Questions tagged [martingales]

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

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Semi-group property of branching processes?

See the edit below! I do have a question about continuous-state branching processes while reading "Fluctuations of Lévy Processes with Applications" by Kyprianou: Let $Y$ be a continuous-...
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42 views

Bounding the variance of a stopping time

Please refer to the exact same setting of this question: Durret problem 4.8.3 - Random walk and optional sampling theorem application I repeat the setting here in any case: Let $S_n = \xi_1+\ldots+\...
1 vote
1 answer
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Probability that a person does not receive a gift

There are two groups $G_1, G_2$ of $n\in\mathbb N$ persons each. Each member of $G_1$ independently gives a gift to a random member of $G_2$. What is the expected number $\mathbb E[L_n]$ of members of ...
1 vote
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How to verify that this process is a local martingale?

In my lecture notes, there is an interesting example for a strict local martingale: For its construction we consider Brownian motion $(W_t)_{t \geq 0}$ (w.r.t. the fitration $(\mathcal{F}_t)_{t \geq 0}...
1 vote
1 answer
56 views

Questions about asymmetric random walk

Let $X_1, X_2,\dots$ be i.i.d random variables with $\mathbb P(X_1=1)=\mathbb P(X_1=-2)=\frac 1 2.$ I need to consider $S_n = \sum\limits_{i=1}^n X_i$ with $S_n = 0$. What is $\mathbb P\left(\sup\...
3 votes
1 answer
198 views

I know that $\exp(\lambda B_{t} -\frac{\lambda^{2}}{2}t)$ is a martingale, how can I use it to prove that the following are martingales

Let $B$ be a standard Brownian motion and further let I know that for $\lambda \in \mathbb R$, we have that $(\exp(\lambda B_{t} -\frac{\lambda^{2}}{2}t))_{t\geq 0}$ is a martingale, but how can I ...
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Showing that a stopping time is finite for an unbalanced betting game

Let $\tau=\inf\{n:S_n=a \text{ or } S_n=-b\}$ be a stopping time and $X_n = Y_1+Y_2+...+Y_n$ a martingale (with $X_0=0$) that describes an unbalanced betting game between two players. $Y_i$ are ...
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1 answer
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Is this stopped martingale uniformly integrable?

let $(X_n)$ be a sequence of i.i.d. random variables with $\Bbb{P}(X_1=1)=p$ and $\Bbb{P}(X_1=-1)=1-p$ (s.t. $p\in (0,1)\setminus\{1/2\}$) then define for $a>0$ the sequence $S_0=a$ and $S_n=S_{n-1}...
0 votes
2 answers
52 views

Is it true that $\Bbb{E}(X_\infty)=\Bbb{E}(X_0)$ if $X$ is a martingale?

Let us consider a martingale $(X_n)_n$. Let me assume that $\lim_{n\rightarrow \infty} X_n=X_\infty$ exists a.s. and in $L^1$. My question is, is it then always true that $\Bbb{E}(X_\infty)=\Bbb{E}(...
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How $M_T$ is well-define in this statement of optional stopping theorem?

I'm reading about optional stopping theorem from this note. In Optional stopping theorem, version 3. below, $a, b>0$ and $$ T=\inf \left\{n \in \mathbb{N}: M_n \leq-a \text { or } M_n \geq b\right\...
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A possible typo in this statement of optional stopping theorem

I'm reading about optional stopping theorem from this note. In Optional stopping theorem, version 2 below, there is no restriction on the stopping times $T_1, T_2$, i.e., $0 \leq T_1 \leq T_2 \color{...
1 vote
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The stopped process $(M'_n, n \in \mathbb{N})$ with $M'_n := M_{T \wedge n}$ is a martingale

I'm tryin to verify that the following stopped process is a martingale, i.e., Theorem Let $(M_n, n \in \mathbb{N})$ be a martingale and $T$ a stopping time with respect to a filtration $(\mathcal{F}...
7 votes
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Has Mathematical Finance education become unnecessarily inaccessible? [closed]

Background I had a very good and bright student with some decent exposure to markets and the standard college math curriculum who got overwhelmed during a mathematical finance course due to ...
2 votes
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Limit of product of Poisson random variables

Let $(X_n)_{n \ge 0}$ be independent random variables, such that $X_n$ has Poisson distribution with parameter $n^2$. I want to find almost sure limit of the sequence $Y_n = \frac{X_1 \cdot X_2\cdot \...
6 votes
2 answers
344 views

Best strategy to reach $500 for a gambling situation in a casino

Suppose a gambler has \$100 to start with. Each time he/she has 0.4 chances of winning and 0.6 chances of losing a bet. If he/she wins he gets twice the money he put in and loses what he bet if he ...
2 votes
1 answer
104 views

How can I show that the stopping time $T=\min\{n\geq 0: S_n=1\}<\infty$?

Let be consider random variables $X_i$ which are independent and such that $\Bbb{P}(X_i=1)=1-2^{-i}$ and $\Bbb{P}(X_i=1-2^i)=2^{-i}$. Then define $S_n=\sum_{k=1}^n X_k$ and $T=\min\{n\geq 0: S_n=1\}$. ...
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If $S$ is a martingale, then alle processes $((HS)_t)_t$ are martingales

I have just started with financial mathematics in discrete times, and while reading about martingales I came across this statement without proof, which I am not sure how I would prove. I would ...
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1 answer
38 views

Sequence of i.i.d random variables and stopping time

Let $(Y_t)$ be a sequence of independent and identically distributed random variables such that $ \mathbb{E}\left[|Y_{t}|\right] < \infty $. 1.Show that if $\tau$ is a random time so that the event ...
4 votes
1 answer
58 views

Intuition behind Filtrations, Martingales and Stopping times

I would like to gain some intuition or context for the study of Martingales. What I have seen so far seemed to be motivated more by measure theory that probability theory. In the context of studying ...
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Showing that a stochastic integral is a Martingale in $M_2^c$

Suppose $\sigma$ is a real valued, bounded and continuous function and $(X_t)_{t\in\mathbb{R}^+}$ a stochastic process, B a standard Brownian motion. I want to show, that the Integral:$$\int_0^t\sigma(...
1 vote
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How to construct a martingale with given covariation?

The following fact is stated in my lecture notes: Let $f: \mathbb{R}_+ \to \mathbb{R}_+$ be an increasing and continuous function satisfying $f(0) = 0$. Then there exists a continuous real-valued ...
1 vote
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23 views

Filtration to define a martingale for the difference of two Brownian motions

Suppose we want to find a filtration, say $(\mathcal{F}_t)$, to ensure that $Z_t := B_{8t}-6C_t$ is a martingale, where $B$ and $C$ are two independent Brownian motions. For this, we want to ensure ...
1 vote
2 answers
45 views

Why can you simplify $E[\int_s^t X_u du |F_s] = X_s \int_s^t du$?

As per the question, I am trying to simplify the expression $$E[\int_0^t X_u du | Fs]$$ where $X_u$ is standard Brownian motion, and $F_s$ is the filtration of the process until time $s$. Now you can ...
4 votes
1 answer
71 views

Martingale convergence theorem and martingale property

Let $X_1,X_2,...$ be independent random variables: $$ \ X_n = \begin{cases} 0 \mbox{ with probability }= 1 - \frac{1}{n} \\ 1 \mbox{ with probability } = \frac{1}{2n} \\ -1 ...
4 votes
4 answers
82 views

Stopping time max $\{n: \ X_1 + \cdots + X_n \le t \}$

Let $\{X_n\}_{n \in \mathbb{N}}$ be iid and non-negative. Let $$N(t): = \max \{n: \ X_1 + \cdots + X_n \le t \}.$$ Is $N(t)$ a stopping moment with respect to the natural filtration $\{\mathcal{F}_n\}...
0 votes
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p-Variation of a Semimartingale

I have two questions regarding the $p$-Variation of a Semimartingale: Let $X_t$ be a semimartingale on $[0,1]$ and $\Pi_n = \{t^n _k = \frac{k}{n}: 0 \leq k \leq n\} $ a partition of $[0,1]$. For $p &...
6 votes
2 answers
172 views

What is the probability that a 2D symmetric random walk will reach a point $(x,y)$ before returning to the origin?

Let $p_n(\vec{x})$ be the probability that a symmetric random walk (SRW) on $\mathbb{Z}^n$ starting at the origin reaches the point $\vec{x}\in\mathbb{Z}^n-\{0\}$ before returning to the origin. What ...
2 votes
1 answer
58 views

Show by contradiction that a martingale converge to zero almost surely.

Theorem 5.2.9. If $X_n \geq 0$ is a supermartingale then as $n \rightarrow \infty, X_n \rightarrow X$ a.s. and $E X \leq E X_0$ 5.2.9. Let $Y_1, Y_2, \ldots$ be nonnegative i.i.d. random variables ...
0 votes
1 answer
41 views

The expected value of the sum of random variables of random length.

Suppose, we first roll an $N$-sided dice and let $X$ be the respective random variable. Next, we roll $X$ times an $X$-sided dice. Let $Y_1,\dots,Y_X$ be the respective random variables of the ...
4 votes
2 answers
2k views

Variance of the time of gambler's ruin

The problem is as follows: Let $\xi_1,\xi_2,\ldots$ be independent with $P(\xi_i = 1) = p$ and $P(\xi_i = −1) = q = 1−p$ where $p < \frac{1}{2}$. Let $S_n = S_0 +\xi_1 +\ldots+\xi_n$. Let $V_0 =...
2 votes
1 answer
1k views

expected winning time in fair gambler's ruin problem using martingale

In fair gambler's ruin problem, we already knew that the expected time of winning is $E(\tau_n|\tau_n<\tau_0)=\frac{n^2-k^2}{3}$, where $k$ is how much money we have in the beginning and $\tau_i$ ...
2 votes
1 answer
44 views

L2-Bounded Martingales

A martingale $\{u_n\}$ is $\mathcal{L}^2-bounded$. Show that: $$ \lim \int (u-u_n)^2 d\mu = 0 $$ $$ \int (u_{j+k} -u_j)^2 d\mu \! \begin{aligned}[t] & = \sum_{l=j+1}^{j+k} \int (u_l-u_{l-1})^2 d\...
2 votes
0 answers
72 views

Stopped cadlag submartingale is integrable

I'm trying to understand, why a stopped submartingale is again a submartingale. In the lecture notes to my lecture this is just stated as a corollary of Doob's Optional Sampling Theorem but I don't ...
12 votes
0 answers
248 views

Hardy's inequality proof using Doob's inquality

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\...
3 votes
1 answer
66 views

Almost sure convergence of a martingale

Let $X_1,X_2,\ldots$ be i.i.d random variables with standard normal distribution $\mathcal{N}(0,1)$. We define $$N_n = e^{n/2}\sin{(X_1+X_2+\ldots X_n)}$$ and after some calculations, it's easy to ...
3 votes
1 answer
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what if the square of a martingale is still a martingale?

Let $(\Omega,\mathcal{F},(\mathcal{F}_t:t\ge{0}),P)$ be a stochastic basis and $M=(M_t:t\ge{0})$ a locally square integrable martingale, which means a stochastic process such that: $M_t\in{L^2(\Omega,...
0 votes
1 answer
137 views

Martingales bounded in $L^1$

From the definition, for $X_n$ to be a martingale we require $E|X_n| < \infty$, for all $n\in\Bbb N$. Which implies that $X_n$ is bounded in $L^1$. However for theorems such as martingale ...
2 votes
0 answers
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Maximal inequality for time-continuous martingales: counter-example

The following can be regarded as the generalization of the maximal inequality for time-continuous martingales: Let$ (X_t)_{t \ge 0}$ be a supermartingale with right-continuous sample paths. Then, for ...
4 votes
1 answer
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Calculating $\mathbb E\left[\sum_{k=1}^TX_k\right]$ where $(X_k)$ where $T$ is a stopping time

Let $(X_k)_{k\in\mathbb N^*}$ be integrable random variables and $(\mathcal F_k)_{k\in\mathbb N^*}$ a filtration such that $(X_k)$ is $(\mathcal F_k)$-adapted. Let $T$ be an integrable $(\mathcal F_k)$...
1 vote
0 answers
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Doob's inequality and supermartingale

Let $(X_{n,k})_{\substack{k\leqslant K \\ n \geqslant 0}}$ be a nonnegative sequence of random variables. Let $\mathcal{F}=(\mathcal{F}_n)$ be a filtration. We assume that for each $k$, the sequence $...
4 votes
1 answer
97 views

Azuma's inequality for a simple case of Polya's urn

Suppose that an urn contains one red ball and one blue ball. A ball is drawn from the urn uniformly at random. After that, the ball is put back into the urn and another ball of the same colour is ...
5 votes
0 answers
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Comparing the inequalities of Azuma and Chernoff

Let $n$ be a positive integer and let $p = p(n) \in (0, 1)$. Let $X$ be the sum of the i.i.d. random variables $Y_1,\ldots, Y_n$, which are $1$ with probability $p$ and $0$ with probability $1-p$. ...
0 votes
0 answers
21 views

Computing covariation of Brownian motion and bounded variation process

Suppose $(B_t)_{t\geq0}$ is a Brownian motion and $(A_t)_{t\geq0}$ is a continuous process of bounded variation. I wish to show that $\langle A,B\rangle =0$. For this, I know that $(B_t-t)_{t\geq0}$ ...
1 vote
1 answer
51 views

Conditional expectation at a single point

Given two random variables, the conditional expectation $E(Y|X)$ is a $\sigma(X)$ measurable random variable whose integration over any set in $\sigma(X)$ agrees with the integration of $Y$ over the ...
0 votes
1 answer
40 views

Stopping time of iid random variable

Let $[\xi_i]_{i\geq 0}$ be i.i.d. random variable with $\mathbb{P}(\xi_1=1)=\frac{p}{2} , \mathbb{P}(\xi_1=-1)=\frac{1-p}{2}$ and $\mathbb{P}(\xi_1=0)=\frac{1}{2}$ for some $p\in(1/2,1)$. Define $X_0 ...
2 votes
0 answers
29 views

Is $X_t=42+t^2W_t^3+tW_t^2+\int_0^t3W_udu+\int_0^tW_u^3dW_u$ a martingale?

I'm really stuck with this problem. Is $X_t$ or $X_t-\mathbb{E}(X_t)$ a martingale, where $$ X_t=42+t^2W_t^3+tW_t^2+\int_0^t3W_udu+\int_0^tW_u^3dW_u ? $$ There is a hint, to find first $dX_t$, and ...
3 votes
1 answer
44 views

Is it true that $\sigma(X_1,X_1+X_2)=\sigma(X_1,X_2)$?

Given a sequence $(X_k)_{k\in\mathbb{N}}$ of real centered r.v.'s, and a sequence $(Y_n)_{n\in\mathbb{N}}$ defined by $$Y_n = X_1 + ..+X_n$$ it is stated, in a chapter on martingales of the book I'm ...
2 votes
1 answer
33 views

Conditional expectation with brownian and exponentials

Let $B$ be a standard brownian motion under the usual conditions. Evaluate $$ \mathbb E\left[(B_s+B_t^2)e^{B_t}|\mathcal F_s\right]$$ I started by separating and adding $t/2$ to the exponential $$ \...
4 votes
1 answer
117 views

Representation of a martingale as a stochastic integral

Let $\xi_{1},\xi_{2},\dots$ be an i.i.d. sequence that takes values $\pm 1$ with equal probabilities. Define the simple symmetric random walk with $X_{0}=0$, and $X_{n}=\sum_{i=1}^{n}\xi_{i}$. Define ...
3 votes
1 answer
43 views

For martigale $\{ X_n \}$ s.t. $E(X_n - X_{n - 1})^2 \le k$, do we have $\forall \epsilon. \lim_{n \to \infty} P(|X_n / n| > \epsilon) = 0$?

Consider a constant $k$ and a martingale $\{ X_n, n \ge 1 \}$ s.t. $EX_1^2 \le k$ and $\forall n \ge 2$, $E(X_n - X_{n - 1})^2 \le k$. Let $\lambda$ denote $E|X_1|$. By Chebyshev's inequality, We can ...

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