Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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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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26 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 ...
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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 $(B_t-t)_{t\geq0}$ ...
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Are these two definitions for local martingales equivalent?

I have the following problem with two different definitions of a local martingale: Let $M$ be adapted and right continuous. My definition is the following: $M$ is $\mathbb{R}^d$-valued local ...
3 votes
1 answer
59 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,...
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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 ...
2 votes
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37 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 ...
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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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1 answer
49 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-...
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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] = \...
3 votes
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=\...
0 votes
2 answers
78 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 ...
1 vote
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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 $$dX_t=-e^{\frac{1}{2}t}\sin(B_t)dB_t,$$ ...
3 votes
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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}\...
2 votes
1 answer
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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 ...
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 $\...
3 votes
1 answer
141 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(...
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Are lower bounded Local Martingales Supermartingales

I found following proof in my Lecture notes. Let M be a continous local Martingale and $M(t)\geq-a \,$ for some $a$. Then I choose a Localizing Sequence $\tau_{k}\,$ for M and I get $\\$ $\lim\...
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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 $X = 0$ ...
4 votes
1 answer
36 views

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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3 votes
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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 ...
1 vote
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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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80 views

Show that $V_t = \|B_t\|^{-1}$ is a true local martingale

I have some difficulties with solving a problem of stochastic calculus related to local martingale. Here is the problem : Let be $B$ a 3-dimensional Brownian Motion starting at $B_0 = (0,0,1)$ and ...
2 votes
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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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2 votes
1 answer
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Inverse of norm of Brownian motion is a semi-martingale.

I would like to show that $f(x_1,x_2,x_3)=\frac{1}{\sqrt{(x_1^2+x_2^2+x_3^2)}}$ is a local martingale. I know that for $B_t=((B_1)_t,(B_2)_t,(B_3)_t)$ the Itô formula reads $$df(t,B_t)=f_t(t,B_t)dt+df(...
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2 votes
1 answer
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How to prove that $(UM_t)_{t \geq 0}$ is a Continuous Local Martingale?

Consider a filtered probability space $(\Omega, \mathcal{F},(\mathcal{F}_t)_{t \geq 0},P)$ and $(M_t)_{t \geq 0}$ a continuous local martingale (CLM) w.r.t $(\mathcal{F}_t)_{t \geq 0}$. Let $U$ denote ...
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covariation of semimartingales

Consider continuous semimartingales $X,Y,Z$, i.e. $X=X(0) + A + M$, where $A$ is a process of finite variation and M a local martingale, both starting in $0$ I want to proof: $\langle X + Y , Z \...
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On different definitions of local martingale

While I think I have acquired the essence of the nature of local martingales, I am still a bit uncomfortable with the several definitions that I come across in the literature. For instance ( I am ...
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How can I make the following expression to be a martingale?

Let $B$ denotes a Brownian motion in a filtration $\mathcal{F}$ and we define $X$ as the following process:$$X_{t}=\int_{0}^{t}h\left(B_{s}\right)ds,$$ where $h:\mathbb{R}\mapsto\left[1,\infty\right)$ ...
1 vote
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Why do we need continuity for the convergence of the localized martingale?

$M(t)$ is a local continuous martingale and $\tau_{k}$ is a localizing sequence $\lim\limits_{n \rightarrow \infty}M(t\wedge \tau_{n})=M(t)$. I know that this is true for continuous martingales, but ...
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Quadratic variation of a stochastic integral w.r.t. a local martingale

I am trying to prove the (seemingly simple) property: for a continuous local martingale $M$ and an $M$-integrable process $H$, the quadratic variation $\langle\int H\,dM\rangle$ of $\int H\,dM$ is ...
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$e^{X_t}$ is martingale using Ito's Formula

$B_t$ is the Brownian process. How to prove that $ Y_t=e^{X_t}$ is a martingale using Itô's formula? Here we have, for $f$ deterministic $$X_t = \int_0^t f(s)dB_s-\frac{1}{2}\int_0^t f^2(s)ds $$ and $...
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4 votes
1 answer
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Potential Local Martingale property derived from its quadratic variation

Suppose we have a continuous local martingale $M$ such that $\langle M \rangle_t =o(t)$ - i.e. $$\lim_{t \rightarrow 0} \frac{\langle M \rangle_t}{t} = 0$$ Does this imply that $$\lim_{t \rightarrow ...
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stochastic exponential of continuous local martingale is local martingale

Consider a continuous local martingale $M$ and the stochastic exponential $Z(t)= \exp\{M(t) - 0.5 \langle M \rangle(t)\}$, where $\langle M \rangle(t)$ is the quadratic variation of $M$. Using Ito's ...
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4 votes
1 answer
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local martingale remains local martingale when lowering the localizing stopping times

Assume we have a stochastic process $(X_t)_{t \geq 0}$ which is a local martingale in respect to some filtration $F=(F_t)_{t \geq 0}$. That means by definition that there exists an almost surely ...
3 votes
1 answer
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Are all local martingales continuous in probability?

We say $X$ is continuous in probability if for any $t_0 \geqslant 0$ fixed, we have $ \mathbb{P}\left(\left\|X_{t}-X_{t_0}\right\|>\varepsilon\right) \rightarrow 0$ as $t \rightarrow ...
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1 vote
1 answer
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local martingale?

Consider the ito-process: $X(t)= \sigma(t) dW(t)$ for $t \in [0,T]$, where $\sigma$ is a predictable process and $\int_{0}^T \sigma^2 dt <\infty$. Consider $X^2(t)$: Appyling Ito's formula to $f(x)=...
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Properties of Quadratic Covariation

I want to show that for any two continuous local martingales and any stopping time $T$, $\langle M^T, N \rangle = \langle M, N \rangle ^T$. The method suggested in Revuz and Yor as an exercise is to ...
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1 vote
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Understanding of definition of a local martingale

In my lecture note, a local martingale is defined as the picture shows. I am a bit confused with the sequence of the stopping time. I understand this as every element in the sequence is a function ...
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Stochastic integral over an $L^2_\text{loc}$ function is a continuous local martingale

Let $B_t$ be any standard Brownian motion and let $f \in L^2_\text{loc}$. Then $W = \int_0^{\cdot} f(\cdot,s)\, dB_s$ is a continuous local martingale. This is stated in my lecture notes right after ...
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Finding a stochastic process from Ito’s lemme on $Y_t$ such that $\int_0^1Y_s\,\mathrm{d}B_s= B_1-2B_1\mathbb{I}_{|B_1|}$

Problem: I would like to zoom in on a particular part of a question as a follow-up on a question I have previously asked, which I feel deserves its own space. I was looking at the random variable $$X=...
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1 vote
1 answer
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How to determine if the process is martingale?

I have that $\{X_t; t=1,2... \}$ is an i.i.d random variables such that $P(X_t=1)=p=1/4$ and $P(X_t=-1)=1-p=3/4$. Let $S_0=0$ and $S_t=\sum_{i=1}^tX_i$ for $t=1,2...$ Let finally $Y_t=a^{S_t}$ for $t=...
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Karatzas&Shreve 5.4.33 - Equivalence of Three Local Martingales

I encoutered this problem while self-studying Karatzas&Shreve.I have worked out the equivalence relation between $\Lambda_t$ and $M_t$, which is setting $C_t = e^{-\alpha t} $ and $C_t = e^{\alpha ...
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