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

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Show this one is a martingale?

On a fixed interval $[0,T]$, let $(W_t)_{0\le t \le T}$ be a Brownian motion, and $ (\gamma_t)_{0\le t \le T} $ a cadlag process. Let $$ M_t = exp ({\int_0^t\gamma_sdW_s - \frac{1}{2}\int_0^t\gamma_s^...
3
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1answer
29 views

Exercise on stopping times

Let $(Y_n)_{n \geq 1} $ be a sequence of independent r.v.'s s.t. $$P(Y_n=y) = {n \choose k } \left(\frac1n\right)^y \left(1-\frac1n\right)^{n-y}\quad {\rm if }\;y \in \{0,1,\dots,n\}$$ How to show ...
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1answer
35 views

Local martingale implies martingale

Let $M$ be a right-continuous local martingale such that $M^*_t \in L^1(P)$ for all $t \in \mathbb{R}_+$. Here \begin{align*} M^*_t(\omega) = \sup_{0 \leq s \leq t} |M_s(\omega)|. \end{align*} Now, I ...
2
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1answer
39 views

Construct a martingale with a given distribution?

Given a random variable Y, is it possible to construct a martingale M such that $$M_1 \stackrel{D}{=} Y$$ I'm not sure how to go about proving that such an M exists under such general conditions, but ...
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1answer
32 views

How many types of martingale related stochastic processes are there?

Previously I had thought that the only concepts in this direction were martingales, submartingales, and supermartingales. However, at least when discussing quadratic variation and stochastic ...
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0answers
32 views

Stopped process not uniformly integrable

I need to construct a counter example such that the process $\{X_n\}_{n \ge 1}$ is uniformly integrable; however, the stopped process $X_{\tau \wedge n}$ where $\tau$ is a stopping time, is NOT ...
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1answer
14 views

Convergence of a Branching process

Consider the Branching process: $\{ \xi_i^n , n \ge 1, i \ge 1\}$ are i.i.d. taking values $0, 1, \ldots$ and $Z_0 := 1, \; Z_{n+1} := \sum\limits_{i=1}^{Z_n} \xi_i^{n+1}$. Assume $\mu := \mathbb{E}[\...
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0answers
25 views

An example in martingales convergence

Let $\xi_1, \xi_2, \cdots$ be i.i.d with $P(\xi_i=1)=P(\xi_i=0)=\frac{1}{2}$. Let $X_0 := \frac{1}{2} \; ;\;\; X_n := \frac{1}{2} \xi_1 + \frac{1}{2^2}\xi_2 + \cdots + \frac{1}{2^{n}} \xi_n + \frac{...
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0answers
23 views

Stopped process of maximum stopping times

Suppose $X$ is an adapted process and $\tau_1, \ldots , \tau_k$ are stopping times such that $X^{\tau_1}, \ldots , X^{\tau_k}$ are all martingales. I want to show that then $X^{\tau_1 \vee \ldots \vee ...
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23 views

Martingale w.r.t. the filtration $\{\mathcal{F}_{t \wedge \tau}\}$

I want to show that any martingale w.r.t. the filtration $\{\mathcal{F}_{t \wedge \tau}\}$ is also a martingale w.r.t. the filtration $\{\mathcal{F}_{t}\}$. So, suppose $(X_n)_{n \geq 0}$ is a ...
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1answer
42 views

Martingale property cannot hold for general random times

Let $\sigma \leq \tau$ be two random times that are no stopping times. I want to create a simple example that shows that for these random times $\mathbb{E}[M_\tau \mid \mathcal{F}_\sigma] = M_\sigma$ ...
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1answer
27 views

Martingale representation theorem , optimal stopping time and the principal agent problem

I am self-learning some Econ papers. Any suggestion will be appreciated. Even though the questions are from an Econ paper, they are math-related. I provide the economic interpretation as background ...
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2answers
37 views

Covariance of Martingales

I have proven that Martingales have orthogonal increments. From this I need to show that $\operatorname{Cov}[M(t),M(s)]$ relies only on $\min\{s,t\}$. I used the expected value definition of ...
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1answer
93 views

Question on part 3 of the Star Trek problem in Williams, Probability with Martingales

Consider this M.SE question, which is E12.3 in Williams. The answer of Robert Israel (and Xoff) seems to give an exponential bound on $R_n$ almost surely. Wouldn't this imply the convergence of $$\...
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34 views

Variation on the classic ABRACADABRA problem

Suppose letters are chosen randomly from the alphabet, one at a time. It is well known that the expected time until ABRACADABRA is spelled out is $26^{11}+26^{4}+26$. The proof uses discrete time ...
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1answer
47 views

Showing a stopping time is finite

Let $T = \inf\{ n : S_n = a \text{ or } S_n = -b\}$ be a stopping time, where $S_n = X_1 + \dots +X_n$ and each $X_n$ is a martingale. I am looking at a proof which shows that $T < \infty$ almost ...
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1answer
23 views

Conditional Expectation of Martingale

Letting $(X_t, F_t)_{t \in \mathbb{R}}$ be a martingale with continuous realizations and $0 \leq s \leq t$, I want to find $E(\int_{0}^{t}X_udu|F_s)$. I understand that $E(X_u|F_s)=X_s$ for $u\geq s$....
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2answers
32 views

Expectation of two successive martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to the filtration $\{\mathcal{F}_n\}_{n \geq 0}$. I want to prove that: \begin{align} \mathbb{E}[M_nM_{n+1}] = \mathbb{E}[M_n^2]. \end{align} ...
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1answer
41 views

Integral w.r.t. a Martingale

Consider the stochastic integral $$ Z_t = 1+\int_0^tZ_{s^{-}}\,dX_s $$ where $X$ is a Martingale. In the textbook by Shreve (see here pages 493-493) it is said that since $Z_{s^{-}}$ is left-...
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0answers
33 views

Product of expectations is a martingale

Consider a probability space $(\Omega, \mathcal{F}, P)$ and random variables $X_0, X_1, \ldots , X_n$ adapted to the filtration $\{\mathcal{F}_t\}_{t\geq0}$. Assume furthermore that each $X_n$ is ...
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0answers
30 views

Martingale demonstration

I have a model where the process followed by an asset is a Geometric Brownian Motion: dSt= μStdt + σStdWt Where μ and σ are positive constant and Wt is a Brownian Motion. I have to consider a bank ...
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1answer
35 views

Prove the tower property for conditional expectation using filters [closed]

While studying about martingales I came about the following proposition : $$\mathbb{E}(\mathbb{E}(X|\mathcal{F}_n)|\mathcal{F}_m) = \mathbb{E}(X|\mathcal{F}_m)$$ if $m \le n$ where $\mathcal{F}_n$ is ...
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0answers
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 ...
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0answers
26 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 $k$...
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 }, \end{...
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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
42 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
20 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 ...
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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
17 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 well....
0
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1answer
31 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 ...
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1answer
33 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
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1answer
44 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
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1answer
44 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
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1answer
42 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 \infty}E(...
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2answers
43 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
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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 $\mathcal{F}...
4
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0answers
48 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 ...
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1answer
43 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) & = E(...
4
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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 E(X^...
0
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1answer
41 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}) = e^{-|r|\sqrt{2\...
2
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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
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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 ...
1
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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
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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
53 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 ...
1
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0answers
29 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 \mathbb{E}(...