Questions tagged [martingales]
For question about discrete or continuous (super/sub)martingales. Often used with [probability-theory] tag.
2
votes
1answer
24 views
Optional sampling theorem with a.s. finite stopping time
I'm trying to prove the following generalized version of Doob's optional sampling theorem:
Let $X$ be a square integrable martingale with respect to a filtration $\mathbb F = \left\{ \mathcal F_n \...
0
votes
1answer
32 views
min & sum of discrete stopping times using alternate definition
Let $A, B$ be stopping times of a process $X_0, ... X_n$ where $A$ is a stopping time iff $I_{A = k}$ is a function of $X_0, ... X_k$ (where $I$ denotes an indicator variable).
We want to show that:
...
1
vote
1answer
21 views
probability with martingales 12.2 sum of zero-mean independent variables in L^2
I am struggling with the following theorem from David Williams, Probability with Martingales:
THEOREM
Suppose that $(X_{k}:k\in\mathbb{N})$ is a sequence of independent random variables such that, ...
0
votes
0answers
18 views
Super martingale and stopping time
I am a little confused about stopping times and martingales. Suppose I have a super-martingale $f(X_t)$ where $X_t$ is a martingale. Also, $f()$ is increasing in $X_t$.
Consider a person looking to ...
1
vote
1answer
35 views
Probability problem (Durrett 4.7.4.) Exchangeable Sequence of Random Variables.
I am having trouble in solving the following problem;
If $X_1, X_2, . . . \in \mathbb{R}$ are exchangeable with $EX_i^2
< \infty$ then $E(X_1X_2) ≥ 0.$
What I know is that the definition of ...
0
votes
0answers
19 views
Conditions for a process to be a martingale
Stumbled upon a problem which raised some questions in my mind, here's the deal :
$
dS = (a + b.S)dt + (c + d.S) dW \\
X = \alpha(t).S + \beta(t)
$
where $a,b,c,d$ are constants.
No conditions were ...
0
votes
0answers
26 views
Martingale to estimate gain or loss of a gambler.
Let $Z_1, Z_2, Z_3, ...$ be a sequence of i.i.d random variables valued in $\{-1, 1\}$ and taking each value with equal probability.
Define $X_n=\sum_{i=1}^n Z_i$ and $Y_n=X_n^2-n$. It can be easily ...
3
votes
0answers
23 views
Central limit theorem for square integrable martingales
Let $(M_t)_{t \in \mathbb{R}^*}$ be a square-integrable martingale. I am looking for a reference for the following convergence result :
$$\frac{M_t}{\sqrt{\langle M_t \rangle}} \overset{d}{\to} \xi$$
...
2
votes
1answer
65 views
Is $\{C^{X_n}\}_{n = 1}^{\infty}$ is a martingale?
Suppose $\{X_n\}_{n = 1}^{\infty}$ is a Galton-Watson branching process with ($P(X_1 = 1) = 1$ and $X_{n+1} = \Sigma_{i = 1}^{X_n} \theta_{i,n}$, where $\{\theta_{i,n}\}_{n = 1}^{\infty}$ are i.i.d. ...
0
votes
1answer
23 views
Join of random variables - relation to max
On Wikipedia in the article Snell envelope author uses 'join' operation on random variables. Following line is relevant:
$$ U_n = X_n \lor \mathbb{E}^\mathbb{Q}[U_{n+1} | \mathcal F_n] \quad \text{...
0
votes
1answer
12 views
Find values of $a$ and $\lambda$ for which $Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale
Find values of $a$ and $\lambda$ for which $Z(t)=Z_{0}e^{at+bW_{t}}-\lambda t$ is a martingale. In here $W_{t}$ is a Brownian motion and $a,b\in\mathbb{R}$ can be positive as well as negative, since $...
1
vote
1answer
22 views
Do non-negative submartingales with bounded second moments converge almost surely?
Suppose $\{X_n\}_{n = 1}^{\infty}$ is a discrete-time submartingale (a sequence of random variables, such that $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$), such, that $\exists C \in \mathbb{R} \forall n \...
0
votes
1answer
24 views
Do non-negative submartingales with finite second moment converge almost surely?
Suppose $\{X_n\}_{n = 1}^{\infty}$ is a discrete-time submartingale (a sequence of random variables, such that $P(E[X_{n+1}|X_1, … X_n] \geq X_n) = 1$), such, that $\forall n \in \mathbb{N} EX_n^2 <...
1
vote
1answer
29 views
Show that $X_t=e^{B(t)}-1-\frac{1}{2}\int_0^te^{B(s)}ds$ is a martingale
The problem tells me to show that $X_t$ is a martingale but I am getting that it is not. Here the assumed filtration is ${\scr F}_t=\{\sigma(X_s):s \le t\}$. Here is what I tried.
$$E(X_t|{\scr F}_s) ...
6
votes
2answers
50 views
Example of a Non-negative Martingale Satisfying Certain Conditions
Question
The question is to find an example of a non-negative martingale $(X_n)$ with $EX_n=1$ for all $n$ such that $X_n$ converges almost surely to a random variable $X$ where $EX\neq 1$ and $\text{...
0
votes
1answer
14 views
Martingale to power
Assume a Martingale $(X_i)$. My Question is what do I know about $(X_i^p)$ with $p\in \Bbb{N}_{\geq 2}$? I understand that I need $X_i\in\mathcal{L}^p$ to have any sort of (Sub-/Super-)Martingale.
0
votes
1answer
29 views
Conditional expectation of integral of Ornstein-Uhlenbeck process
Given that $X(t)$ is an Ornstein-Uhlenbeck process with $X(0) = x_0$, which is a Markov process, but not a Martingale, how could I go forward if I would like to calculate
$E[\int_0^T X(s)ds | \...
1
vote
0answers
28 views
Martingale convergence theory question
I'm confused on what this proof would look like.
I know that by Martingale Convergence Theorem:
$$X= \lim Z_n / \mu^n$$ exists a.s.
I also know that Kesten Stigum Theorem:
Assume that $E[L] &...
1
vote
1answer
18 views
Doob's Optional Stopping Theorem: $\xi_\tau$ vs $\xi_{\tau\land n}$
I have some troubles in understanding the Optional Stopping Theorem by Doobs.
I have a bit of confusion about the following (Brzezniak, Zastawniak - Basic Stochastic Processes p. 58-59):
Let $\...
0
votes
0answers
34 views
Finding out the sequence as Martingale.
Consider the sequence $\{X_n\}_{n\geq 1}$ of independent random variables with law $N(0, \sigma^2)$. Define the sequence $Y_n= \exp \bigg(a\sum_{i=1}^n X_i-n\sigma^2\bigg), n\geq 1,$ for $a$ a real ...
1
vote
1answer
27 views
Prove that $\{S_n^2-\sigma^2n\}$ is a martingale.
Let $S=\{S_n\}$ be a mean zero random walk with $EX_1^2=\sigma^2$. Prove that $\{S_n^2-\sigma^2n\}$ is a random walk.
So we need that
$\{S_n^2-\sigma^2n\}$ is integrable, (easy)
$\{S_n^2-\sigma^2n\...
0
votes
0answers
20 views
Reference Request: Martingales - Convex Analysis (-Harmonic Analysis) relationship
During my undergraduate studies I've encountered martingales, convexity (and also partial differential equations). Fast forward a few years, as a PhD student in applied math, I often find myself ...
2
votes
1answer
25 views
Stopped random walk is not uniformly integrable
I know that in general Doob's Optional Stopping Theorem doesn't hold for unbounded stopping times, but that it does when the system up to the stopping time is uniformly integrable.
One counter ...
0
votes
0answers
14 views
Interpreting Doob's Up-Crossing Inequality
Doob's upcrosssing inequality states that, for martingale $X_n$, if $U_n(a,b)$ counts the number of upcrossings through $(a,b)$ up to time $n$, then $E[U_n(a,b)]\leq \frac{1}{b-a}E[(a-X_n)_+]$.
We ...
1
vote
1answer
33 views
Continuous local martingale $M$ with $\langle M\rangle_t=t$ is a martingale
I am currently reading a proof of the Levy characterization of Brownian motion, and it seems it uses the following result. Let $(M_t,\mathcal F_t)_{t\geq 0}$ be a continuous local martingale with ...
0
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0answers
18 views
exponential martingale, expectation of stopping time [duplicate]
Let $\{B(t): t \ge 0 \}$ be a linear Brownian motion.
Show that, for $\sigma >0$, the process $\{exp(\sigma B(t)-\sigma^2 t/2): t \ge 0 \}$ is a martingale.
2.Show, by taking derivatives $$\...
1
vote
1answer
12 views
Doob's decomposition theorem relaxation for nonuniqueness
Doob's decomposition theorem states that, given any sequence of adapted, integrable functions $X_n$ (adapted to $\mathcal{F}_n$), we get an almost surely unique decomposition $X_n=A_n+Y_n$, where $Y_n$...
1
vote
0answers
26 views
Checking for martingale and Itô's formula
Assuming that $\{ W ( t ) | t \geq 0 \}$ is a Brownian motion, I'm trying to check whether the process
$$X ( t ) = W ( t ) + 4 t$$
is martingale with respect to filter $\mathcal{F}_t$. For this we ...
0
votes
0answers
10 views
Square integrability of martingale?
So we had the following example in class:
Assume $(\Omega, ℱ,(ℱ_{t}),ℙ)$ a filtered probability space and $X: \Omega \to \mathbb{R}$ a random variable such that
$$
\mathbb{E}[|X|^{2}] < \infty \...
0
votes
0answers
18 views
Density of hitting time of absolute value of a Brownian motion
I am interested in the probability density of
$$
\tau =\inf\{t\geq 0: \vert W(t)\vert = 1\}
$$
where $W(t)$ is a standard Wiener process. I have two approaches in mind:
First approach: I could ...
1
vote
0answers
31 views
$B_t^3 - 3t B_t$ is a $L^2$ martingale ($B_t$ being a standard Brownian motion)
By Itô's formula I get that
\begin{align}
d(B_t^3 - 3t B_t) &= (3 B_t^2 dB_t + 3 B_t dt) - 3(B_t dt + 3 t d B_t) \\
&= (3 B_t^2 + 3t) d B_t
\end{align}
which seems related to martingale ...
-1
votes
0answers
17 views
Geometric Brownian Motion and Martingales?
Let Gt denote the canonical filtration of a Brownian motion Wt. Show that for any λ ∈ R, the S.P. Mt(λ) = exp(λWt − λ^{2}t/2), is a continuous time martingale with respect to Gt. Explain why its kth ...
2
votes
1answer
21 views
Convergence of a cadlag martingale to a square-integrable limit
Let $(M_t)_{t\geq 0}$ be a cadlag uniformly integrable (UI) martingale with $M_t\rightarrow M_\infty$ a.s. as $t\rightarrow\infty$. If $\Bbb E[M_\infty^2]<\infty$, is it true that $\Bbb E[M_t^2]<...
2
votes
0answers
44 views
An example of a submartingale $X=\{X_n\}$ such that $\{X_n^2\}$ is a supermartingale.
A submartingale is a real-valued stochastic process $X=\{X_n\}$ adapted to a filtration $\{\mathcal{F}_n\}$ such that
$$E[X_{n+1}\mid \mathcal{F}_n] \geq X_n.$$ For a supermartingale just reverse the ...
1
vote
0answers
29 views
Distribution of inter-arrival time of non-homogenous Poisson Process
What is the distribution of inter-arrival time of a non-homogenous Poisson Process?
In other words, if $T_n$ is a non-homogeneous Poisson process with intensity $\lambda(s)$, and $S_{n+1} = T_{n+1} - ...
0
votes
0answers
27 views
Conditional law of Poisson process
1)What does the notation for $G_n$ and $H_n$ mean?
2)How would $G_n$ and $H_n$ look like for a Poisson process?
3)How to show that $v$ would be the compensator of a Poisson process?
Thanks!
0
votes
1answer
55 views
Prove the infinum of supermartingales is still a supermartingale
In Tomas Bjork's Arbitrage Theory Continous Time (2009), Proposition 21.20 claimed that, in discrete time, the (pointwise) infinum of an arbitrary family of supermartingales is still a supermartingale....
2
votes
1answer
43 views
martingale $X(t)=e^{-\lambda t}f(B(t))- \int_{0}^{t} e^{-\lambda s} (\tfrac{1}{2}\Delta f(B(s))- \lambda f(B(s)))ds$
Let $f: \mathbb{R}^d \longrightarrow \mathbb{R}^d$ be twice continuously differentiable and $\{B(t): t \ge 0 \}$ be a d-dimensional Brownian motion. Further suppose that, for all $t>0$ and $x \in \...
3
votes
1answer
59 views
$V\in L^2$, $U$ is standard normal, $U,V$ independent$\implies$ $\lim_{\sigma \to \infty} E[V| V+\sigma U]=E[V]$ a.s.
How to show that
\begin{align}
\lim_{\sigma \to \infty} E[V| V+\sigma U]=E[V], a.s.
\end{align}
where $E[V^2]<\infty$ and $U$ is standard normal and $U$ and $V$ are independent.
I was thinking ...
1
vote
0answers
12 views
Doob martingale concentration with approximate expectations
Consider a set of random variables:
$$ \vec{X} = X_1,\ldots,X_n$$
and a function $f(\vec{X})$ used to construct the Doob martingale with elements
$$ B_i = E_{X_{i+1},\ldots,X_{n}}[f(\vec{X}) | X_1,\...
3
votes
1answer
49 views
Markov chain and martingale: Taking one step to the right or going back to zero
Consider the Markov chain on nonnegative integers that does the
following: from any site $x \geq 0$ it jumps to $x + 1$ with
probability $p$ and to 0 with probability $1 − p$. Is this chain
...
2
votes
1answer
38 views
If a process is natural increasing, does that imply that the stopped process is natural increasing?
Let $A = \{A_t\}_{t\geq 0}$ be a right-continuous stochastic process on the filtration $\mathcal F = \{\mathcal{F}_t\}_{t\geq 0}$ and let $T$ be a stopping time. Suppose also that $A_t$ is natural ...
1
vote
2answers
85 views
Restriction martingale to a measurable set
If we have that $M_n$ is a martingale adapted to $\mathscr{F}_n\subset\mathscr{F}$ on a probability space $(\Omega,\mathscr{F},\mu)$ then if $A$ is an $\mathscr{F}_0$ measurable set we have $$M_n 1_A=...
0
votes
0answers
22 views
Compute the exit time of $M_t = W_t^2 - t$ from $[-R, R]$
I have a question in my textbook that states:
Apply the optional stopping theorem for the martingale $ M_t = W_t^2 -t $ to show that $E[\tau] = R^2$.
This exercise came right after the section on ...
0
votes
1answer
24 views
Expected value- $E(1/3)^t$
$X_1, X_2, ... $ are independent random variables.
And $P(X_n=1)=P(X_n=-1)=1/2$.
$t=inf (n: X_1+X_2+...+X_n=1)$
Find $E(1/3)^{t}$.
I tried to do it from the definition of expected value:
$E(1/...
2
votes
1answer
23 views
Submartingale characterization
is there a characterization of submartingales in terms of stopping times similar to the martingale case in A martingale characterization.
Especially is the following true:
If $X$ is an adapted ...
1
vote
0answers
20 views
Limit of an indicator martingale
Consider the Lebesgue measure on $[0,1]$.
We define $I_k^n=[k2^{-n},(k+1)2^{-n})$, for $0\leq k\leq 2^n-1$, and $F_n=\sigma(\{I^n_k:k\})$.
Finally $M_n=1_{I_0^n}2^n$.
How do we show that $M_n$ is a ...
0
votes
0answers
33 views
Prove that $P(\exists n\in N:S_n=n-2008)=1$
Let $X_n$, $n=1,2,...$ be independent random variables. $P(X_n=1)=\frac{1}{2}=P(X_n=-1)$
$S_n=\sum_1^nX_k$, $T=inf[n:S_n=n-2008]$
a) Prove that $P(\exists n\in N:S_n=n-2008)=1$
b) Prove $ET\...
0
votes
1answer
18 views
Varying definitions of a martingale
I frequently see that, over some filtration $\mathscr{F}_{n}, \{X_{n}\}$ is defined as a martingale if $E[X_{n+1}|\mathscr{F}_{n}]=X_{n}$. Sometimes, however, I see this extended to $E[X_{n+s}|\...
0
votes
0answers
33 views
In 'Probability: A Graduate Course' by Allan Gut, it is written '[…]in a sense life itself is a martingale.' Can someone explain how?
I hope this question doesn't infringe any rules.
Original text (p. 480, 2nd edition):
If we interpret a martingale as a game, part (ii) states that, on average, nothing
happens, and part (i) ...
