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Questions tagged [local-martingales]

For questions about local martingales (in continuous time).

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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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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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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 ...
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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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Local martingale with constant stopping time

Let $M_t$ be a continuous local martingale (there exist an almost surely divergent sequence of stopping times $(T_n)$ such that $M^{T_n}$ is a square integrable martingale). Is it true that for each ...
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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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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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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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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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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, $\...
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49 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 ...
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39 views

Orthogonal decomposition of local martingales

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complate and right-continuous filtration on $(\Omega,\mathcal A,\...
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51 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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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).
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Prove that a martingale with a spatial parameter is differentiable

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\operatorname P)$ $M:\Omega\...
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Find local-martingales, equations

Find $x_{1}, x_{2}, x_{3} \in \mathbb{R}$ and local-martingales $H_{1}, H_{2}, H_{3}$ (i.e. $H^1, H^2, H^3 \in \mathcal{L}(W)$ where $W$ is a standard Brownian Motion) such that $$x_1+\int_{0}^{T}H_{...
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A process that has zero drift term containing itself in dB term

I want to show the following is a martingale process.($Z(t)$ continuous) $$dZ(t)=\theta_{t}Z(t)dB_{t}$$ where $\theta_{t}$ is deterministic function $$\theta_{t}=\sigma t$$ My trial:$$E[\int^{T}_{0}(...
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Stopped continuous local martingales

Suppose we have a continuous local martingale $M$. Then, by definition, there exists a sequence of stopping time $(T_n)$ s.t $T_n\uparrow \infty$ and that the stopped process $M^{T_n}$ is a uniformly ...
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How can we show that this is a stopping time?

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A,\...
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Proving martingale property of $N_t = Z(M_{t\wedge s} - M_{t \wedge r})$ for martingale $M$

(Stochastic calculus and Brownian motion, LeGall, page 80). Suppose $M = (M_t)$ is a martingale. Also, let $Z$ be a bounded random variable which is $\mathcal{F}_r$ adapted. Then we like to show ...
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Is the space of continuous local martingales equipped with the topology of uniform convergence on compact sets complete?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $M_{c,\:\text{loc}}(\mathcal F,\operatorname P)$ denote the set of ...
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A non-negative local submartingale of the class DL is a submartingale

Can somebody explain why a non-negative local submartingale of the class DL is a submartingale? I was reading the proof here https://almostsure.wordpress.com/2009/12/24/local-martingales/ in Lemmma 3 ...
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Prove that local martingale M is a martingale

$M$ is a local martingale. Also for stopping times $T$ and $S$, $S \leq T$, if $E(M_{T}| F_{S})< \infty$, than $E(M_{T} | F_{S}) = M_{S}$. Prove that $M$ is a martingale. I'm trying to prove it ...
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is product of two martingales a martingale with common filtration? [closed]

hello please answer my question. 1)is product of two martingales respect of their natural filtration, martingale? 2)is product of two martingale respect to common filtration, martingale? thank for ...
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find 2 local martingales M(t) and N(t) such that M(T)=N(T),but for t<T,M(t) $\ne$ N(t)

question: find 2 local martingales M(t) and N(t) such that M(T)=N(T),but for t < T,M(t) $\ne $ N(t). I can only find one is local martingale and the other one is martingale. can you help me ...
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Associativity of an integral against a function with finite variation

I'm studying continuous martingales and there is this thing which worries me: Let $a$ be of finite variation and $f$ be $a$–integrable. Then the function $(f · a)$ is right-continuous and of finite ...
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Is it true that every continuous local martingale is a true martingale

Given a continuous local martingale $X_t$ starting at 0, i.e. $X_0=0$, by the continuity of the paths, we can deduce that $\sup_{0\leq s\leq t}|X_s|<\infty$ P-a.s. Hence, $E[\sup_{0\leq s\leq t}|...
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Martingale property of a stochastic integral w.r.t. a local martingale.

I have some problems with the martingale property of a stochastic integral with respect to a continuous local martingale M. I know that if $X \in L^2(d[M])$ then $$ \int XdM$$ is a local martingale. ...
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Local Martingales in J. Michael Steele (Stochastic Calculus and Financial Applications )

In Steele's Stochastic Calculus and Financial Applications the Itô integral is defined at a first stage for integrands belonging to the set $\mathcal{H}^2$ comprised of all $f\colon\Omega\times[0,T]\...
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Show that $(S_t - M_t) \phi(S_t) = \Phi(S_t) - \int_{0}^{t} \phi(S_s) dM_s$

Show that $(S_t - M_t) \phi(S_t) = \Phi(S_t) - \int_{0}^{t} \phi(S_s) dM_s$ (This is from Le Gall's book, Brownian Motion, Martingales, and Stochastic Calculus.) Here, $M$ is a continuous local ...
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178 views

How can a martingale be a density process?

Let $(\Omega,\mathcal F,\mathbf P,\{\mathcal F_t\}_{t\ge 0})$ be a filtered probability space, and let $\mathbf Q$ be a probability measure on $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\ge 0})$ such that ...
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1answer
91 views

Exponential martingales and changes of measure

Suppose $X$ is a subordinator (an increasing Levy process) with Laplace exponent $\Phi$, i.e. $$ \exp(-\Phi(\lambda)) = E(\exp(-\lambda X_1)). $$ Let $\mathcal{F} = (\mathcal{F}_t)$ denote the ...
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I've found two definitions of a localizing sequence. Does one of them have some drawbacks?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I\subseteq\mathbb R$ be an interval (of any form) $(\mathcal F_t)_{t\in I}$ be a filtration of $\mathcal A$ I've found two ...
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1answer
120 views

Inverting the cumulative distribution function to solve for boundary conditions

Given a function: $$f[x]=a\, \Phi \left[-x+\sigma \sqrt{\tau}\right]-\left(b+c\, e^{-d \tau}\right)\Phi \left[-x\right]$$ where $\Phi$ is the cumulative density function of the standard normal ...
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1answer
183 views

Martingale property for stochastic time changed Martingale processes

I came across a Lemma and proof in Tankov & Cont, financial modelling with jumps. The lemma and proof can be seen here: I have a hard time understanding the reason for the relation in the ...
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1answer
88 views

Characterization for strongly orthogonal martingales

In Limit Theorems for Stochastic Processes by Jacod and Shiryaev we have the following statement: For two square-integrable martingales $X,Y$ starting in zero we have $$ \langle X,Y\rangle =0 \...
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If $M$ is a local martingale and $τ:=\inf\left\{t≥0:|M_t|≥ε\right\}$, then $\text P[[M]_∞≥δ]≤\text P[τ<∞]+\text P[[M]_τ≥δ]$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a right-continuous filtration of $\mathcal A$ $M$ be an almost surely continuous local $\mathcal F$-...
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Integrability of inverted Brownian Motion

I'm trying to understand the counterexample to the claim that every $L^2$-bounded local martingale is also a true martingale. For this I'm considering the local martingale $$ X_t:=\frac{1}{\vert B_t +...
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493 views

Is a stopped local martingale a local martingale?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$ $M$ be a local $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)...
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Example of a random variable $X$ that is an $\mathscr{F}_t$-local martingale, but not an $\mathscr{F}_t^X$-local martingale.

This is a problem from Ethier and Kurtz' Markov Processes. The book introduces some theorems on local martingales but they all involve the process being right continuous. I think this problem must be ...
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If $M$ is a local martingale and $τ:=\inf\left\{t\ge0:\left|M_t\right|\ge\varepsilon\right\}$, then $M^τ$ is a martingale

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$ $M$ be a continuous local $\mathcal F$-martingale on $(\Omega,\mathcal A,\...
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Is an $L^2$-bounded continuous local martingale already a strict martingale?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $M$ be a continuous local $\mathcal F$-martingale on $(\Omega,\...
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Breaking a semimartingale into non-semimartingale parts

I know that if $X_t$ is a semi-martingale then $X_t$ can be written as the sum of an martingale $M_t$ and a nice predictable process $A_t$. I was wondering if the converse is false. That is ...
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Show that semimartingale related to Brownian motion is local martingale

I got stuck on another problem regarding whether certain process is a local martingale. Set $T_\epsilon=\inf\{ t\ge 0:B_t=\epsilon\}$, $\lambda >0$ and $\alpha\neq 0$ Show that the process $$ ...
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144 views

Is a local continuous martingale square integrable.

As the headline suggests, I'm wondering whether a local continuous martingale is actually necessarily square integrable. In the text I'm reading this is only mentioned without explanation (the ...
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139 views

Semimartingale property of function of Brownian motion

I have stumbled upon the exercise in Jean-François Le Gall's book. I would be highly thankful for any comments: Let $f$ be twice differentable and $g$ be a continuos function. Verify that the process ...
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202 views

Martingale representation theorem and Markov property

Martingale representation theorem says that every local martingale on $[0,T]$ adapted to a Brownian filtration is a Ito integral with respect the the Brownian motion generating the filtration. The ...
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Confused about step in proof about quadratic variation

In this link, Theorem 5, starting with "For the converse statement", the proof begins: Let $X$ be a continuous local martingale. Set $\tau_n = \inf\{t : |X_t| \geq n\}$. Then $X^{\tau_n}$ is a local ...
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76 views

Show that $X \in \mathcal{M}^{loc}_c$ implies: $X$ is locally square summable and locally bounded.

If $X$ is a continuous local martingale with $X_0= 0$ a.s., then: 1) the process $X$ is a locally bounded martingale; 2) $X$ is locally square integrable. Localising sequence definition. There ...
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141 views

Show that $M-M_0$ is an $L^2$ bounded martingale if $E([M]_\infty)<\infty$.

Let $M$ be a continuous local martingale, and let $[M]_\infty=\lim\limits_{t\rightarrow \infty}[M]_t,$ and if $E([M]_\infty)<\infty$. Show that $M-M_0$ is an $L^2$ bounded martingale. Since $M$ is ...