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

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49 views

What is the variance of the expected time until one can construct an ABRACADABRA sequence?

Suppose that I am interested in the expected time until one can type out a sequence ABRACADABRA, assuming that each letter has $\frac{1}{26}$ probability of occurring. If we use a martingale $$X_n = n ...
0
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0answers
21 views

Probability of a repeated substring in a randomly selected string

Let $k , n , s \in \mathbb{N}$. Let $X$ be a string of length $n$ selected uniformly at random from an alphabet of size $s$. What is the probability that $X$ contains a repeated substring of length ...
2
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1answer
32 views

Indicator function minus probability on an event is a martingale

Define \begin{align} \epsilon_j = \mathbb{1}_{A_j} - \mathbb{P}(A_j) = \begin{cases} 1- \mathbb{P}(A_j) \qquad &\text{if } \omega \in A_j\\ - \mathbb{P}(A_j) \qquad &\text{otherwise }, ...
1
vote
2answers
44 views

How can I show that the stochastic process $M_t = W_t^3 – 3t W_t$ is a martingale $\mathbb{E}[M_u|F_t]$? [closed]

How can I show that this stochastic process $M_t$ is a martingale $\mathbb{E}[M_u|F_t]$? $W_t$ is a Brownian Motion. $$M_t = W_{t}^3 – 3t W_t$$
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2answers
40 views

Martingale convergence theorem for Poisson.

Let $\{A_n\}_n$ be random variables such that $A_0=1$ and given $\{A_j, j=0,\dots , n-1\}, A_n \sim Poisson(A_{n-1}).$ It is straightforward that $\mathbb{E}[A_n| A_1, \dots , A_{n-1}]=A_{n-1}$, and ...
2
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0answers
19 views

Understanding Azuma's inequality.

my question might be naive, but if I'm asked to apply Azuma's inequality on a Doob's martingale $E[X|X_1 \dots X_n]$ with independent random variables $X_j$, isn't it the same as finding the ...
0
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1answer
23 views

Martingales with bounded increments

It is known that if $T$ is a stopping time such that $E[T] < \infty$ and $(M_n)$ is a martingale with bounded increments, i.e. $\lvert M_n - M_{n-1}\rvert \leq K < \infty$ for every $n$, almost ...
1
vote
1answer
52 views

Determine probability of counting problem

At time $0$, an urn contains $1$ black and $1$ white ball. At each time $1,2,3, \ldots $ a ball is chose at random from the urn and is replaced together with a new ball of the same colour. Just after ...
2
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0answers
16 views

Multiplicative Super-martingales

Let $\{X_n\}$ be a stochastic process which is strictly positive, i.e. $X_n > 0$ almost surely for all $n$. It then follows that $\{Z_n = \log(X_n) \}$ is a well-defined stochastic process as ...
0
votes
1answer
28 views

Supremum of a supermartingale

For a nonnegative supermartingale $(M_n)_{n\in\mathbb{N}}$ I would like to prove that $$E[\sup_n M_n] \leq E[M_0]$$ I have that $E[M_n] \leq E[M_0]$ for every $n$. This combined with nonnegativity ...
1
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1answer
26 views

How does the sum of squared martingale difference sequence concentrate?

Let $X_i$ be a continuous, real-valued martingale such that $E[X_i] < \infty$ and $E[X_i | H_i] = X_{i-1}$ where $H_i = \{X_1, X_2, \dots, X_{i-1}\}$. Also, let $Y_i = X_i - X_{i-1}$ be a ...
1
vote
1answer
43 views

Martingale property for two stochastic processes

Let $(\Omega,F,P)$ be a probability space with filtration $\left\{F_{t}\right\}_{t\geq 0}$ generated by one dimensional Brownian motion $(B_{t})_{t\geq 0}$ defined on $(\Omega,F,P)$, assume that ...
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0answers
49 views

Modification of a submartingale $(M_t)_{t}$ that is determined by rational limits approaching from the right

Suppose $(M_t)_{t \in [0,\infty)}$ is a submartingale (on a complete probability space $(\Omega, \mathcal{F},P)$) with respect to a filtration $(\mathcal{F}_t)_{t \in [0,\infty)}$ that satisfies the ...
0
votes
1answer
43 views

If $(M_t)_{t \in [a,b]}$ is a martingale, then $t \mapsto E [ M_t ]$ is continuous.

Suppose $(M_t)_{t \in [a,b]}$ is a stochastic process. Denote $(\mathcal{F}_t)_{t \in[a,b]}$ to be the natural filtration generated by the process $(M_t)_{t \in [a,b]}$. Moreover, suppose $(M_t)_{t ...
2
votes
1answer
41 views

Limit of random variables with martingale

Let $\{(X_n,\cal F_n)\}$ be a supermartingale, and suppose $X_n$ converges almost surely to some $X_{\infty}\in L^1$. Are the following true: For a fixed $n$, Is it true that $\lim_{m\to ...
1
vote
2answers
37 views

If $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion?

If I have that $B_t$ is a standard brownian motion process, is $B_t^2 - \frac{t}{2}$ a martingale w.r.t. brownian motion? I know that $B_t^2 - t$ is but can't see it for the latter.
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0answers
14 views

Showing time inversion of a Brownian Motion $X_t = tB_{1/t}$ is continuous at $t=0$ USING the fact $X_t$ is BM on $\mathbb{Q}$? [duplicate]

I am reading the following paper on a rigorous construction of Brownian Motion: Brownian Motion. In the paper, they give a peculiar proof of the fact that the time inverted Brownian Motion is ...
3
votes
1answer
31 views

Are there two distinctly separate definitions for the Optional Stopping Theorem?

I have been reading a book called Stochastic Calculus by Steele. Inside, they have a theorem they state as the "Optional Stopping Time Theorem": If $M_n$ is a martingale with respect to ...
4
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0answers
42 views

For a simple random walk $S_n$ and for a stopping time $\tau$, what is the intuitive interpretation of $P(\tau < \infty) = 1$?

Suppose we have a simple random walk $S_n$ and we define a stopping time to be $\tau = min\{n: S_n = A \ \text{or} \ S_n = -B\}$. That is, we stop the first time we hit $A$ or $-B$. With this, I have ...
0
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1answer
38 views

If $B_t$ is standard Brownian Motion, how to show that $X_t = B_t^2-t$ is a martingale?

If $(B_t, \mathcal{F}_t)$ is standard Brownian Motion, I would like to show that $X_t = B_t^2-t$ is a martingale. My attempted proof works as follows: \begin{align} E(X_{t+1}|\mathcal{F}_t) & = ...
4
votes
1answer
84 views

If $X_n, Y_n$ are martingales, $E(X^2_{n+1})-E(X^2_n) \geq \frac{[E(X_{n+1}Y_{n+1})]^2}{E(Y^2_{n+1})} - \frac{[E(X_{n}Y_{n})]^2}{E(Y^2_{n})}$

If we have that $X_n, Y_n$ are square integrable martingales where $Y_0 \neq 0$, I'd like to show that for $n = 0,1,2,\ldots$ $$ E(X^2_{n+1}) - \frac{[E(X_{n+1}Y_{n+1})]^2}{E(Y^2_{n+1})} \geq ...
0
votes
1answer
38 views

For Brownian Motion $B_t$, and stop time $\nu = \inf\{t: B_t = r\}$, how to show $E(e^{-\alpha\nu}) = e^{-|r|\sqrt{2\alpha}}$?

If we have that $B_t$ is a Brownian Motion process, and we define a hitting time as $\nu = \inf\{t: B_t = r\}$ where $r \in \mathbb{R}$, how can I show that: $$ E(e^{-\alpha\nu}) = ...
2
votes
0answers
26 views

Prove convergence of quadratic variation.

Let $Z$ be a (not neccesarily cadlag) process adapted to some filtration $\mathbb{F}_t$. Assume that $T_n,T$ are stopping times so that $T_n<T_{n+1}$ and $T_n\uparrow T$. Assume that $Z^{T_n}$ is a ...
2
votes
2answers
29 views

If we have that $\tau \leq \nu$ are stopping times and that $X_n$ is a submartingale, how to show $E(X_{\tau \wedge n}) \leq E(X_{\nu \wedge n})$ [duplicate]

If we have that $\tau, \nu$ are stopping times and that $\tau \leq \nu$, if $X_n$ is a submartingale, how can I show that $E(X_{\tau \wedge n}) \leq E(X_{\nu \wedge n})$? Is there a way to ...
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0answers
51 views

Showing a predictable gambling strategy gives a supermartingale

I am stuck on the following problem. Your winnings per unit stake on game $n$ are $\epsilon_n$, where the $\epsilon_n$ are IID RVs with $$ P(\epsilon_n=1)=p, \ P(\epsilon_n=-1)=q, \text{ where ...
0
votes
1answer
30 views

Why is this stopping time the result of intersections instead of unions?

On page 54 of the book "Basic Stochastic Processes" of Brzezniak and Zastawniak, author proposes this example: A coin is tossed repeatdely and you win or lose 1 pound depending on which way it lands. ...
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0answers
51 views

If $\tau \equiv \min\{n:X_1+ X_2 +\cdots+ X_n > k\}$, how to show $\mathbb{E}(\tau) = 1 + {k \over \mu}$ for exponential random variables?

Let $X_1, X_2, \ldots$ be i.i.d where $X_i \sim \mathrm{Exponential}(\frac{1}{\mu})$, and suppose $k$ is a positive constant. Define $\tau \equiv \min\{n:X_1+ X_2 +\cdots+ X_n > k\}.$ I would like ...
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0answers
28 views

If $\{M_n\}$ is a sub-martingale that's bounded, and $\nu, \tau$ are bounded stop times with $\nu \leq \tau$, how to show $E(M_\nu) \leq E(M_\tau)$?

Suppose that $\{M_n\}$ is a submartingale that is bounded and we have that $\nu, \tau$ are bounded stopping times and that $\nu \leq \tau$. I would like to show that $\mathbb{E}(M_\nu) \leq ...
1
vote
1answer
20 views

How to show that a sequence of random variables is non-anticipating?

Let $\{M_n,\mathcal{F}_n\}$ be a martingale with $\mathbb{E}(M_n^2) < \infty$ for all $n$. Then, let us define the following relation: $$ M_n^2 = N_n + A_n $$ where ${N_n, \mathcal{F}_n}$ is a ...
0
votes
1answer
17 views

Equivalent definitions of $\mathrm{BMO}_p$ martingales

I'm working through exercise 3.16 in Revuz and Yor. Assume $Y$ is a continuous UI martingale and $1\leq p<\infty$. Then these are equivalent $\exists C\ \forall T$ stopping time ...
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1answer
28 views

How to show that a function $A_n$ is monotone increasing when $M_n^2 = N_n + A_n$, where $M_n, N_n$ are martingales

Let $\{M_n,\mathcal{F}_n\}$ be a martingale with $\mathbb{E}(M_n^2) < \infty$ for all $n$. I would like to show that we can write $$ M_n^2 = N_n + A_n $$ where ${N_n, \mathcal{F}_n}$ is a ...
0
votes
0answers
22 views

If we have that $Y_n = y^{S_n}$, with $S_n$ the sum of total gambling winnings at time $n$, what roots of $y$ cause $Y_n$ to be a martingale?

Suppose we have a game with the following: the player loses $\$1$ with probability $\alpha = 0.52$,wins $\$1$ with probability $\beta = 0.45$ and $\$2$ with probability $\gamma= 0.03$. At each round, ...
3
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0answers
41 views

Holder Continuity of a Continuous Stochastic Process

I have recently read the proof that the Brownian Motion and Fractional Brownian motion are almost surely Holder Continuous. I was wondering if this can be extended to a higher class of continuous ...
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0answers
50 views

Intersection of two simple random walks

Suppose that $X_n$ and $Y_n$ are independent, symmetric, one-dimensional simple random walks, where $X_0 = 0$ and $Y_0 = N$ for some $N \in \mathbb{N}$ where $N$ is even. I would like to show that the ...
4
votes
2answers
72 views

Stopping time on an asymmetric random walk

Suppose that we are given an asymmetric random walk whose step is defined as $P(\xi_i = 1) = p$ and $P(\xi_i = -1) = 1-p$, where $p >1/2$. The hitting time, $T_x$ is defined as $\inf{\{n : S_n = ...
0
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1answer
161 views

Help with Durrett 5.7.8

I've been working on Durrett Exercise 5.7.8 without any avail: Let $S_n$ be the total assets of an insurance company at the end of year $n$. In year $n$, premiums totaling $c>0$ are received and ...
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0answers
67 views

Martingale Concentration with Random Variable Bounds

I am seeking a concentration inequality for a sequence of martingales $\{M_t\}_{t=1}^n$. Let there be other random variables $C_1, ..., C_n$ such that $P(M_t \leq C_t) = 1$ (so they are not ...
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1answer
45 views

Check that an Ito integral is a martingale.

Before presenting my problem I will introduce some notation. Time index $t\in [0,T]$. $$C_t = \begin{cases} Z_n = B_{t_{n-1}}, & \text{if $t=T$} \\[2ex] Z_i = B_{t_{i-1}} , & \text{if ...
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1answer
56 views

Submartingale and stopping time

Let {$X_1, \dots, X_n$} be a submartingale, and let $T$ be a stopping time for {$X_i, 1\leqslant i \leqslant n$}. Show that $ E(\mid X_T \mid) \leqslant 2E(X_n^+)-E(X_1)$. The corresponding result for ...
0
votes
1answer
33 views

Martingale w.r.t different filtrations

$X_n,n\geq1$ is a martingale w.r.t to $\mathcal{G}=\mathcal{\{G}_n\}_{n=1}^{\infty}$. Let $\mathcal{F}=\mathcal{\{F}_n\}_{n=1}^{\infty}$. Where $F_n=\sigma(X_1,X_2, \ldots,X_n)$. ...
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0answers
57 views

There exists a real number so that $X_n$ is a martingale

I am working on the following problem: Let $Y_n$ be a sequence for which there exists constants $\alpha$ and $\beta$ with $$ E(Y_{n+1}\mid \mathcal{F}_n)=\alpha Y_n +\beta Y_{n-1} $$ for each ...
0
votes
1answer
20 views

Prove that a process with independent increments and a constant mean is a martingale

how to prove that a process with independent increments and a constant mean is a martingale? in a solution to this problem i found this : $X_t - X_s$ is independent from $F_s$ hence : ...
1
vote
1answer
20 views

Martingale $c^{W_t}$ where $W$ is Brownian motion

I have the process $c^{W_{t}}$ where $c$ is a constant and $W$ is Brownian motion. I would like to check if $\mathbb E[c^{W_{t+1}}|F_t]=c^{W_t}$. Dividing the right site yield $\mathbb ...
1
vote
1answer
34 views

Example of a non square-integrable martingale?

Are there (simple) examples of martingales which aren't square integrable?
3
votes
0answers
35 views

Application of Doob's optional stopping theorem to an elementary probability problem

The elementary probability problem is as follows. Let $(X_k)_{k\in\mathbb{N}}$ be a sequence of i.i.d. random variables such that $X_k \sim U(0,1)$ for each $k$. Define $\tau := \inf\{n\geq 0: ...
0
votes
1answer
38 views

What is the difference between a martingale and doob's martingale?

Every sequence that was termed as a doob's martingale, I was able to deduce that it was also a martingale. So here are few of my questions: 1) Is it correct to say that every doob martingale is also ...
0
votes
0answers
25 views

Sufficient unconditional moment condition for the convergence of $\sum_n (X_n - E[X_n])$

Let $\mathcal F_n$ be a filtration and $X_n$ be $\mathcal F_n$ measurable. Then $M_n = \sum_{k=1}^{n} (X_k -E[X_k])$ is a $\mathcal F_n$ measurable martingale. Let's assume that it is a square ...
0
votes
1answer
27 views

Absolute expectation of stopped martingale

Let $M_0,M_1,\dots$ be a martingale with respect to $X_0,X_1,\dots$ and $T$ be a stopping time with respect to $X_0,X_1,\dots$ Define $T_n=\min\{n,T\}$ and let $M_{T_n}$ be the stopped martingale. By ...
2
votes
1answer
31 views

Martingale Transform counterexample

I am studying discrete time martingale theory and came across the classical "You can't beat the system" theorem: given a martingale $M$ and a previsible process $C=(C_n)_{ n \ge 1}$ such that $C_n$ is ...
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0answers
26 views

Modelling the ballot theorem as a martingale.

The page 19 in the link http://www.imada.sdu.dk/~jbj/DM839/FL15.pdf provides the explanation of what a ballot theorem is and how we can prove that it is a martingale. It takes a random variable $S_k$ ...