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Questions tagged [quadratic-variation]

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

For a continuous , the process is the unique process which, when subtracted from the square of the martingale once again yields a martingale. Quadratic variation may be realised as the limit of the sum of squared increments over a refining partition with vanishing mesh.

244 questions
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Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. I am aware that they are equivalent in the continuous…
Alfie
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4 answers

Quadratic variation of Brownian motion and almost-sure convergence

Say that $W(t)$ is a Brownian motion. The quadratic variation $[W,W](t)$ is defined in terms of a partition $\Pi = \{0 = t_0 < t_1 < \cdots < t_n = t\}$ by $$ \begin{split} [W,W](t) &= \lim_{|\Pi|\to 0} \sum_{j=0}^{n-1} \Big( W(t_{j+1}) - W(t_j)…
James Davidoff
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2 answers

Quadratic Variation of Brownian Motion

Quadratic Variation of a Brownian motion $B$ over the interval $[0,t]$ is defined as the limit in probability of any sequence of partitions $\Pi_n([0,t])=\{0=t^n_0<\cdots
TheBridge
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Quadratic variation of the Ornstein-Uhlenbeck process

Let $(X_t)_{t\geq 0}$ be the zero-mean Ornstein-Uhlenbeck process such that $X_0 = 0$ almost surely, i.e. $$X_t = \sigma e^{-\alpha t}\int_0^t e^{\alpha s}\,dB_s \quad \qquad (\triangle)$$ On the other hand, $(X_t)$ is the unique process that…
Calculon
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Why is it true that the continuous local martingale with quadratic variation "t" is a square integrable continuous martingale?

I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $M_t$ be a continuous local martingale. On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in \mathcal{M}_2^c$", where $\mathcal{M}_2^c$ means the…
Mark
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1 answer

Continuous Square integrable martingale Quadratic Variation

We know that given a continuous square integrable martingale there exists unique (up to indistinguishability) continuous, natural and increasing process which is quadratic variation process of the given martingale. I would like to know the converse…
chandu1729
  • 3,871
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2 answers

Quadratic variation of ito integral

Is it true in general that if $f$ is a deterministic function, and $W$ is brownian motion, then the quadratic variation of $\int_0^t f(W_s) dW_s$ is $\int_0^t f^2(W_s) ds$? Is it also true in general that the quadratic variation of $\int_0^t…
user428487
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Quadratic variation of $X_t=\int_0^t B_s \, ds$

Let $B$ be a standard brownian motion and $$ X_t=\int_0^t B_s \, ds. $$ What is the quadratic variation $[X]_t$ of $X$? I see $dX_t$ as an sde with drift term $B_t$.
user14108
8
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1 answer

Optional quadratic variation and predictable quadratic variation

What's the difference between optional quadratic variation (which sometimes is denoted by $ [M]$) and predictable quadratic variation (i.e $\ < M > $) of a stochastic process?
learningmath
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Quadratic Variation of Diffusion Process and Geometric Brownian Motion

I'm looking to find out the stochastic differential equation satisfied by the quadratic variation of Geometric Brownian Motion, Diffusion Process. For example, for a diffusion process that satisfies $dX_t = \mu(t, X_t) dt + \sigma (t, X_t) dW_t$,…
ul15524
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1 answer

Calculation of the quadratic variation of an Itô process.

Assume we have an Itô process of the form : $$X_t=X_a+\int_a^t f(s)dB(s)+\int g(s)ds$$ (or $dX_t=f(t)dB(t)+g(t)dt$). I would like to calculate the quadratic variation of the process using the definition: $$\sum_i (X_{t_{i}}-X_{t_{i-1}})^2=\sum_i…
Chaos
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Example of a continuous function with discontinuous quadratic variation

Let $f: [0,\infty)\to \mathbb{R}$. The quadratic variation of $f$, if it exists, is defined as the function $\langle f\rangle: [0,\infty) \to \mathbb{R}$ with $$ \langle f\rangle_t := \lim_{n\to \infty} \sum_{t_i \in \pi_n(t)} \left( f(t_{i+1}) -…
Gnomeo
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Trying to compute the quadratic covariation

Hi everyone this is my first question here. I have the following question on my exercise sheet: let $B$ be a brownian motion, $\phi$ a progressively measurable function such that $E \int_0^T \phi_t dt \lt\infty $, $\forall T>0$. Set $X_t:= \int_0^t…
Uskebasi
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0 answers

Quadratic variation of a Brownian motion up to time $T$ converges to $T$ in $L^2$?

In Stochastic Calculus for Finance II: Continuous-time Models by Steve Shreve, Theorem 3.4.3. Let $W$ be a Brownian motion. Then $[W, W](T) = T$ for all $T > 0$ almost surely. where $[W, W](T)$ is the quadratic variation of $W$ up to time…
Tim
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2 answers

$L^1$ bounded martingale

If $(M_t)_{0\leq t<\infty}$ is continuous martingale and it is $L^1$ bounded, does it imply that quadratic variation $\langle M\rangle_\infty$ is finite a.s. ?
chandu1729
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