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

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2
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1answer
33 views

Expectation of martingales [closed]

If I know that $\{M_t\}_t$ is a martingale, we know that $$\mathbb{E}({M_tM_s})=\mathbb{E}(M_{\min({t,s})}^2)$$ Is there something I can say about $\mathbb{E}(M_tM_sM_r)$?
0
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0answers
45 views

martingale property of function of compensated martingale

The process $((N_t - \lambda t)^2 - \lambda t ) $ where $t \in \mathbb{R}_+$ and $N_t$ a Poisson process. I succeed to show that the expectation of the absolute value $ E \left[ |((N_t - \lambda t)^2 -...
1
vote
1answer
19 views

proving a random variable is a martingale

I am on the final part. I have shown all the properties of martingales, except for the fact that $E|N_n| < \infty$. The solutions state $|N_n|$ is bounded, but I don't see how it is as $S_n$ is not ...
1
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2answers
39 views

Checking if $X(t) = \exp(t/2)\cos(W(t))$, with $W(t)$ a Wiener process, is a martingale

This is what I've done: Let $s < t$ and $F_t$ be a filtration adapted to $W(t)$ $$E[e^{t/2}\cos(W(t))|F_s] = e^{t/2} E[\cos(W(t)) - \cos(W(s)) + \cos(W(s))|F_s]$$ $$= e^{t/2} [E[\cos(W(t)) - \cos(...
3
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0answers
36 views

Show $M_n=X_1+X_2+…+X_n-n\theta$ is a martingale w.r.t ${X_n}$

Show $M_n=X_1+X_2+...+X_n-n\theta$ is a martingale w.r.t ${X_n}$, given that $X_i$ are i.i.d. random variables with $\mathbb{E}[X_i]=\theta$ this is what I've done: $$\mathbb{E}[M_{n+1}|X_{\le n}]=\...
3
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1answer
45 views

Martingale convergence for UI martingales

I started reading this paper (Lamb, Charles W.. “Shorter Notes: A Short Proof of the Martingale Convergence Theorem”. Proceedings of the American Mathematical Society 38.1 (1973): 215–217) today. In ...
1
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1answer
24 views

If $Y_n=\min\{M_n,7\}$ and $\{M_n\}$ is a martingale wrt ${X_n}$, show that ${Y_n}$ is a supermartingale wrt ${X_n}$

If $Y_n=\min\{M_n,7\}$ and $\{M_n\}$ is a martingale wrt ${X_n}$, show that ${Y_n}$ is a supermartingale wrt ${X_n}$ I tried doing cases for $M_n<7$ and for $M_n>7$, but I couldn't get that $E[...
3
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1answer
58 views

Attempting to show $P(|S_n| <1)$ for a martingale $(S_n)$

Now, I am stuck on the last part of the question. I managed to find the solutions, but I don't udnerstand them completely. What I don't understand is: How they got that indicator function, and why ...
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0answers
24 views

Show $X_t$ is a martingale, if $(M_k)_{k=0}^n$ is a discrete-time martingale, and $X_k$ = $M_k$ − $M_{k−1}$ for (k = 1, . . . , n).

If $X_1, \dots, X_n$ are independent random variables, then using the equation $$\operatorname{Var}\biggl(\sum_{i=1}^\infty X_i\biggr) = \sum_{i=1}^n \operatorname{Var}(X_i)$$ Show that this is also ...
0
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1answer
28 views

showing a sequence of random variables is a martingale

I am trying to show that $S_n$ is a martingale: $E(S_n | \mathcal{F}_{n-1}) =now, I don't know what to do, as we don't know anything about $S_n$.
3
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1answer
50 views

Determine $E\sum_0^\infty X_n1_{(T=n)}$

$X_T = \sum_0^\infty X_n 1_{(T=n)}$ where $T$ is a stopping time and $(X_n)$ is a martingale. Show that if $T$ is bounded then $EX_T = EX_0$: $T \leq N$, and then consider $X_T = X_{T\wedge N} = \...
0
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1answer
23 views

Is the Stock Prices in a Perfect Market martingale or not?

Stock Prices in a Perfect Market Let Xn,, be the closing price at the end of day n of a certain publicly traded security such as a share of stock. While daily prices may fluctuate, many scholars ...
2
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1answer
35 views

Almost sure and $\mathcal{L}^1$ convergence of $Y_n=(\cos a)^{-n}\cos(a(Z_1+\cdots+Z_n))$ with $(Z_n)$ i.i.d. Bernoulli

Let $(X_i)$ be i.i.d. with $P(X_i= a)=P(X_i = -a)=\frac{1}{2}$, for some $a$ such that $2a \notin\pi\mathbb Z$. Let $$Y_n=\frac1{\cos^n(a)}\cos\left(\sum_{i=1}^n X_i\right).$$ Check whether $(Y_n)$ ...
0
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0answers
32 views

Expected value of the infinite product of indepedent random variables

We assume that $Y_{n}$ are independent random variables and we let $Y_{n}$ have the values $\frac{3}{2}$or $\frac{1}{2}$ with probability $\frac{1}{2}$ each. We let $X_{n}=Y_{1}\cdot \cdot \cdot Y_{n}$...
0
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2answers
28 views

Right-continuity of the expectation of a supermartingale

I start with a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),P)$. I assume that the filtration is right-continuous. On this probability space I define a supermartingale $M$. Now ...
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0answers
19 views

Martingale implies moment generating function exists

Let $X_T = \ln(S_T/S_0)$ where $S_T$ denotes the stock price at time $T$ and $S_0$ is the spot price. There is a well known relationship between the moments of $X_T$ and the characteristic function $\...
0
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1answer
32 views

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
30 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 ...
0
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1answer
36 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 ...
1
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1answer
36 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
34 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 ...
1
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1answer
15 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
26 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{...
2
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1answer
33 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 ...
2
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0answers
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 ...
1
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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 $\mathbb{E}[M_\tau \mid \mathcal{F}_\sigma] = M_\sigma$ ...
1
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1answer
28 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 ...
2
votes
2answers
41 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 ...
4
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1answer
94 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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0answers
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 ...
1
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1answer
58 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
33 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} ...
0
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1answer
42 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
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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
50 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
31 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
votes
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 ...
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
votes
0answers
19 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
votes
1answer
35 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
vote
1answer
35 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
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 $\...