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

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Brownian Motion with Levy's Characterization 2

Let W be a $\mathbb{R}$-valued Brownian motion. To prove that $(B_t)_{t\geq 0}$, where: $B_t:=W_t-\int_0^t\frac{W_u}{u}du$, is a Brownian Motion with respect to $\mathcal{F}^B$, I showed $[B]_t=t$ and ...
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41 views

Friedman's urn is a supermartingale or a submartingale?

Here is the urn model: At time zero there are $r$ red and $g$ green balls in an urn. At each time-step, we draw out a ball at random and replace it along with $c$ of the same color and $d$ of the ...
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0answers
39 views

Brownian motion with Lévy’s Characterization

I want to show that: if for all $\lambda \in \mathbb{R}$ the process $(exp(\lambda X_t-\frac{\lambda ^2}{2}t))_{t\geq0}$ is a $\mathcal{F}^X$ local martingale, then the $\mathbb{R}$-valued process X ...
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1answer
46 views

Gambling strategy for coin-flip with 51% chance to win.

So I'm curious if there is a decent gambling strategy for this site I play on. Essentially it's a coin flip where you can have anywhere from 49% to 51% chance to win against the other player. ...
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20 views

Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?

Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
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0answers
18 views

showing a processes is martingale using ito's lemma

Let $Y(t) = t^2W_t - 2 \int_0^t sW_s \ ds$ where $W_t$ is brownian motion. I am trying to show it is a martingale by showing it is driftless. I set $Z(t,W_t) = t^2W_t$ and ito's gives $dZ = 2tW_t \ dt ...
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Proof for Yor's Formula [on hold]

Given that X and Y are two semimartingales, how can it be proved that the following statement is true? $\varepsilon(X)\varepsilon(Y)=\varepsilon(X+Y+[X,Y])$
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22 views

Show that X is a supermartingale [on hold]

I need to prove that X, being a positive local martingale, is also a supermartingale. How can this be done?
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1answer
13 views

Conditional expectation: when does $X_t=E[X_t\mid \mathcal{F}_s]$ for $s<t$

I came across a calculation (1$^\circ$ calculation, 2$^{nd}$ step) that stated, for $s<t$ $$E[B_s(B_t^2-t)]=E[B_sE[(B_t^2-t)\mid\mathcal{F}_s]]$$ I know the expectation here is zero, however, I ...
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1answer
23 views

Martingale and local martingales

I have to show that $e^{B_t^1}cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-diemensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}cos(B_t^2)=1+\int_0^t ...
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41 views

Martingal-property of stochastic Integral w.r.t. Brownian Motion

To Show that $(e^{B_t^1}cos(B_t^2))_{t \in \mathbb{R_+}}$ (where: $B=(B_s^1,B_s^2)$ is a 2-dimensional Brownian Motion) is a Martingal I used Ito's Lemma and showed that this is equal to: $ 1+ ...
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1answer
31 views

How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?

I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is: \begin{align*} E\left((X(t)-\lambda t)^2 ...
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0answers
54 views

Probability of the martingale staying non-negative.

Here is a question on martingales (given after third graduate lecture on the subject). Let $X_n$ a martingale with respect to the natural filtration and such that $X_0 = 0$, assume that $\frac{1}{2} ...
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1answer
16 views

Bounding expectation of a supremum process

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is ...
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2answers
42 views

Show $(X_n+a)^2$ is a submartingale

Let $(X_n)$ be a martingale, and let $EX_n^2 < \infty$ - then I am told to show $E(X_n+a)^2 $ is a sub martingale. I wrote $$(X_n+a)^2 = ((X_{n-1} + a) + (X_n - X_{n-1}))^2 $$ then $$E((X_n+a)^2 | ...
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26 views

How to prove this continuous martingale converges?

Suppose $B = (B_t, t \geq 0)$ is standard Brownian motion. Let $M^\lambda_t := \exp(\lambda B_t - \frac{\lambda^2 t}{2})$ (I have previously shown that this is a martingale). How do I prove that $$ ...
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14 views

Azuma's inequality conditional version

Let $(\Omega, \mathcal{F},P)$ be a probability space. Consider a martingale $M_n$ with filtration $\mathcal{F}_n$. Let $B \in \mathcal{F}$. On $B$, $a_n \leq |M_n - M_{n-1}| \leq b_n$ a.s. Can we ...
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29 views

Predictable quadratic Variation Intervals of Constancy

From Revuz and Yor - Continuous Martingales and Brownian Motion 1999 Chapter IV Proposition 1.13 it is proven, that for a continuous local martingale $M_t$ the intervals of ...
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1answer
26 views

A Property of Martingale of Sum of i.i.d. Random Variables

I am trying to solve the following problem: Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variables with finite mean. Let $F_n =\sigma(Y_1,...,Y_n)$. Let $\tau$ be a stopping time ...
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26 views

Ito's formula application

Let $ \alpha, \beta \in R$ and define $$ N(t)=e^{\beta t} \cos(\alpha W (t)) $$ I need to use Ito formula to compute $dN(t)$ Suppose $\alpha$ is fixed. What should $\beta$ be so that $N$ is a ...
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0answers
27 views

show that a function with brownian motion is a martingale

Let $B=(B^1,B^2)$ be a two-dimensional Brownian motion w.r.t. the Filtration $\mathcal{F}^B$. Show that $(M_t^2)_{t\in \mathbb{R}_{+}}:=(e^{B_t^1} \cos(B_t^2))_{t\in\mathbb{R}_{+}}$ I've tried it ...
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1answer
46 views

Proving martingale property

I have problem in proving a martingale property. Assume a random variable $\tau$ with a exponential distribution with parameter $\lambda$. Then define $Y_t = 1_{\{\tau \leq t \}}$. We then can prove ...
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1answer
26 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 $\mathbb{E}(M_tM_sM_r)$?
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42 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 ...
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1answer
18 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 ...
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2answers
31 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)) - ...
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0answers
31 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 ...
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1answer
31 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 ...
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1answer
21 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 ...
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1answer
55 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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18 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 ...
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1answer
26 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 $S_n$.
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1answer
49 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} = ...
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1answer
21 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 ...
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1answer
30 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 $(Y_n)$ ...
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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 ...
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2answers
24 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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16 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 ...
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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 - ...
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1answer
25 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
24 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 ...
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1answer
35 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
26 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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16 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
7 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 := ...
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0answers
22 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 + ...
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0answers
19 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
17 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
37 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
22 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 ...