?

Let $(\Omega,\mathcal{A},P)$ be a probability space. Let $\Bbb F=(\mathcal{F}_n)_n$ be a filtration wrt this space and $(Z_n)_n$ an $\Bbb F$-martingale. Now, basic properties of the conditional ...
0
votes
1answer
18 views

Finite quadratic variation leads to finite covariation

I'm trying to prove that if two functions have finite quadratic variation then their covariation is finite. I've seen that $2|[X,Y]_{t}| \leq [X]_{t}+[Y]_{t}$ but I can't see how to get there. It ...
2
votes
1answer
30 views

Do paths of a continuous time martingale always have a left limit?

I'm looking at this theorem: "Let $(X_t, \mathcal{F}_t)_{t \in \mathbb{R}_+}$ be a submartingale, where $\mathcal{F}$ is right-continuous and complete, and the function $\mu(t) := E[X_t]$ is ...
2
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0answers
40 views

Sufficient condition for martingale property

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be a filtered probability space and $M=(M_t)_{t\geq 0}$ an $\mathcal{F}_t$-adapted stochastic process. If $$ \forall t<s, \ ...
2
votes
1answer
42 views

Uniqueness in Doob's Decomposition Theorem

I'm a little uneasy about one step in the uniqueness proof for Doob's Decomposition Theorem. Let $(X_n)_{n \geq 0}$ be a submartingale, $(M_n)_{n \geq 0}$ a martingale, and $(A_n)_{n \geq 0}$ be an ...
0
votes
1answer
16 views

Let $S_n$ be a Simple Random Walk. What is $E[S_m|S_n]$ if $m < n$?

Let $S_n = W_1 + ... + W_n$ be a simple random walk with $W_i$ IID and $P[W_i = 1] = P[W_i = -1] = 1/2$. Find $E[S_m | S_n]$ when (a) $m > n$ and (b) $m < n$. For part (a), I get the answer of ...
0
votes
1answer
9 views

Modification of a local martingale

I am quite curious to know if the following is true, which comes up to my mind when reading a paper on SLE: For any local martingale $(X_t)_{t \geq 0}$ and stopping time $\tau$, is it true that $$ ...
1
vote
1answer
46 views

Verifying a proof of martingales.

I am trying to prove the following: Let $T$ be a stopping time bounded by $c$, and let $(X_n)$ be a martingale, then $E(X_T)=E(X_0)$. Here is what I did: $\int ...
2
votes
2answers
100 views

Bounding profits of gambler by Azuma Inequality

A gambler plays the following game: In each round, he can pay any $0 < p < 1$ dollars, and win 1 dollar with probability p (independently). Show that the probability that the gambler's net ...
-1
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1answer
54 views

Calculate expected value for a lazy Random Walk

Calculate the mean of time needed for a lazy random walk on $[0,n]$ which starts on $0<k<n$ to hit $0$ or $n$ if in each step the walk stays in probability $\frac 1 3$, goes to the right in ...
2
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0answers
43 views

An irreducible Markov chain is a martingale

Let $\{X_n\}$ be an irreducible Markov chain. Does exist example of such $\{X_n\}$ which is also a martingale given that: a. $\{X_n\}$ is recurrent with finite number of states (but bigger ...
3
votes
1answer
43 views

Prove that if $E(X\log X)<\infty$ then $E(\sup_n |S_n|/n)<\infty$.

This is part 2 of a two part question. In the first part, we were asked to show that if you had a non-negative sub martingale $M_n$ then $$\sup_n E(\sup_{k\leq n} M_k)\leq \sup_n 2E(M_n \log M_n)+2$$ ...
2
votes
1answer
28 views

Equivalent condition for $ (X_n, \mathcal{F}_n) $ to be a martingale

I've encountered an interesting problem and am not quite able to solve it. It is to prove the following statement ($ X_n $ denotes a sequence adapted to a filtration $\mathcal{F}_n) $: $$ ...
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vote
2answers
83 views

Give simpler examples of $L^p$ unbounded yet converging martingales

I read the following example in the book Counterexamples in Probability and Real Analysis by Gary L. Wise and Eric B. Hall: Does anyone know simpler examples? I do have one! I would be glad to ...
4
votes
1answer
75 views

Martingale with respect to a decreasing filtration

I am trying to solve problem 2.16 from the book "Continuous Martingales and Brownian Motion" by Revuz and Yor. There are two things that confuse me from the exercise so hopefully someone can shed some ...
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votes
1answer
54 views

Probability of a nonnegative submartingale converging to zero [closed]

Suppose that $\{X_k\}$ is a nonnegative submartingale, and $\Pr(X_1 = 0) = 0$. Then could we conclude that $\Pr(\liminf X_k=0) = 0$? What about $\Pr(\lim X_k=0) = 0$? Thanks a lot. Some background ...
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vote
1answer
15 views

Semimartingale jumps question

I am reading a statement which contains $\Delta X \cdot Y$ where $X$ is a semimartingale and $Y$ is a finite variation process and the notation means the lebesgue stieltjes integral. My problem is ...
0
votes
1answer
48 views

Exponential of Brownian motion with negative drift

I am reading a text on Brownian motion and don't understand the following: Let $X_t = \exp \{ W_t - \frac{t}{2} \}$, where $W$ is a standard Brownian motion on $\mathbb{R}$. Let $T_n = \inf \{ t \geq ...
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0answers
44 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
3
votes
0answers
66 views

Condition for $L^p$ convergence of backwards martingale

Is there any condition that is known to be sufficient for $L^p, 1<p<\infty$ convergence of a backwards martingale (and why is it sufficient)? I couldn't find anything else than the normal $L^1$ ...
2
votes
1answer
25 views

Doob Decomposition is $L^1$ bounded

Suppose $X_n$ is a martingale that is $L^p$ bounded for some $p > 1$. Then the problem asks to show that the Doob Decomposition of the submartingale $|X_n|^p = M_n + A_n$ where $M_n$ is a ...
0
votes
1answer
47 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...