Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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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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38 views

Apply Ito’s formula to $F(X_t,Y_t,Z_t)$, where $Z_t = X_t Y_t$.

Let $(X)_{t \geq 0}$ and $(Y)_{t \geq 0}$ be continuous semimartingales with values in $\mathbb{R}$. Apply Ito’s formula to $F(X_t,Y_t,Z_t)$, where $Z_t = X_t Y_t$ and $F$ is a $C^2$ function. ...
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35 views

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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40 views

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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34 views

Expectation of the Inverse Bessel-3 Process

I am studying the inverse Bessel-3 process. I understand that there have been several posts already about this process, but I have a conceptual question about why this process is (strictly) a local ...
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127 views

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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24 views

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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37 views

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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61 views

Inequality with local martingale (find sharp estimate for $\mathbb{P}(M_t\geq \alpha)$)

I am considering a local martingale $M_t$ with continuous sample paths (i.e. $M_t\in\mathcal{M}_{C,\,\text{loc}}$) and with quadratic variation $\langle M\rangle_t$ (and also $M_0=\langle M\rangle_0$ ...
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21 views

Semimartingales and continuously differentiable functions

Let $f$ be continuously differentiable on $[0,1]$. This gives us that $f$ is a semimartingale. I would like to understand why this is. The definition of a semimartingale is a process that can be ...
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44 views

$M$ local martingale with $M_0>0$, $B$ Brownian motion s.t. $M_t=B_{\langle M\rangle_t}$ $\implies$ $\langle M\rangle_{\infty}=\inf\{u,B_u=0\}$

Let $M$ be a continuous, nonnegative local martingale with $M_0=m>0$ a real constant and $$M_{\infty} =\lim_{t\to \infty}M_t=0 \quad\text{a.s.}$$ I am doing exercese 4.25 in chapter 3 of "...
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25 views

How to prove that this stochastic process converges in mean square as $t \to \infty$

I have that $\{ X_t ; t=0,1...\}$ is a martingale with finite second moments (meaning that $E(X_t^2) < \infty$). Assume that $E(X_t^2) \to C \quad $ as $t \to \infty$ where $C < \infty$ is a ...
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43 views

Show tilting measure is martingale

I am doing an exercise regarding the Girsanov theorem, $B_t$ is standard brownian motion. I already find that $$M_t=\exp(Y_t)\quad Y_t=\int_{0}^{t}-2mB_sdB_s-\frac{1}{2} \int_{0}^{t}4m^2B_s^2ds $$ $...
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55 views

Local times and quadratic variation

Suppose we have a continuous local martingale $(M_t)_{t \geq 0}$, where $M_0 = 0$. Let $L_{t}^{0}$ denote the local time of this process at zero. I would like to prove the simple statement that: \...
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Exit and hitting times for the Bessel process $\textrm{d}X_t=\frac{n-1}2\frac{\textrm{d}t}{X_t}+\textrm{d}B_t$

I am trying to analyse the exit time $T_1:=\inf\{t:X_t\notin[\alpha,2]\}$ and hitting time $T_2:=\inf\{t:X_t=0\}$, where $\alpha<1$ is a constant, and $X_t$ follows the Bessel process defined by ...
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22 views

Showing that the expectation of a local martingales are equal is to zero

I have difficulty understanding the proof of Theorem 16 from Chapter 3 of Protter's Stochastic Integration and Differential Equations. Here, $\mathbb{L}$ is the space of caglad functions, and $\tilde{...
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38 views

$X$ a continuous square integrable martingale, $Y$ a square integrable bounded variation martingale, then $[X,Y]=X_0Y_0$ and $XY$ is a martingale

This is a sentence from page 73 of Protter's Stochastic Integration and Differential Equations that I am struggling to prove. One can also easily verify (as a consequence of Theorem 23 and Corollaries ...
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116 views

Question about Protter's proof that a Cadlag, locally square integrable local martingale is a semimartingale

This is Corollary $1$ in Chapter $2$ of Protter's Stochastic Integration and Differential Equations. Theorem 8 states that each $L^2$ martingale (martingales $X$ such that $X_0 = 0$ and $E[X_\infty^2]...
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95 views

Using various inequalities to prove this martingale inequality?

I’ve got another martingale inequality that I would be grateful for a kickstart on. Suppose $X_t$ is a local martingale such that $|X_t|$ and $\langle X_t\rangle\leq c$ $\forall$ $t\geq 0$ and for ...
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24 views

Is a mixture of two cadlag local martingales again a cadlag loc. mart.?

Exercise IV.1.37 of Revuz--Yor asks: Let $\mathbf W = C(\mathbb R_+, \mathbb R)$, $X$ be the coordinate process and $\mathcal F_t^0 = \sigma(X_s, s \leq t)$. Prove that the set of probability ...
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39 views

Covariation of density process in Girsanov's Theorem

I am reading a proof of the following theorem concerning Girsanov's Theorem. Let be $M$ a continuous local $P$- martingale. Then $M-\langle M,Y\rangle$ is a continuos local $Q$ martingale. Here $Y_t=...
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151 views

Is the exponential martingale of Brownian motion uniformly absolutely continuous?

Let $M = \{M_t\}_{t\ge0}$ be the exponential martingale of Brownian motion $W= \{W_t\}_{t\ge0}$, that is, $$ M_t = \mathcal E(W)_t = \mathrm{exp} \left( W_t - \frac{t}{2}\right). $$ Question: Is $M$ ...
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190 views

Interpretation of Gisarnovs theorem

Ive found a statement of Girsarnovs theorem that looks as follows "Every $P$-semimartingale is a $Q$ semimartingale, in particular if $M$ is a local martingale then $\hat{M}_{t}=M_{t}-D_{t}^{-1}[...
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18 views

Localization of localizations of previsible processes is a localized process from Rogers Wiliams

Below is an exercise and hint from Rogers and Williams' Diffusions, Markov Processes, and Martingales Volume 2. Here, we are trying to show that if $H$ is a previsible process in $l(l(\mathscr{H}))$ ...
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83 views

An example to be a local martingale but not a martingale

What is a good example for $\varphi_{s}$, to $$\int_{0}^{t}\varphi_{s}dW_{s}$$ be a local martingale, but not a martingale? A simplier question: what should I choose for $\varphi_{s}$, if I don't want ...
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Write the martingale part of this semimartingale as an Itō integral process with respect to a one-dimensional Brownian motion

Let $H$ be a separable $\mathbb R$-Hilbert space and $(X_t)_{t\ge0}$ be an $H$-valued semimartingale with $$\int_0^t\left\|X_s\right\|_H^2\:{\rm d}s<\infty\;\;\;\text{almost surely for all }t\ge0\...
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Estimate for continuous Ito semimartingale

I'm reading an article in which one defines a continuous Ito semimartingale of the form $$\hat{V}_{i \Delta_n}' - \bar{V}_{i \Delta_n} = \frac{2}{k_n \Delta_n} \sum_{j=1}^{k_n} \int_{(i+j-1)\Delta_n}^{...
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57 views

Quotient of continuous local martingale with quadratic variation

Consider a local martingale $(M_t)_{t\ge 0}$ with continuous paths and $\lim_{t\rightarrow\infty}[M]_t=\infty$ a.s. I want to show, that $$\lim_{t\rightarrow\infty}\frac{M_t}{[M]_t}=0\quad\text{a.s.}$$...
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66 views

A martingale related proof, need help

could anyone help me with this one line proof? I couldn't understand the proof in Chapter 1, 4.50 proposition, part(c),'only if' direction (page53) in Jean Jacod and AN. Shiryear's Limit Theorems for ...
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34 views

Stochastic Fubini

Provided a stochastic basis $(\Omega,\mathfrak A,\mathfrak F,P)$ supporting a continuous local martingale $M$ with time horizon $T>0$, and a measure space $(E,\mathfrak E,\mu)$, let $F: \mathbb [0,...
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86 views

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 martingale ...
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32 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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51 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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154 views

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 $X$ is a ...
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$X=M+A$ 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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70 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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30 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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1answer
42 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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35 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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51 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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51 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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72 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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75 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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229 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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45 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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198 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
60 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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103 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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94 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 ...