Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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About continuous local martingales, question on Le-Gall's book

Background Hello, I'm working on question 4.24 on Le-Gall's Brownian motion(...) and I would ask you to check if my ideas are correct. The question is as follows: $(M_t)$ is a cont. local ...
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1answer
13 views

Applyin Itô's formula in function of quadratic variation

I am learning some basic stochastic calculus and came across the following exercise: Consider a local martingale $M$ with continuous trajectories. Let $Z_t = \exp(M_t −0.5[M]_t)$. Show that Z ...
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21 views

Is it a good example for local martingale, but not for martingale?

If $B$ is a Brownian-motion in the $\mathcal{F}$ filtration, then the following process is a good example for being a local martingale, but not a martingale?$$S_{t}=\int_{0}^{t}\frac{1}{1-s}dB_{s},\;\;...
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+50

Stochastic exponential of Ito procss local martingale iff the Ito process is

Let $dX_t= \mu_t dt + \sigma_t d B_t$ be an Ito process and define its stochastic exponential by $$\mathscr E X_t= e^{\int_0^t \mu_s ds + \int_0^t \sigma_s dB_s - \frac{1}{2} \int_0^t \sigma_s^2 ds} $...
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Showing some properties of local martingales in Karatzas, Shreve Problem 1.5.19

The goal is to show the following statements (given in Karatzas, Shreve, Chapter 1, 5.19 Problem): (i) A local martingale $X$ of class DL is a martingale. (ii) A nonnegative local martingale $...
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14 views

$X=M+A$ bounded semimartingale, then $M$ and $A$ are bounded? Counterexample?

Let be $X=M+A$ a semimartingale with $M$ being a local martingale and $A$ an adapted process a finite variation. If $M$ and $A$ are bounded, then of course $X$ is bounded as well. Is the converse true?...
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26 views

Prove that process is a local martingale.

$ \{X_{t}\}_{t\geq 0}, \{Y_{t}\}_{t\geq 0} $ - Ito processes. From Ito's formula we have got: $$ X_{t}Y_{t} = X_{0}Y_{0} + \int_{0}^{t}X_{s}dY_{s} + \int_{0}^{t}Y_{s}dX_{s} + \int_{0}^{t}dX_{s}dY_{s} ...
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1answer
25 views

$\int_0^t(g(s)X_1(s)+f(s)X_2(s))dWs$ - local martingale

How to show that process $\int_0^t(g(s)X_1(s)+f(s)X_2(s))dWs$ is a local martingale where $X_1(s)=\int_0^sf(u)dW_u$ and $X_2(s)=\int_0^sg(u)dW_u$? where $f,g\in P_{[0,T]}^2=\{f:[0,T]\times \Omega\...
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32 views

Process with multiple local properties as process with locally multiple properties

This is motivated by reading the notion "locally square integrable local martingale". Parsing that I arrive at a process $X$ for which two sequences of stopping times $(\sigma_n)$ and $(\tau_n)$ exist,...
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23 views

Equivalent conditions for local martingales

I am having some trouble solving this task: Consider the solution $X$ of the SDE in $\mathbb{R}^n$ $$\mathrm{d} X_t = b (X_t) \mathrm{d} t + \sigma (X_t) \mathrm{d} B_t,$$ where $B$ is a $n$-...
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2answers
39 views

Is next question right? And why?

Let $X(t)=(X^1(t),...X^d(t)$ be a d-dimensional ($\mathcal{F_t}$)-semi-martingale such that (1) $M^i(t) =X^i(t)-X^i(0) \in M_2^{c\ loc}$ and (2)$\langle M^i,M^j\rangle(t)$=$\delta_{...
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28 views

d-dimensional Brownian Motion

Let $B_{t} = (B_{1}(t), \ldots B_{d}(t))$ be a d-dimensional Brownian motion with $d>3$. Fix $0 < r < R < \infty$. Define $$A_{r}(R) = \{x \in \mathbb{R}^{d} : r < |x| < R\},$$ where,...
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32 views

Independence of two local martingales implies zero quadratic variation

I'm self-studying about martingales, but my book sadly only lists this result as an exercise I'm unable to figure out. I'm trying to prove: Let $M$ and $N$ be local continuous martingales, ...
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23 views

When is a this stochastic integral a Martingale, stopping time in the integral.

There are lots of questions on this site asking when Stochastic Integrals are Martingales. I have a similar question : Fix $0<T<\infty$. Given a Brownian Motion $W_t \in \mathbb{R}^d$ on a ...
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17 views

Assertion on Local martingale problem in relation to SDE solution

I have a rather simple question regarding the following theorem which can be found in Karatzas's "Brownian Motion and Stochastic Calculus". Assume $(X,W),(\Omega,\mathcal F,P),\{\mathcal F_t\}$ ...
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35 views

Could an integral w.r.t a local martingale be a proper martingale?

The following is taken from "Stochastic differential equations and diffusion processes" by Ikeda & Watanabe. Let $Y,X\in \mathcal M_2^{c,loc}$ (the space of continuous, square integrable, ...
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55 views

Difference Between Local Martingale and Martingale

I would like to try and distinguish between the two concepts of martingale and local martingale. I have read this answer in Martingale / local martingale : some confusion which was a good start and ...
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41 views

Prove that $(Y_{t})_{t\in[0,\epsilon]}$ is a martingale.

I am struggling the following exercise: Processes $X$ and $Y$ are defined as: $$X=\Bigg( \int_{0}^{t}e^{W_{s}^{2}}\mathrm{d}W_{s}\Bigg)_{t\ge 0},$$ $$Y=\Bigg(\int_{0}^{t}W_{s}^{5}\mathrm{d}X_{s}...
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131 views

How to show this is not a martingale.

Be advised that I cross-posted this question on MathOverflow. You can find it in this link: https://mathoverflow.net/questions/352152/show-that-this-process-is-not-a-martingale Assume we have the ...
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1answer
29 views

Local martingale property for stochastic integrals.

First we start with a Brownian Motion $B(t)$, $a\leq t\leq b$ and an admissible filtration for the BM $\{\mathcal F_t\}$. Let $f(t,\omega)$ be a stochastic process satisfying: $f(t)$ is adapted to ...
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40 views

expectation involving quadratic variation inequality

Let M be a local martingale and let $\tau_n$ be a localising sequence. $1)$ $\liminf_nE[ \langle M \rangle_{\tau_n \land t}] \le E[\langle M_t \rangle]$, is this actually an inequality or an equality?...
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33 views

Quadratic covariation and stopping time

Let $\pi_n: 0 = t_0^n < . . . t_{k_n}^n=t$ be a sequence of partitions of $[0,t]$ with $|\pi_n| \to 0$. Let M,N be two a.s. continuous local martingale . I think that $ \langle M^\tau,N^\tau \...
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2answers
88 views

A specific local martingale

Knowing that M and N are two local martingales. Can I do the following simple reasoning for claiming that $N^\tau(M^\tau - M)$ is a local martingale? : Since a stopped local martingale is still a ...
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2answers
64 views

Representation of stochastic integral w.r.t. semimartingale with finitely many jumps in finite time

In the book "Semimartingale theory and Stochastic Calculus" from He, Wang and Yan I found the following integration by parts formula for semimartingales (Theorem 9.33): If $X$ and $Y$ are ...
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36 views

Showing a randomly scaled Brownian motion is a (local) Martingale.

Assume we work on a filtered probability space $(\Omega, \mathscr{F},(\mathscr{F})_{t\geq0}, \mathbb{P} )$ Let $B_t$ be a standard $\mathscr{F}_t$-Brownian motion and let $X$ be a positive random ...
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2answers
75 views

Compare Doob décomposition for stochastic process $(X_t)_{t\geq 0}$ and $(X_n)_{n\in\mathbb N}$

If know that if $(X_n)_{n\in\mathbb N}$ is a stochastic process, then $X_n$ can be written as $X_n=M_n+A_n$ where $(M_n)$ is a Martingale and $(A_n)$ is an adapted and predictable process. I know ...
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1answer
98 views

l1-bounded local martingale is uniformly integrable martingale

Let $M$ be a cadlag local martingale, i.e. there exists a sequence of stopping times $\{ \tau_k\} $ such that $\tau_k \to \infty$ and $M^{\tau_k}$ is a u.i. martingale for all $k$. Let $M_t$ be also $...
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1answer
68 views

Stopping Time with local martingale

I am working on Karatzas&Shreve Brownian Motion and Stochastic Calculus P.36, Problem 5.19. • (i) A local martingale of class DL is a martingale. • (ii) A nonnegative local martingale is a ...
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1answer
54 views

Covariation of product of local martingales

Let $(M_t)$ and $(N_t)$ be continuous local martingales, and let $[M,N]_t$ be their continuous covariation. Show that $(MN - [M,N])$ is a local martingale. I have tried the following. Since $M_tN_t = ...
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1answer
32 views

Expectation of a local martingale

How do I prove that $\mathbb{E}[\int_o^t B^4_s(\omega) ds ]$ is finite for any $ t \geq 0$? Here $B_t$ is the standard Brownian Motion. If I can change the expectation with the integral the result ...
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0answers
52 views

Limiting expectation of a uniformly integrable local martingale.

Assume that $M = (M_t)_{t\geq 0}$ is a uniformly integrable continuous local martingale with $M_0 \in L^1$ and $\lim_{t\to\infty} M_t = M_\infty$ almost surely. By Fatou's Lemma we know that $E[M_\...
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1answer
15 views

When is the limit of a parameterized family of martingales also a martingale?

Suppose I have a family of martingales continuous martingales $(X^\alpha)_{\alpha \in (0,\infty)}$, that is $X^\alpha = (X^\alpha_t)_{t \geq 0}$ is a continuous martingale, for each $\alpha$. Are ...
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1answer
42 views

Bound on supremum of local martingale

Let $M$ be a continuous local martingale starting at $0$. How can I prove $$ P(\sup_{s\leq t}M_s>a,\langle M\rangle_t\le b)\leq 4\frac b{a^2} $$ for all $a,b>0$ and $t>0$?
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105 views

Martingality of a Doleans-Dade exponential local martingale.

A paper I read recently seems to make the following statement: if $\gamma_t$ is a progressively measurable process, and that $\exp\left(\int_0^T\gamma_s dW_s\right) \in L^p$ for some $p>1$, then ...
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1answer
30 views

If $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for $X_t$ a local martingale, is $X_t$ a proper martingale?

Let $X_t$ be a continuous local martingale. Suppose $\mathbb{E}[\underset{0 \leq s \leq t}{\sup} \rvert X_s \rvert] < \infty$ for any $t > 0$. Is it true that $X_t$ is a proper martingale? We ...
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1answer
34 views

Does $\mathbb{E}[\langle X_{\tau} \rangle^{\frac{1}{2}}] < \infty$ imply $X_t$ is a martingale, given that it is a continuous local martingale

Let $X_t$ be a continuous local martingale. Suppose $\langle X_{\tau} \rangle$ satisfies $\mathbb{E}[\langle X_{\tau} \rangle^{\frac{1}{2}}] < \infty$ for any stopping time $\tau$. Is it true that $...
2
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1answer
42 views

Equality of the stochastic integral under two probability measures

This questions is very short. Under the Girsanov Theorem assumptions we have two equivalent probability measures $\mathbb P$ and $\mathbb Q$ and a measurable space $(\Omega,\mathcal F)$, right? We ...
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68 views

Every local martingale with respect to the Brownian filtration has its continuous version.

I would appreciate some help on the following. In class, we said that Every local martingale with respect to the Brownian filtration has its continuous version. To prove this, it is apprently enough ...
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1answer
543 views

Show that a process is a local martingale

We have the following setting We have a 2-dimensional Brownian motion $(X,Y)$, and we define the process $M_t$ as $$M_t=e^{X_t}\cos(Y_t)$$ The problem is to show that the process $M_t$ is a local ...
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155 views

Second order derivatives in Ito formula for Brownian motion and local martingale

Itô's formula for a $\mathcal{C}^2$ function of two variables F reads: \begin{align} F(X_t, Y_t) &= F(X_0, Y_0) + \int_0^t \frac{\partial F}{dx}(X_s, Y_s) \, dY_s + \int_0^t \frac{\partial F}{dy}...
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45 views

Function and local martingale

Is there exist non-negative continuously differentiable function $g$, such that the process $(g(t)W_t^2)_{t \geq 0}$ is local martingale? I know that i need Ito formula, but how i know which function ...
4
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1answer
188 views

How to prove that the process $Y_t$ is martingale?

We have SDE $$dX_t = X_t(1-X_t)dW_t$$ where $W$ is standard Brownian motion and $X_0 = x_0 \in (0,1)$. Assume that holds $P(X_t \in (0,1))=1.$ For any $u$, we can define $f(x) = (\frac{x}{1-x})^u \...
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0answers
17 views

Prove that a process (given through rsdes) is a martingale.

i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely, let $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t $ ...
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1answer
183 views

Show that this process is not a martingale

I have to show that the following process $(X_t)_{t\in [0,\infty)}$ is no martingale. Let $Y_n$ be a sequence of independent random variables with $$P(Y_n=n)=\frac{1}{2n^2},\quad P(Y_n=-n)=\frac{1}{...
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0answers
39 views

Show $[\int _0 ^t B_{2,s} dB_{1,s}]^2 - \int_0^t (B_{2,s})^2 ds$ is a martingale

I want to show the following: $$[\int _0 ^t B_{2,s} dB_{1,s}]^2 - \int_0^t (B_{2,s})^2 ds$$ is a martingale where $ (B_{2,s},B_{1,s}) $ is a two dim'l Brownian motion. My attempt: By Ito formula, ...
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1answer
106 views

How can I show that the stochastic integral of a jump process w.r.t. Brownian motion is a local martingale by using this special localizing sequence?

Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<\infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then \begin{equation} M_t=\int_0^...
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0answers
42 views

Wiener space, probability measure, local martingale

I hope you can help me the following problem. Assume a Wiener space, that means a probability space $(\Omega,\mathbb F,\mathbb P)$, where $\Omega = C([0,\infty))$, $X$ is the coordinate process, $\...
2
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1answer
125 views

Is there a process that is not square-integrable but still gives rise to a martingale when integrated w.r.t. Brownian motion?

When an adapted process $X$ satisfies $\int_0^TX_t^2dt<\infty$ a.s. but not $E\int_0^TX_t^2dt<\infty$, the stochastic integral $\int_0^tX_sdB_s$, $0\le t\le T$, is only guaranteed to be a local ...
2
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1answer
104 views

Non-negative uniformly integrable local martingale

I am not sure if the following is true or not: "A non-negative uniformly integrable continuous local martingale is a (true) martingale." Is this statement true? Or can someone give a counter-example?...
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1answer
734 views

Uniformly integrable local martingale

Can someone give me an example of a uniformly integrable local martingale that is not a martingale? Or are all U.I. local martingales true martingales (continuous, of course).