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

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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 ...
1 vote
23 views

### 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 $[X,X]_t$ ...
• 75
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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 ...
• 131
1 vote
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• 461
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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 ...
• 1,640
117 views

### 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 \...
• 121
200 views

### 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. ...
• 3
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• 1,023
1 vote
31 views

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 ...
• 2,956
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 ...
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 $b-a$ ...
• 1,904
1 vote
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### 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 ...
• 9,967
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