Questions tagged [martingales]

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

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Inequality in probability theory.

Let X be a nonnegative random variable and let $(\mathcal{H}_i)$ be a sequence of increasing $\sigma$ algebras. Let $(A_i)_{0 \leqslant i \leqslant N}$ be a sequence of pairwise disjoint events. Do we ...
1 vote
1 answer
37 views

Show $D_{n} : = X_n - X_{n-1}, n \in \mathbb{N}$ are pairwise uncorrelated, where $X_n$ is a square integrable martingale

Assume that $(X_n)_{n \in \mathbb{N}}$ is a square integrable martingale w.r.t. the filtration $\mathcal({F}_n)$. Let $$D_{n} : = X_n - X_{n-1}, n \in \mathbb{N}$$ Show that $D_n$ are pairwise ...
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1 vote
2 answers
57 views

How to show that $X_t = (1+t)^{-1/2} \exp \biggl( \frac{B_t^2}{2(1+t)} \biggr)$ for a Brownian motion $B$

Show that $X_t = (1+t)^{-1/2} \exp \biggl( \frac{B_t^2}{2(1+t)} \biggr)$, where $B$ is a Brownian motion, is a martingal. I understand that we need to show that $\mathbb{E}(X_t \vert \mathcal{F}_s) = ...
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1 vote
0 answers
18 views

Find C so that $C^n\lambda^{S_n}$ is a martingale

Let $ X_1,X_2,... \sim i.i.d.$ $P(X_n=1)=p=1-P(X_n=-1)=1-q, n=1,2,...$, $S_n=\Sigma_{k=1}^{n}X_K$ and $F_n$ is filtration generated by $X_n$. For fixed $\lambda>0$ we aim to find cosntant C that $...
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0 votes
1 answer
28 views

Is $S_n-nEX_1$ a martingale

Let's take $X_1,X_2,...\sim i.i.d.$ with finite expected value. We have $S_n = \Sigma_{k=0}^{n}$ and filtration $\mathcal{F}_n$ that is generated by process $X_n$,( can we write that $ \forall_n ~ \...
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0 answers
23 views

Discrete harmonic function martingale

Let $A \subset \mathbb{Z}^2$ be a finite set of points in the square lattice, and let $B$ (the boundary of $A$) be the set of points in $\mathbb{Z}^2 \setminus A$ with at least one (horizontal or ...
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2 votes
1 answer
50 views

Hypothesis Testing Two Measures Coin Tosses Martingale

Consider a sequence of iid tosses of a coin with $X_i$ denoting the outcome of the $i^{th}$ toss. These random variables are defined on some $(\Omega, \mathcal{F})$ on which we have two probability ...
0 votes
1 answer
35 views

Stopped martingale bound in van Handel's notes

I am reading Ramon van Handel's lecture notes on stochastic calculus and came across a confusing claim. Suppose we define the martingale $M_n = M_0 + \sum_{k = 1}^n \xi_k$, where $\{\xi_k\}_{k = 0}^\...
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4 votes
1 answer
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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 $$Y_n = X1 + ..+X_n$$ it is stated, in a chapter on martingales of the book I'm ...
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+50

A version of Kazamaki's condition

We suppose that $X$ is a continuous local martingale such that $X_0=0$ and $$\forall u \in \mathbb{R}_+,\sup_{r\in[0,u]}E[e^{X_r/2}]<\infty.$$ Prove or disprove $e^{X-[X]/2}$ is a martingale. This ...
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1 vote
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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
0 answers
24 views

Ito representation theorem

By the Ito representation theorem for a $\mathcal{F}_T$-measurable random variable $X$ there exists a $L^2$ process $\eta$ such that $$ X = \mathbb{E}[X]+\int_0^T\eta_s\,dB_s $$ where $B$ is a $\...
2 votes
1 answer
50 views

finite variation process

Problem: Let M be a continuous and bounded martingale, and let A be a bounded increasing process. Show $E A_{\infty}M_{\infty}=E\int_0^{\infty} M_t dA_t$. Lemma: if $f:[0,T] \rightarrow \mathbb{R}$ is ...
3 votes
1 answer
49 views

A bound for martingale difference

Let $n,d\in \mathbb N$ such that $d\leq n$. Suppose we have a sequence of $n$ i.i.d tosses of a coin with $P[H]=p$, denoted by $X_1, ..., X_n$, where $H$ means head. For $1\leq i \leq n-d+1$, we call ...
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2 votes
1 answer
66 views

How can I show that $(f(X_n))_n$ is a martingale?

Let me assume $X=(X_n)_n$ is a Markov chain with values in $E$. Assume further that $Q$ is it's transition matrix and $\nu$ the initial distribution. Let $f:E\rightarrow \Bbb{R}_+$ be a harmonic ...
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2 votes
1 answer
66 views

Discounted GBM being a martingale

I am trying to show that the discounted Geometric Brownian Motion SDE is a martingale, but I must be doing something wrong. GBM SDE is given by $$X_t=X_0+\int_{h=0}^{h=t}X_hr dh+\int_{h=0}^{h=t}X_h\...
0 votes
1 answer
28 views

$ A \in \mathcal{F}_{m}$and$\mathbb{E}[x_m \mathbb{1}_{A}]= \mathbb{E}[X_n \mathbb{1}_{A}]$ for $m<n$ then $(X_n)_{n \in \mathbb{N}}$ is a martingale

Show that $ A \in \mathcal{F}_{m} $ and $\mathbb{E}[X_m \mathbb{1}_{A}]= \mathbb{E}[X_n \mathbb{1}_{A}]$ for $m<n$ then $(X_n)_{n \in \mathbb{N}}$ is a martingale. Considering a sequence $(X_n)_{n \...
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0 votes
1 answer
29 views

Functional equation related to the supercritical Galton-Watson process

Consider the Galton-Watson process $(Z_n)_n$ and the martingale $M_n = \mu^{-n}Z_n$. We assume that $\mu > 1$, i.e. our Galton-Watson process is supercritical and $\mathbb{E}(M_\infty) < \infty$....
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1 vote
1 answer
38 views

Max of Product of Random Variables vs replacing one rv with its mean

Apologies for the uninformative title. A paper I am reading states the following fact. If $X$ and $Y$ are independent random variables, assume all well-behaved, with means $\bar{X}, \bar{Y}$ then $$ E ...
0 votes
1 answer
39 views

Does $(H \cdot X)$ being martingale imply $X$ is martingale?

I know that if $H_t$ is a bounded and predictable/previsible (sub)martingale, and $X_t$ is a (sub)martingale, then $(H \cdot X)$ is also a (sub)martingale, and this is quite easy to show by using the ...
0 votes
1 answer
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Supermartingale question

I'm wondering if the following statement is true: Let $(M_n)_{n \in \mathbb{N}}$ and $(N_n)_{n \in \mathbb{N}}$ are non-negative random processes, and let $(\mathcal{F}_n)$ be a filtration. Let $\...
0 votes
1 answer
40 views

Is $E[X_sX_t^2]=0 \,\,s<t$ for a MDS?

Let $(X_t,\mathcal F_t)$ be a stationary martingale difference sequence (MDS). Can we say that $$E[X_sX_t^2]=0 \quad s<t \quad ?$$ For $s>t$ we can use the the law of iterated expectations and ...
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1 vote
1 answer
58 views

How to Apply Martingale Convergence Theorem to Standard Gaussian Random Variables?

I am fairly new to stochastic process so my question might be a little fundamental.... Let $f_n$ denote an orthonormal basis of $H_0^1(D)$, where $D$ is an open, simply connected, proper subset of $\...
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1 vote
0 answers
25 views

Hitting time estimate for Vector valued process

We denote $v>w$ for two vectors $v, w \in \mathbb{R}^{m}$, if $$v^{i}>w^{i}, \quad \forall 1\leq i \leq m,$$ and write componentwise expectation as $$\mathbb{E}[v | \mathcal{F}]=(\mathbb{E}[v^{1}...
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2 votes
1 answer
42 views

Determine $(a_n)_n$ and $(b_n)_n$ such that $(S_n^2-a_nS_n-b_n)_{n \ge 0}$ is a martingale

Consider the assymtric random walk $(S_n)_n$ on $\mathbb{Z}$. Determine $(a_n)_n$ and $(b_n)_n$ such that $(S_n^2-a_nS_n-b_n)_{n \ge 0}$ is a martingale. Remark: In lecture I have already seen that $(...
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2 votes
1 answer
48 views

Expected value of an exponential martingale

Consider the symmetric random walk $(S_n)_{n \ge 0}$ on $\mathbb{Z}$. For real $u \ne 0$ consider the exponential martingale $Z_n = e^{uS_n}cosh(u)^{-n}$ of the symmetric random walk $(S_n)_{n \ge 0}$ ...
  • 2,988
2 votes
0 answers
23 views

Ruin probability for exponential martingale of a random walk

Let $(S_n)_n$ be an assymetric random walk on $\mathbb{Z}$. A player starts with $a$ dollars capital and plays until he either wins $b$ dollars or lost his entire capital $a$. We know that $$T_{a,b} :=...
  • 2,988
4 votes
0 answers
45 views

Follow-up to "Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for..."

I don't understand the accepted answer to the following question: Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every ...
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2 votes
1 answer
41 views

Proving $M_n := e^{S_n - n/2}$ does not converge in $L^2$

Let $S_n = \sum_{i=1}^nX_i$, where $X_i \sim N(0,1)$ i.i.d. Let $M_n := e^{S_n - n/2}$ and $\mathcal{F}_n = \sigma(X_1, \dotsc, X_n)$, then $M_n$ is a martingale w.r.t. $\mathcal{F}_n$. I need to show ...
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-1 votes
0 answers
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Limit of the difference between two consecutive elements of a supermartingale process

Deterministic case: Let $(x_k)$ be lower bounded decreasing sequence. Then $(x_k)$ has a limit and $x_{k+1}-x_{k} \to 0$ as $k \to \infty$. This fact is useful if we know a constant $c \geq 0$ has to ...
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0 votes
1 answer
42 views

Gambler's problem

Let N$\geq 2$ be an integer. Consider a gambler who starts with i$<N$ Euro. For each successive gamble the gambler either wins 1 Euro with probability p or loses 1 Euro with probability q=1-p. ...
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3 votes
2 answers
78 views

Is $Z_n = X_n Y_n$ a martingale?

Suppose $X_n$ and $Y_n$ are independent martingales with respect to filtration $\mathcal{F}_n$. Is $Z_n$ a martingale with respect to the same filtration, where $Z_n = X_n Y_n$ and we know that $\...
1 vote
1 answer
31 views

Proving martingale inequality $\mathbb{E}[\sup_t M_t^2] \leq 4\mathbb{E}[Z^2]$

Let $Z$ be a square int'ble rv. on a filtered probability space with filtration $(\mathfrak{F}_t)_{t \in \mathbb{R}_+}$, we define the martingale $M_t = \mathbb{E} [Z|\mathfrak{F}_t] $. I now want to ...
1 vote
1 answer
23 views

Change of Numeraire for GBM

Change of Numeraire for GBM Let $N_1$ and $N_2$ be numeraires specified by, $$ \begin{align} \frac{dN_1(t)}{N_1(t)}&=\mu_1\:dt+\sigma_1\:dB_1(t)\\ \frac{dN_2(t)}{N_2(t)}&=\mu_2\:dt+\sigma_2\:...
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2 votes
1 answer
56 views

Understanding the idea behind martingale betting setup to solve expected value problems

Learning martingale theory I've come across a method that uses gambling strategies in martingale betting to calculate expectations. It's clever and I understand the solution but I'm not sure how they ...
2 votes
1 answer
74 views

How can I prove that $X_n$ is a martingale iff $\Bbb{E}(X_T)=0$ for all bounded stopping times $T$

Let me consider $(\Omega, F,(F_n)_n, \Bbb{P})$. Let $(X_n)_n$ be an adapted process at $0$. I need to show that $(X_n)_n$ is a martingale iff $\Bbb{E}(X_T)=0$ for all bounded stopping times $T$. My ...
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0 votes
1 answer
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Pointwise second moment of continuous local martingale

Let $ (M_t)_{t \ge 0}$ be a continuous local martingale (as defined in LeGall). Let $\mathbb{E}$ denote expectation. Is $\mathbb{E}[M_t^2] < \infty$ for all $t \ge 0$? If yes, then how does one ...
1 vote
1 answer
32 views

Is the sequence of conditional expectations of a convergent sequence of random variables still convergent?

Let $(\mathcal{F}_n)_{n \geq 0}$ be a filtration of a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and let $\mathcal{F}_\infty$ be the $\sigma$-algebra generated by the $\mathcal{F}_n$. If a ...
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1 vote
2 answers
45 views

Brownian motion does not vary more than $\sqrt{n}$ on each interval $[n,n+1]$

Let $(B_t)_{t\geq 0}$ be a Brownian motion in $\mathbb{R}$ defined on a measure space $(\Omega,\mathscr{F},\mathbb{P})$. Can you give me some hints on how to prove the following: for a.e. $\omega\in\...
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0 votes
0 answers
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et $P(i)$ be the probability that the chain eventually enters state $N$ given that it starts in state $i$. Show $\{P(X_n), n\geq 0\}$ is a martingale.

Consider the Markov chain $\{X_n, n \geq 0\}$ with $P_{NN} = 0$. Let $P(i)$ denote the probability that this chain eventually enters state $N$ given that it starts in state $i$. Show that $\{P(X_n), n\...
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0 votes
0 answers
22 views

Expectation of Brownian Motion adapted for Filtration

I just started learning about Brownian Motions and martin gales and have the following issue. If $X_t$ is a brownian motion, I cannot understand how the results below are different when adapting for ...
1 vote
0 answers
13 views

Prove that a polynomial $(P(S_n,n))$ is a martingale [duplicate]

Let $(S_n)_{n\geq 0}$ be the simple symmetric random walk in $\mathbb{Z}$: $$\begin{cases} S_0=0 \\ S_n = X_1 +\dots + X_n ; n>0 \end{cases} $$ where $(X_i)_{i\geq 1}$ is an i.i.d. sequence with $\...
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0 votes
0 answers
32 views

Let $\{W_t\}_{t \ge 0}$ be a Brownian motion. Determine whether $Y_t=e^{W_t^2 - t}$ is a martingale.

Let $\{W_t\}_{t \ge 0}$ be a Brownian motion. Determine whether $Y_t=e^{W_t^2 - t}$ is a martingale. Attempt: For $0\le s<t$, \begin{align*} E[Y_t|F_s] &= E[e^{W_t^2-t}|F_s] \\ &= E[e^{(W_t-...
1 vote
1 answer
26 views

Determine whether the stochastic process $Y_t=10+t^2+e^{W_t}$ is a martingale

Let $\{W_t\}_{t \ge 0}$ be a standard Brownian motion on a probability space $(\Omega,\cal{F},\Bbb P)$ and let $\{\cal F_t\}_{t \ge 0}$ be the filtration of the Brownian motion. Determine whether the ...
0 votes
1 answer
20 views

Filtration with same adaptable and predictable processes

Let us have a toss of two coins and our $\Omega=(hh,tt,ht,th)$. 1) give a filtration so that all stochastical processes are also adaptable and also predictable. 2)given the filtration $F_{0}= trivial,...
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0 votes
1 answer
57 views

How do I show that $Z_n$ is a martingale?

Let me assume $(X_n)$ is an i.i.d. sequence of random variables and define $S_n=X_1+...+X_n$. Assume further that $X_1\sim N(0;1)$. I want to show that $Z_n=\left(e^{\frac{\lambda S_n-\lambda^2 n}{2}}\...
  • 1,525
2 votes
0 answers
33 views

Uniformly Integrable Random Variables are Uniformly Bounded Almost Everywhere?

I'm a beginner in stochastic processes(and measure theory). I am trying to prove(or disprove) the fact that, if $\xi_n$ are uniformly integrable random variables, then they are uniformly bounded by ...
2 votes
0 answers
38 views

Is there a connection between Brownian motion and Hermite polynomials?

Given a standard one dimensional Brownian motion (starting in 0) $(B_t)_{t \geq 0}$ I am supposed to proof that $B_t$, $B_t^2 - t$ and $B_t^3 - 3tB_t$ are martingales, which I was able to do without a ...
0 votes
0 answers
33 views

Is sum of products of random variables is Martingale

Problem We are given two sequences of i.i.d random variables $A_1, A_2,\dots $ and $B_1, B_2, \dots$ with known expected values $\mu_A$, $\mu_B$ and variances $\sigma^2_A$, $\sigma^2_B$. Define new ...
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0 answers
21 views

Martingal Problem and path-wise unique strong solution

Assume we have a path-wise unique strong solution $X_t$ to an SDE $D$. And the following equivalence: A process $Z_t$ is a weak solution to $D$ if and only if it is a solution to local martingale ...

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