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Itō isometry from Wikipedia:

Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is adapted to the natural filtration $\mathcal{F}_{*}^{W}$ of the Wiener process. Then $$ \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \, \mathrm{d} t \right], $$ where $\mathbb{E}$ denotes expectation with respect to classical Wiener measure $\gamma$. In other words, the Itō stochastic integral, as a function, is an isometry of normed vector spaces with respect to the norms induced by the inner products $$ ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) = \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma (\omega) $$ and $$ ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega). $$

I was wondering what the two normed spaces are and what their norms are, so that the two normed spaces are isometric wrt their norms?

Thanks and regards!

Tim
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1 Answers1

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Let $(W_t)_{t \geq 0}$ a Wiener process on a probability space $(\Omega,\mathcal{A},\mathbb{P})$. Denote by $X \bullet W_T$ the stochastic integral $$X \bullet W_T := \int_0^T X_t \, dW_t$$ Then Itô Isometry states $$\|X \bullet W_T\|_{L^2(\mathbb{P})}^2 = \|X\|_{L^2(\lambda|_{[0,T]} \times \mathbb{P})}^2$$ where $\lambda|_{[0,T]}$ denotes the Lebesgue measure on $[0,T]$.

This means that the mapping $L^2(\lambda|_{[0,T]} \times \mathbb{P}) \ni X \mapsto X \bullet W_T \in L^2(\mathbb{P})$ is an isometry between the normed space

  • $$L^2(\mathbb{P}) := \left\{f: \Omega \to \mathbb{R}; f \, \text{measurable}, \int_{\Omega} f(\omega)^2 \, d\mathbb{P}(\omega)< \infty\right\}$$ endowed with the norm $$\|f\|_{L^2(\mathbb{P})} := \left( \int_{\Omega} f(\omega)^2 \, d\mathbb{P}(\omega) \right)^{\frac{1}{2}} $$

and the normed space

  • $$L^2(\lambda|_{[0,T]} \times \mathbb{P}):= \left\{ f:[0,T] \times \Omega \to \mathbb{R}; f \, \text{measurable}, \int_{\Omega} \int_0^T f(t,\omega)^2 \, dt \, d\mathbb{P}(\omega)< \infty\right\}$$ endowed with the norm $$\|f\|_{L^2(\lambda|_{[0,T]} \times \mathbb{P})} := \left(\int_{\Omega} \int_0^T f(t,\omega)^2 \, dt \, d\mathbb{P}(\omega) \right)^{\frac{1}{2}}$$

In the Wikipedia article, they consider $W$ as canonical Wiener process, i.e. $$\Omega := C_{(0)} := \{w:[0,\infty) \to \mathbb{R}; w \, \text{continuous}, w(0)=0\}$$ and $\mathbb{P}=\gamma$ is given by the Wiener measure.

saz
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  • Thanks! Is $\gamma$ a probability measure on $\Omega$? In the Wikipedia article, it is the classical Wiener measure, which I think is a probability measure on the sample path space $\mathbb R^{[0,T]}$, instead of a probability measure on $\Omega$? Also I think integral $\int_\Omega \cdot d \gamma(\omega)$ should be $\int_\Omega \cdot d P(\omega)$ instead, where $P$ is the probability measure on $\Omega$ in the definition of the Weiner process $W: \Omega \times [0,T] \to \mathbb R$ and the stochastic process $X: \Omega \times [0,T] \to \mathbb R$? If I am wrong, what am I missing? – Tim Feb 19 '13 at 19:03
  • @Tim You are right: $\mathbb{P}$ is (in general) a probability measure on $\Omega$. The point is that in the wikipedia article, they consider the canonical Wiener process and this implies that $\mathbb{P}$ is given by the Wiener measure $\gamma$ on $C_{(0)} := {w: [0,\infty) \to \mathbb{R}; w , \text{continuous}, w(0)=0}=:\Omega$. – saz Feb 19 '13 at 19:10
  • Thanks! (1) The Wikipedia article says "$\mathbb{E}$ denotes expectation with respect to classical Wiener measure $\gamma$" which I think is on $C_{(0)}$ or $\mathbb R^{[0,T]}$. Is it wrong, because I think it should be "$\mathbb{E}$ denotes expectation with respect to $\mathbb P$ on $\Omega$"? Okay, I now see that you equate $C_{(0)} = \Omega$. Why is that? (2) May I also ask what are the differences between a Weiner process and a canonical Weiner process, and the differences between a Weiner measure and a canonical Weiner measure? – Tim Feb 19 '13 at 19:28
  • (3) Is the law of a stochatic process $\Omega \times [0,T] \to \mathbb R$ defined on the product Borel sigma algebra on $\mathbb R^{[0,T]}$? But the (canonical) Weiner measure is defined on $C_{(0)}$ which is a proper subset of $\mathbb R^{[0,T]}$? WHat is the sigma algebra on $C_{(0)}$? – Tim Feb 19 '13 at 19:30
  • (1) As I already wrote: In the Wikipedia article $W$ is a canonical Wiener process and this means automatically $\Omega = C_{(0)}$, $\mathbb{P}=\gamma$ (Wiener measure) [see (2)]. If you want to consider an arbritary Wiener process, you can replace $\gamma$ simply by an probability measure $\mathbb{P}$. (I edited my answer, so you can see it above.) – saz Feb 19 '13 at 19:41
  • (2) Let $(B_t){t \geq 0}$ a Wiener process on a (arbritary) probability space $(\Omega,\mathcal{A},\mathbb{P})$. Then you can define a probability measure $\gamma$ on ($C{(0)},\mathcal{B}(C_{(0)})$ by $$\gamma(\Gamma) := \mathbb{P}(B_{t_1} \in C_1,\ldots,B_{t_n} \in C_n)$$ where $\Gamma := {w \in C_{(0)}; w(t_1) \in C_1,\ldots,w(t_n) \in C_n)}$ (a so-called cylinder set- they are a generator of $\mathcal{B}(C_{(0)})$). Then you can show that the family of projections $(\pi_t){t \geq 0}$ defines a Wiener process on $(C{(0)},\mathcal{B}(C_{(0)},\mu)$-the so-called canonical Wiener process. – saz Feb 19 '13 at 19:45
  • ... and $\mu$ is called (canonical) Wiener measure (and no, there's no difference as far as I know). But if you are not familar with that, it would be the best to read a book. (3) The $\sigma$-algebra on $C_{(0)}$ is given by $\mathcal{B}^{[0,T]}(\mathbb{R}) \cap C_{(0)}$ where $\mathcal{B}^{[0,T]}(\mathbb{R})$ denotes the product Borel sigma algebra, i.e. it's the trace $\sigma$-algebra. Alternatively, you can define the metric of locally uniform convergence on $C_{(0)}$ (and this metric generates the same $\sigma$-algebra). That's why I denoted it by $\mathcal{B}(C_{(0)}$. – saz Feb 19 '13 at 19:46
  • Thanks for the clarification! I now recall I have seen the projection from the sample path space to each state space is called "the first canonical process" in a note (I haven't seen it in a book yet). Do you happen to know if there are concepts like "the second canonical process"? What books do you recommend? Thanks again! – Tim Feb 19 '13 at 19:55
  • @Tim No, I haven't heard of "second canonical processes" yet. (But since I'm still a student, that doesn't mean a lot.) I read "Brownian motion - An introduction to stochastic processes" - René L. Schilling/Lothar Partzsch. It concentrates on the Brownian motion (aka Wiener process) and does therefore not contain that much about the "general" theory of stochastic processes. As an introduction to the topic of stochastic processes/analysis it's quiet readable. – saz Feb 19 '13 at 20:03