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Questions tagged [quadratic-variation]

Questions on quadratic variations of stochastic processes. (Not to be confused with functions of bounded variation.)

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Quadratic covariation of martingale transforms to simple processes.

Let $X$ and $Y$ be simple processes, that is $X_t=\sum_{n=0}^\infty\xi_n{1}_{(t_{n},t_{n+1}]}(t)$ for a uniformly bounded sequence $(\xi_n)_{n\in\mathbb{N}}$ of random variables so that $\xi_n$ is $\...
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Can we find the solution of $3^m - 2^n = 1$ [duplicate]

For the equation, $3^m - 2^n = 1$ is there any way to find the possible solution. What I feel is that there are two variables in the equation so we must have at least two equations to get the solution....
Shinnaaan's user avatar
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Intuition: Why is quadratic variation finite for martingales

(Disclaimer: I'm not well-read in this topic, so might be getting some details wrong. Hopefully not wrong enough to make my question for intuition moot) For any martingale $(X_t)_{t \geq 0}$ with $X_0=...
Bananach's user avatar
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A continuous local martingale $M$ is constant on an interval if $\langle M\rangle$ is

Problem Let $M\in\mathbb{M}^{loc}_C$, i.e. $M$ is an (a.s.) continuous local martingale with $M_0=0$. Show that $M$ is constant on an interval $[a,b]$ with $0\leq a < b$ if $\langle M\rangle$ is ...
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Approximation of quadratic variation of martingales

Let $\{X_t\}$ be a square integrable martingale, $X^n_t\rightarrow X_t$ in $L^2(\Omega,\mathcal{F},P)$ for each $t$ (i.e., in the sense of mean square). Do we have $[X^n,X^n]_t$ converges to $[X,X]_t$ ...
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Definition of first-order and second-order (quadratic) variation

In Shreeve's book on finance in continuous time, he "defines" the following. He says on p. 99: In general, to compute the first-order variation of a function up to time $T$, we first choose ...
herbhofsterd's user avatar
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Quadratic variation of integral of martingales

I have a question regarding the quadratic variation of a martingale. Assume for each $x \in \mathbb{R}$ there exists a martingale $(M_t(x))_{t \geq 0}$ with known quadratic variation $\langle M(x) \...
mathematico's user avatar
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Derivative of the quadratic variation of Levy process

Let $L(t)$ be a n-dimensional Levy process having the decomposition $$ L(t) = \int_{B} x \widetilde{N}(t,dx) $$ where $B=\{ |x|<1 \}$ and $\widetilde{N}(dt,dx) = N(dt,dx) - \nu(dx)dt$ is the ...
MrIncandenza's user avatar
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Prove that $L^2$ martingales with bounded increments that converge almost surely to a finite limit has converging quadratic variation

If $X_n$ is a sequence of $L^2$ martingales with bounded increments (i.e. $|X_n-X_{n-1}|<K$ for some $K>0$) such that $X_n$ converge almost surely to a finite limit, prove that the quadratic ...
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Is this a valid counterexample that $B$ is not a brownian motion?

Let $U,V$ be two Brownian motions. In class we have seen that if $U$ and $V$ are independent, then $B:=xU+\sqrt{1-x^2}V$ is a Brownian motion for $x\in [-1,1]$. Now I asked myself if the independence ...
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Proof that the quadratic variation of a semi martingale is the quadratic variation of its local martingale part

I am asking this is that I am trying to show that the quadratic variation of a continuous semi martingale is just the quadratic variation of its local martingale part. That is if $X_t = X_0 + M_t+A_t$ ...
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Why is the quadratic covariation of two independent brownian motions zero?

In my notes there is a remark saying that if $B$ and $B'$ are independent Brownian motions, then $\langle B,B'\rangle = 0$. I know the following properties of quadratic covariation: $\langle M, N \...
Jamal's user avatar
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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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On the quadratic variation of the Brownian motion

Since it is a martingale, it is easy to prove that $\mathrm{Var}[B]=\mathbb{E}([B,B]_t)$, where $[B,B]_t$ denotes the quadratic variation. But what is $[B,B]_t$ equal to, or equivalently what does \...
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Quadratic Variation Mixed Process Poisson and Brownian Motion

I am trying to solve this problem where we're asked to compute the quadratic variation of a process. I assume that it is necessary to apply Ito's formula but not sure how to get the right solution. ...
Niko's user avatar
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Compute the Integral of $z_t^{(n)}=\sum_{i\geq0}2W_{t_i^n}\mathbb{1}_{]t_i^n,t_{i+1}^n]}(t)$ w.r.t. the Brwonian Motion $W_s$

Let $z=2W$, where $W$ is a Brownian Motion. For $\{t_0^n,t_i^n,...\}=D^n$ (enumeration of dyadic partition), it is possible to show that the processes $$z_t^{(n)}=\sum_{i\geq0}2W_{t_i^n}\mathbb{1}_{]...
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Are there some upper bounds of $|\langle X,Y \rangle_t|$ in terms of $\langle X \rangle_t$ and $\langle Y \rangle_t$?

$\newcommand{\diff}{\, \mathrm d} \newcommand{\and}{\quad \text{and} \quad}$Let $(\Omega, \mathcal F, \mathbb F, \mathbb P)$ be a filtered probability space where $\mathbb F = (\mathcal F_t, t \ge 0)$ ...
Akira's user avatar
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When is the quadratic covariation between two Poisson processes zero?

Let us consider a probability space $(\Omega,\mathscr{F},\mathbb{P})$ endowed with a filtration $\mathbb{F}$, and let $N$ and $M$ be two independent Poisson process with different parameter $\lambda$ ...
Morris Fletcher's user avatar
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p-Variation of a Semimartingale

I have two questions regarding the $p$-Variation of a Semimartingale: Let $X_t$ be a semimartingale on $[0,1]$ and $\Pi_n = \{t^n _k = \frac{k}{n}: 0 \leq k \leq n\} $ a partition of $[0,1]$. For $p &...
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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 $(B_t-t)_{t\geq0}$ ...
Milly Moo's user avatar
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Convergence of quadratic variation of Brownian Motion

Let $(\Omega,\mathcal{F},\mathbb{F}=\{\mathcal{F}_t:t>0\},P)$ be a filtred probability space and $B=\{B_t:t>0\}$ a Brownian Motion. Say $0=t_0^k<...<t_{N_k}^k=t$ is a partition of $[0,t]$ ...
Roberto Palermo's user avatar
4 votes
1 answer
172 views

$L^2$ convergence in the product measure implies convergence when the quadratic variation is absolutely continuous.

(This question is partially related to another one on this forum.) In Karatzas and Shreve, II edition, Chapter 3, we see in equation (2.2) the definition of the following measure on the product space $...
AlmostSureUser's user avatar
2 votes
2 answers
602 views

Ito isometry for correlated Brownian motions

This question Ito isometry with two independent Brownian motions asks for an Itô isometry for two independent Brownian motions $V_t,W_t:[0,T]\times\Omega\rightarrow\mathbb R$. It turns out that the ...
Syd Amerikaner's user avatar
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Quadratic Variation for càdlàg and càdlàd function

A càdlàg function is a function $f:[a,b]\rightarrow\mathbb R$ such that i) the left limit $\lim_{t\uparrow x}f(t)$ exists, and ii) the right limit $\lim_{t\downarrow x}f(t)$ exists and equals $f(x)$. ...
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Computing the expected quadratic variation of a Brownian bridge

Consider the Brownian bridge process $$B_t = W_t - tW_1,$$ where $W_t$ be a Brownian motion on $[0,1]$. What is the expected quadratic variation of $B_t$? Definition: The co-variation of two ...
Quertiopler's user avatar
1 vote
1 answer
66 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-...
marbrath's user avatar
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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^...
AlmostSureUser's user avatar
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1 answer
68 views

First order variation of the predictable quadratic variation

In the book "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve it is said that if $X\in\mathcal{M}_2^c$ (i.e. $X$ is a continuous square integrable martingale) the predictable ...
AlmostSureUser's user avatar
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98 views

Quadratic variation time-changed Brownian motion

Hi we known that the quadratic variation of the one dimensional Brownian motion $B_t$ is $t$, so $d[B]_t=dt$. Let $\sigma_t$ be a subordinator (https://en.wikipedia.org/wiki/Subordinator_(mathematics))...
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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 $(\Omega,\mathcal{F},P)$...
neveryield's user avatar
3 votes
1 answer
176 views

Quadratic variation of linear combination of two processes

Suppose I have two process $X_t:=\int_0^t f(X_s,s)dB_s$ and $Y_t:=\int_0^t g(Y_s,s)dB_s$. Let both $g,f$ be continuously differentiable functions. I am aware of the result that implies that the ...
Nobody's user avatar
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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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$\mathbb{E}[(\int_{0}^{t} V_s^2: d\langle M^{c}\rangle_{s})^{\frac{1}{2}}]<\infty$ implies $N_t:= \int_0^t V_s \cdot d M_{s}^{c}$ is a martingale.

Suppose that $M$ is a continuous martingale and $V_s$ is a progressive process with $\mathbb{E}\left[\left(\int_{0}^{t} Tr(V_sV_s^T d\left\langle M\right\rangle_{s})\right)^{\frac{1}{2}}\right]<\...
Flashhh's user avatar
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1 answer
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Expectation of Quadratic variation of Brownian motion

For $V_n$ = $\sum_{i=0}^{n-1}|W(t_{i+1})-W(t_i)|^2$ where $W(t)$ is a Brownian motion, I know the following: $$ E[V_n] = \sum_{i=0}^{n-1} E[(W(t_{i+1}) - W(t_i))^2] = \sum_{i=0}^{n-1} (t_{i+1} - t_i) =...
Coco's user avatar
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1 vote
0 answers
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Proof of the Kunita-Watanabe Identity

We discussed the following version of the Kunita-Watanabe identity in a recent lecture on stochastic analysis, and I don't quite understand how the proof works. Let $X, Y$ be semimartingales and $H$ ...
LSK21's user avatar
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2 votes
1 answer
486 views

Quadratic variation and measure change

Let $W_t$ be a Brownian motion defined on probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and assume $X_t$ is a process given by SDE $$ dX_t=W_tdW_t, W_0=0 $$ i.e. $X_t=\int_0^tW_sdW_s$. With ...
Mushtandoid's user avatar
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1 answer
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About the formula for the quadratic variation of Brownian motion

The attached document (page 17 - remark 3.4.2) fuente shows a formula that is derived from the mean value theorem. However, as much as I have searched for reference to said formula, I cannot find how ...
Christian's user avatar
2 votes
1 answer
66 views

Clarification on bounded variation of the terms in the Ito's formula as per Ikeda and Watanabe's book

Let $X(t) = X(0) + M(t) + A(t)$ be a continuous semi-martingale where $M \in \mathscr {M}$ and $A \in \mathscr A.$ Let $F: \mathbb R \to \mathbb R$ be $C^2$ be a function of class $C^2$. Then $$F(X(t)...
Error 404's user avatar
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How can I get the integral process with its quadratic variation?

If I know that the quadratic variation of a stochastic integral process is $t^3$, how can I find this process. I tried to do it with the Itô's formula, $[X]_{t}=\int_{0}^{t} \sigma_{s}^{2} d s$ then $\...
The πzza man's user avatar
2 votes
1 answer
204 views

Cross Variation of Ito Integral with $L^2$ bounded continuous martingale

I am trying to prove the identity: $$ < I(K), N_{\cdot}>_t = \int_0^t K_s \ d < B,N_{\cdot}>_s $$ where $I(K)_t$ is by definition $ \int_0^t K_s \ d B_s $, and we take $K$ as a ...
Marine Galantin's user avatar
4 votes
1 answer
437 views

Understanding total, quadratic, and $\Phi$ variation of functions

I've started to study stochastic calculus on my own recently (I'll read Fima's book for a simpler introduction and then Steele's for a maybe more formal approach). I've come across the definition of ...
YetAnotherUsr's user avatar
5 votes
1 answer
448 views

A function with cubic variation

Let $W(\omega,x):\Omega\times\mathbb{R}\rightarrow\mathbb{R}$ be a Brownian motion where $\Omega$ is the sample space. Recall that the quadratic variation of $W$ over the interval $[a,b]$ equals $b-a$ ...
Mathew's user avatar
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1 vote
1 answer
36 views

What is the relationship between local differentiability and quadratic variation?

Let $f$ be a continuous function. The quadratic variation of $f$ on $[a,b]$ is the usual definition: if $S$ is the set of Riemann-style partitions of $[a,b]$ directed by refinement, then the ...
Jacob Manaker's user avatar
1 vote
1 answer
242 views

Differential in Ito's formula and little o notation

Let $Z_\pi$ be the continuous semimartingale defined by $$d(\log Z_\pi(t)) = \gamma(t) dt + \sum_{j=1}^n a_j(t) dW_j(t) $$ where $W_j, j =1, ..., n$ is a standard $n$-dimensional Brownian motion and $$...
qp212223's user avatar
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Finding the Quadratic Variation Process

Let $X = X_t$ be an ito process $X_t = X_0 + \frac{1}{3}t^3+\int_0^t e^s dB_s , t \geq 0$ and let $Y = Y_t$ also be an ito process $Y_t = Y_0 + te^t+\int_0^t s^2 dB_s ,t \geq 0$ How would I find the ...
StocCalculusHelpPlease's user avatar
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77 views

Convergence behaviour of Quadratic Variation of Brownian Motion

Let $(B_{t_i})_{t_i \in [0, t]}$ be a Brownian Motion process and $Q_n(t) := \sum^{n}_{i=1} (\Delta_i B) ^2$ where $\Delta_i B = B_{t_i} - B_{t_{i-1}}$. How is it that $Q_n(t) := \sum^{n}_{i=1} (\...
Darby Bond's user avatar
2 votes
1 answer
289 views

The quadratic variation of the following process...

Let $B$ denotes a Brownian motion, and a stochastic process $X$ is definied as follows: $$X_{t}=e^{3B_{t}}+\int_{0}^{t}B_{s}ds.$$ What is the quadratic variation of $X^2$? I got the following result: ...
Kapes Mate's user avatar
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2 votes
2 answers
318 views

Quadratic variation of a product

I have been struggling to figure out the quadratic variation of the following product: $$e^{3B(t)}\int_0^tB(s)ds$$ I know that the integral of the Brownian motion w.r.t time is finite variation so its ...
Khimaira's user avatar
4 votes
0 answers
76 views

Bounding $\mathbb E(X|\mathcal F_s) - \mathbb E(X|\mathcal F_t)$ on Brownian filtration

Let $W=\{W_t\}_{t\in[0;T]}$ be a Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets. Let $X \in L^\infty(\mathcal F_T)$. I want to prove that the ...
Kolodez's user avatar
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2 votes
0 answers
134 views

When does $p$-variation mesh size tend to $0$?

Let $u: [a,b] \to \mathbb{R}$ and define its $p$-Wiener variation as $$V_p(u) := \sup \left\{ \left(\sum_i |u(x_i)-u(x_{i-1})|^p\right)^{1/p}\right\}, $$ where the supremum ranges over finite ...
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