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Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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Showing that $M^T(N-N^T)$ is a continuous local martingale for all stopping times $T$ and continuous local martingales $M,N$

Given is that a cadlag adapted process $X=(X_t)_{t\geq 0}$ is a martingale if and only if $\mathbb{E}X_T=\mathbb{E}X_0$ and $X_T\in L^1$ for every bounded stopping time $T$. Now let $M,N$ be two ...
Daan's user avatar
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Can we always find a martingale part in a càdlàg supermartingale?

Abstract: By Doob-Meyer decomposition theorem, for any càdlàg supermartingale $Z$, there exists a unique predictable increasing process $A$ starts from $A_0=0$ such that $Z+A$ is a local martingale ...
Hirofumi Shiba's user avatar
3 votes
1 answer
60 views

When does a local supermartingale become a proper supermartingale?

Abstract: When a local supermartingale is bounded from below, is it a proper supermartingale? Question: In remark 4.2 (p.16) of the lecture notes by Martin Hairer, the conditions when $$ [0,\infty)\ni ...
Hirofumi Shiba's user avatar
2 votes
2 answers
45 views

Show that this stopped process converges ucp to the original process

Question Let $M$ be a continuous local martingale with null at zero. Let $\tau_n=\inf\{t:|M_t|>n\}$ be a stopping time. Does $M^{\tau_n}\to M$ u.c.p (uniformly on compacts in probability)? Here, a ...
Mingzhou Liu's user avatar
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33 views

Are processes derived from deterministic and square-integrable functions martingales?

Problem Let $f:\mathbb{R}_{\geq 0}\to\mathbb{R}$ be a deterministic and square-integrable function, i.e. $\int_0^\infty f(t)^2\mathrm{d}t < \infty$, and define the process $X=(X_t)_{t\geq 0}$ by $...
Wilfred Montoya's user avatar
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2 answers
55 views

An (a.s.) continuous process $(X_t)_{t\geq 0}$ is a Brownian motion if $(e^{i\lambda X_t + \frac{1}{2}\lambda^2 t})_{t\geq 0}$ is a local martingale

Problem Let $X=(X_t)_{t\geq0}$ be an (a.s.) continuous $\mathbb{R}$-valued process with $X_0=0$ such that $(e^{i\lambda X_t + \frac{1}{2}\lambda^2 t})_{t\geq 0}$ is a $\mathbb{C}$-valued local ...
Wilfred Montoya's user avatar
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2 answers
84 views

Is $(B_t^2 - t^2)_{t\geq 0}$ a local martingale?

Let $B$ be a standard $\mathbb{R}$-valued Brownian motion. It is very easy to show that $(B_t^2 - t^2)_{t\geq 0}$ is not a martingale by checking the martingale condition. First, note that $(B_t^2 - t)...
Wilfred Montoya's user avatar
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A continuous local martingale $M$ is constant on an interval if $\langle M\rangle$ is

Problem Let $M\in\mathbb{M}^{loc}_C$, i.e. $M$ is an (a.s.) continuous local martingale with $M_0=0$. Show that $M$ is constant on an interval $[a,b]$ with $0\leq a < b$ if $\langle M\rangle$ is ...
Wilfred Montoya's user avatar
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If we discretize a continuous-time local martingale, do we still get a local martingale?

On the filtered probability space $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t\geq 0}, P)$, let $Z=(Z_t)_{t\geq 0}$ be a continuous-time local martingale (assume $Z_0=0$). i.e. there exists a sequence ...
NXWang's user avatar
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Does the stopping time sequence given by the hitting time of $n$ reduce any continuous local martingale?

I stumbled upon this, but it doesn't feel right. Is the following true? Let $X$ be a cont. local martingale, $S_n= \inf \{ t | |X_t|\geq n \} $. Then $X^{S_n}$ is a martingale for each $n\in \mathbb{N}...
J.R.'s user avatar
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If $T_k$ localizes $X$, then so does $S_k \wedge T_k$?

There is a fact mentioned in the answer of this question that I can't seem to show. It's probably simple but I'm not seeing it. Essentially it says: If $X$ local martingale with localizing sequence $...
J.R.'s user avatar
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Prove that a bounded local martingale is a martingale

I would like to know if my proof of this fact is correct : A bounded local martingale is a martingale. The sequence of stopping time has the following property : it is increasing and it converges to $+...
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Prove that for a martingale, any increasing stopping time is sufficient to be a local martingale

I would like to prove the following Let $(X_t)_t$ be a continuous martingale. Prove that any sequence of increasing stopping times (with values in $\mathbb{R}_{+}$) is sufficient to get a local ...
G2MWF's user avatar
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Are local martingals always have locally integrable quadratic variation?

I am reading the book: "P. Protter, Stochastic Integration and Differential Equations. Second edition, Springer-Verlag, (2004)", where the following are mentioned: (1) Local martingales are ...
p3256c's user avatar
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1 answer
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Is every local martingale right continuous?

Is every local martingale right càdlàg (i.e. right continuous with left limits)? At the university, in the definition of martingale we assume martingales to be right càdlàg processes. We call an $X$ ...
Kapes Mate's user avatar
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Running maximum = local martingale?

Let $X$ be a Brownian Motion with drift $\mu>0$ and $X^*$ its running maximum. Is $X^*$ a local martingale? If I construct the sequence of stopping times $\tau_k = \inf \{s >\tau_{k-1}, X(s)>...
Santiago's user avatar
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Why is every bounded local martingale a true martingale? [duplicate]

A cadlag process is a local martingale $\iff$ there exists a sequence of stopping times $\tau_n\to\infty (n\to\infty)$ a.s. and the stopped process $M^{\tau_n}$ is a true martingale for all $n\in\...
Uhmm's user avatar
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2 answers
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define stochastic integral that does not start from 0

This problem occurred when I was trying to solve Exercise 3.2.30 of Karatzas & Shreve's book, the exercise stated: For $M\in\mathcal{M}^{c,loc},X\in\mathcal{P}^*$, and $Z$ an $\mathcal{F}_s$-...
tfatree's user avatar
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Integratable local martingale is martingale

Show that an integratable local martingale $(X_t)_{t=1,...,T}$ is martingale. Is this true in continuous time $[0,\infty)$? Let $(\tau_n)$ be a localising sequence for X (so $(\tau_n)$ stopping times ...
user1049882's user avatar
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245 views

Covariation of independent semimartingales is zero

Let $X$ and $Y$ be independent continuous semimartingales on a probability space. I know that we should have $[X, Y] = 0$. I am able to prove that if $M$ and $N$ are independent continuous local ...
George C's user avatar
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Show that a process is a local martingale (Brownian bridge)

Let $W=\{W_t\}_{t\geq 0}$ be a Brownian motion and $\{X_s\}_{t\leq s\leq1}$ be a Brownian bridge. Let we have a value function $V^*:[0,1)\times\mathbb{R}\cup\{(0,1)\}\rightarrow \mathbb{R}$ given by \...
what_456's user avatar
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Symmetric random walk inside a continuous local martingale

Consider a continuous local martingale $\{X_t\}_{t\geq 0}$ with $X_0=0$, and $\limsup_{t\to\infty}X_t = \infty$ and $\liminf_{t\to\infty}X_t = -\infty$ with probability one. Then, I want to show that ...
mathmd's user avatar
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Le Gall's Lemma 5.14: how to obtain $B \subset C$ a.s. from $B \subset \{X^{(n)}_a = X^{(n)}_b\}$ for all $n$?

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a completed filtration. Let $M$ be a real-valued continuous local martingale w.r.t. $\mathcal G$....
Analyst's user avatar
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Assume $M^{\tau_n}$ is uniformly integrable. Is $X^{\tau_n}$ uniformly integrable?

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space and $\mathcal G = (\mathcal G_t, t \ge 0)$ a filtration. Let $M$ be a real-valued continuous local martingale w.r.t. $\mathcal G$. Let $(\...
Akira's user avatar
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1 answer
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A proof of Lévy's theorem: how to obtain the independence $X_t-X_s \perp \mathcal{F}_s$?

I'm reading about Lévy's theorem at page $42$ of these notes, i.e., Let $X$ be a continuous local martingale such that $X_0=0$ a.s. and $\langle X\rangle_t=t$ a.s., $\forall t \in \mathbb{R}_{+}$. ...
Akira's user avatar
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1 answer
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Local martingale: how is $\sup_{s \in[0, t]}\left|M_s\right|$ measurable?

I'm reading about local martingale from this Wikipedia page, i.e., Let $M_t$ be a local martingale. In order to prove that it is a martingale it is sufficient to prove that $M_t^{\tau_k} \rightarrow ...
Akira's user avatar
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1 answer
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If $B$ is a Brownian motion and $f \in \mathcal C^2(\mathbb R^n)$ harmonic, then $f(B)$ is a continuous martingale

I'm reading about Itô's formula for a multi-dimensional Brownian motion at page $39$ of these notes, i.e., Let $\underline{B}$ be a standard $n$-dimensional Brownian motion and $f \in \mathcal{C}^2\...
Akira's user avatar
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1 vote
1 answer
86 views

Given a local martingale $M$ the running supremum $N_t=\sup_{0\leq s\leq t}|M_s|$ is locally integrable

I am trying to understand a detail of the proof of Theorem VI.82 of the book by Dellacherie and Meyer. Theorem Let $M$ be a local martingale and let $N_t=\sup_{s\leq t}\left|M_s\right|$. Then $N$ is ...
AlmostSureUser's user avatar
2 votes
0 answers
53 views

Proving that a Stochastic integral is a Martingale

I am trying to prove the following: $$\lim_{n\rightarrow \infty} E\left(\int_{0}^{t \wedge \tau_{n}} e^{-\delta s} v(X_{s})X_{s} dW_{s}\right)=0$$ where $\tau_{n}$ is a sequence of stopping times with ...
Ethan Davitt's user avatar
1 vote
0 answers
37 views

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}...
julian2000P's user avatar
1 vote
0 answers
66 views

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 ...
julian2000P's user avatar
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0 answers
31 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 $(B_t-t)_{t\geq0}$ ...
Milly Moo's user avatar
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3 votes
1 answer
657 views

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,...
Roberto Palermo's user avatar
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1 answer
48 views

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 ...
sixtyTonneAngel's user avatar
2 votes
0 answers
110 views

Check that a process is a local martingale

Assume $\sigma:\mathbb{R}^d \times E$ is $C^2$ (on both parameters) and that $X_t=\int_0^t \sigma(X_s,I_s)dW_s$, where $(I_t)_{t \geq 0}$ is some discrete Markov Chain with finite state space $E$ that ...
Barreto's user avatar
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1 vote
0 answers
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Why is $\ln(\det(B^TB))$ a local martingale for B a Brownian matrix?

I am studying stochastic calculus on my own so this should be a basic question. I saw in https://link.springer.com/article/10.1007/BF01259552 the statement that for $B$ an brownian matrix (with ...
DerHutmacher's user avatar
1 vote
1 answer
69 views

Differential form of local martingale and its quadratic variation

I have a question regarding the differential form of a local martingale and its quadratic variation (the source of the question is p. 136-137 in https://galton.uchicago.edu/~mykland/paperlinks/I.A.1-...
marbrath's user avatar
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0 answers
43 views

Convergence of total quadratic variation to predictable quadratic variation for continuous martingales: proof clarification.

Suppose that $X\in\mathcal{M}^2_c$ is a continuous square-integrable martingale. By the Doob-Meyer decomposition there exists an increasing predictable process $\left<X\right>_t$ such that $X_t^...
AlmostSureUser's user avatar
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1 answer
127 views

Grisanov theorem application

I have the following Ito process: $X_t=e^{B_t-\frac{t}{2}}$ with $\{B_t\}_{t\geq 0}$ is a $\mathbb{P}$-Brownian motion and I want to find under which probability measure $\mathbb{Q}$ the process $X_t^...
Pefok's user avatar
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4 votes
0 answers
260 views

chapter 1 ex 4.22 from Brownian motion, martingales, and stochastic calculus by Jean-Francois Le Gall

This is ex 4.22 from chapter 1 of''Brownian motion, martingales, and stochastic calculus by Jean-Francois Le Gall''. exercise 4.22: Processes on defined on a probability space $(\Omega,\mathcal{F},P)$...
neveryield's user avatar
0 votes
1 answer
91 views

Criterion for an integral to be a martingale

Let $H$ be predictable and bounded. According to my lecture notes, for Brownian motion $W$, $\int HdW$ is a martingale. Question Is it true that for any $M\in\mathcal{H}_{0,\mathrm{loc}}^{2}:=\{\text{...
Vivian's user avatar
  • 398
1 vote
0 answers
82 views

Why is conditional expectation of brownian motion with a negative sign?

I have been reading up on the below thread: conditional expected value of a brownian motion but I cannot understand how $$\mathbb E\left[B_s - \frac{s}{t} B_t + \frac{s}{t} B_t\ \mid B_t\right] = \...
user1060812's user avatar
3 votes
1 answer
175 views

Inequality on expectation of exponential martingale

Let $X$ be a continuous local martingale with $X_0=0$. Define the exponential local martingale $$\mathcal{E}(X)=e^{X-\frac{1}{2}[X]}.$$ For any $p,q>1$, establish the identity $$\mathcal{E}(X)^p=\...
verygoodbloke's user avatar
0 votes
2 answers
449 views

Expected value of exponential Brownian motion

I want to prove that $$E[e^{2B_t}] = e^{2t}$$ where $B_t$ is a Brownian motion. I have been reading up on Mean of exponential Brownian motion but it does not show how the rest of the log-normal ...
user1060812's user avatar
1 vote
0 answers
55 views

Attempt to show that local martingale is a true martingale

Consider the process $X_t=e^{\frac{1}{2}t}\cos(B_t)$, where $B$ is a Brownian motion in $\mathbb{R}$. Using Ito's formula (unless I'm mistaken) implies that $$dX_t=-e^{\frac{1}{2}t}\sin(B_t)dB_t,$$ ...
verygoodbloke's user avatar
3 votes
0 answers
215 views

Relation between supremum of quadratic variation expectation with expectation of supremum of martingale

Let $X$ be a local martingale. Then the quadratic variation $[X]$ is such that $X^2-[X]$ is a local martingale. I am tasked with showing that $$\sup_{t\geq0}\mathbb{E}([X]_t)<\infty\iff\mathbb{E}\...
verygoodbloke's user avatar
2 votes
1 answer
128 views

A local martingale subtract half its quadratic variation tends to negative infinity

Let $M$ be a continuous local martingale with $M_0=0$ and $[M]_\infty=\infty$ almost surely. I am required to show that $M_t-\frac{1}{2}[M]_t\to-\infty$ almost surely as $t\to\infty$. Unfortunately I ...
verygoodbloke's user avatar
5 votes
1 answer
246 views

Probability of stopping time being finite.

Let $X$ be a continuous non-negative local martingale with $X_0=1$ and $X_t\to0$ almost surely as $t\to\infty$. For $a>1$, let $\tau_a=\inf\{t\geq0:X_t>a\}$. I am tasked with showing that $\...
verygoodbloke's user avatar
3 votes
1 answer
743 views

Show that a local martingale is a true martingale if and only if it is a process of class DL

Let $M$ be a local martingale. Show that $M$ is a (true) martingale if and only if it is a process of class DL. Quick definitions: $\mathscr{S}_a$ is the class of all stopping times $T$ such that $P(...
clay's user avatar
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1 vote
1 answer
229 views

Is there a problem of the proof about local martingale is Martingale?

In the definition of local martingale, an adapted process $M=(M_t)_{t\ge 0}$ with continuous sample paths and vanishing at $0$ is called a continuous local martingale if there exists a non-decreasing ...
Hermi's user avatar
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