Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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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 ...
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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}...
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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 $...
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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 ...
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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 ...
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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 $X$ ...
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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)>...
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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\...
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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 $\mathcal{F}_s$-...
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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 ...
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Integral of $M^\text{*}-M$ the difference of running maximum and current value with respect to $M^\text{*}$ is 0 for a continuous local martingale $M$

For $(M_t)$ a continuous local martingale, and $M^\text{*}_t := \sup_{0 \leq s \leq t}M_s$ the running maximum of the process, then I'm seeking to prove $$ \int_0^T (M^\text{*}_s - M_s) \, \text{d}M^\...
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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 ...
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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 \...
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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 ...
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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. $\mathcal G$....
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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 $(\...
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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., $\forall t \in \mathbb{R}_{+}$. ...
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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 ...
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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\...
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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 ...
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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 ...
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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}...
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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 ...
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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 $(B_t-t)_{t\geq0}$ ...
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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,...
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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 ...
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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 ...
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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 ...
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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-...
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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^...
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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^...
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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 $(\Omega,\mathcal{F},P)$...
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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{...
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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] = \...
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1 answer
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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
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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 ...
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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 $$dX_t=-e^{\frac{1}{2}t}\sin(B_t)dB_t,$$ ...
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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
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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
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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
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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(...
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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 ...
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Show that if $(M_t)_{t \in [0,T]}$ is a non-negative and strict local martingale on [0,T] we have $\mathop{\mathbb{E}}[M_T] < M_0$.

I was stuck in proving this statement. There is a hint: By contradiction, using the tower property and the fact that if a random variable $X \leq 0$ a.s. and $\mathop{\mathbb{E}}[X] = 0$ then $X = 0$ ...
math_killer's user avatar
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1 answer
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The expected squared increment of a continuous local martingale

Suppose $M=\{ M_t\}_{t\geq 0}$ is a continuous local martingale, and $M_0=0$. Then I often see the following equation $$\mathbb{E}M_t^2=\mathbb{E}\sum_i(M_{t_{i+1}}^2-M_{t_i}^2)=\mathbb{E}\sum_i(M_{t_{...
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How to show that if for every stopping time T $\mathbb{E}[X_T] = \mathbb{E}[Y_T]$, then $(X_t-Y_t)_{t\geq 0}$ is a martingale that starts in $0$.

I'm trying to understand a proof for the Kunita-Watanabe identity.       Theorem (Kunita-Watanabe identity): Let $X,Y$ be local martingales that starts in $0$ and $H$ $X$- and $\langle X,Y\rangle$- ...
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Preservation of the local martingale property under the right-continuous filtration

While reading the proof of Theorem 3 from the following post : https://almostsuremath.com/2010/04/01/continuous-local-martingales/ I have come across a subtle point that is not clear to me. So the ...
nomadicmathematician's user avatar
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A discrete local martingale which is integrable for some random $T>0$ is a true martingale up to time $T$.

Let $X$ be a discrete local martingale such that $X_T$ is integrable for some non-random time $T > 0$. I am tasked with showing that $(X_t)_{0≤t≤T}$ is a true martingale. The hint is to show that $...
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Why can we use limit inferior to calculate the expected value of a stopped process?

Consider ($\tau_n$) a diverging sequence of stopping times (e.g. $\inf\{t: X_t>n\}$). We can write the stopped local martingale $X_t^{\tau_n}$ = $X_{t\wedge \tau_n}$, which yields $\lim_{n\...
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How to prove CLT for predictable quadratic variation $\langle X_n \rangle$ for a martingale $X_n$

Im interested in higher order inference (like Edgeworth expansions for iid case) of martingales, and reading series of paper by Per Mykland (1993, 1995, etc). In it, the core condition for the higher ...
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