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Let

  • $T>0$
  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $(\mathcal F_t)_{t\in[0,\:T]}$ be a right-continuous filtration on $(\Omega,\mathcal A)$
  • $B$ be a Brownian motion on $(\Omega,\mathcal A,\mathcal F,\operatorname P)$

Note that $$\mathcal E_0:=\left\{\Phi\in\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]}):\Phi\text{ is }\mathcal F\text{-progressively measurable}\right\}$$ is a closed subspace of $\mathcal L^2(\operatorname P\otimes\left.\lambda^1\right|_{[0,\:T]})$, $$\mathcal E:=\left\{\Phi\in\mathcal E_0:\Phi=\sum_{i=1}^n\zeta_{i-1}1_{(t_{i-1},\:t_i]}\text{ for some }n\in\mathbb N\text{, }0\le t_0<\cdots<t_n\le T\text{ and }\zeta_0,\ldots,\zeta_{n-1}\in\mathcal L^0(\operatorname P)\right\}$$ is a dense closed subspace of $\mathcal E_0$, $$\mathcal M^2:=\left\{X\subseteq\mathcal L^2(\operatorname P):X\text{ is an almost surely right-continuous }\mathcal F\text{-martingale with }\operatorname P\left[X_0=0\right]=1\right\}$$ equipped with $$\left\|X\right\|_{\mathcal M^2}^2:=\sup_{t\in[0,\:T]}\left\|X_t\right\|_{\mathcal L^2(\operatorname P)}^2=\left\|X_T\right\|_{\mathcal L^2(\operatorname P)}^2$$ is a complete semi-normed space and $$\mathcal M^2_c:=\left\{X\in\mathcal M^2:X\text{ is almost surely continuous}\right\}$$ is a closed subspace of $\mathcal M^2$. Let $$(\Phi\cdot B)_t:=\sum_{i=1}^n\zeta_{i-1}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)\;\;\;\text{for }t\in[0,\:T]$$ for $\Phi\in\mathcal E$ of the form as in the definition of $\mathcal E$. Then $\Phi\cdot B\in\mathcal M^2_c$ and $$\mathcal E\to\mathcal M^2_c\;,\;\;\;\Phi\mapsto\Phi\cdot B\tag 1$$ is a linear isometry. Thus, $(1)$ can be uniquely extended to a linear isometry $\mathcal E_0\to\mathcal M^2_c$. Let $$\mathfrak T:=\left\{\left\{t_0,\ldots,t_n\right\}:n\in\mathbb N\text{ and }0=t_0<\cdots<t_n=T\right\}$$ and $$\left|\varsigma\right|:=\max_{1\le i\le n}(t_i-t_{i-1})\;\;\;\text{for }\varsigma=\left\{t_0,\ldots,t_n\right\}\in\mathfrak T\;.$$

Question:$\;\;\;$Let $\Phi\in\mathcal E_0$, $\varsigma=\left\{t_0,\ldots,t_n\right\}\in\mathfrak T$, $\lambda\in[0,1]$, $$\tau_i:=(1-\lambda)t_{i-1}+\lambda t_i\;\;\;\text{for }i\in\left\{1,\ldots,n\right\}$$ and $$\Phi_t^{\varsigma,\:\lambda}:=\sum_{i=1}^n\Phi_{\tau_i}1_{(t_{i-1},\:t_i]}(t)\;\;\;\text{for }t\in[0,T]\;.$$ If $$\Phi_t\in\mathcal L^2(\operatorname P)\;\;\;\text{for all }t\in[0,T]\tag 2$$ and $$\left(t\mapsto\Phi_t\right)\in C^0\left([0,T],\mathcal L^2(\operatorname P)\right)\;,\tag 3$$ then $$\left\|\Phi^{\varsigma,\:\lambda}-\Phi\right\|_{\mathcal E}\xrightarrow{\left|\varsigma\right|\to 0+}0\;.\tag 4$$ If $\lambda=0$, then $(4)$ implies $$\left\|\left(\Phi^{\varsigma,\:0}\cdot B\right)_T-\left(\Phi\cdot B\right)_T\right\|_{\mathcal L^2(\operatorname P)}=\left\|\Phi^{\varsigma,\:0}\cdot B-\Phi\cdot B\right\|_{\mathcal M^2}\xrightarrow{\left|\varsigma\right|\to 0+}0\tag 5$$ by definition of $\Phi\cdot B$. $(5)$ is the reason why many books define the Itō integral as the "mean-square limit" of $$\sum_{i=1}^n\Phi_{t_{i-1}}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)\;,$$ i.e. the $\mathcal L^2(\operatorname P)$-limit of $(\Phi^{\varsigma,\:0}\cdot B)_T$, as $|\varsigma|\to0+$ in the first place. I don't like this definition, cause it obscures what's actually happening and unnecessarily restricts the class of integrands.

I want to study the relation between the Itō (which corresponds to the choice $\lambda=0$) and the Stratonovich integral (which corresponds to $\lambda=1/2)$ and the way we can convert one into the other. My problem is, that I can't find any book which introduces the Stratonovich integral rigorously. Motivated by the last paragraph, all authors simply define the Stratonovich integral to be the mean-square limit of $$\sum_{i=1}^n\Phi_{\tau_i}\left(B_{t_i\:\wedge\:t}-B_{t_{i-1}\:\wedge\:t}\right)$$ as $|\varsigma|\to0+$ with $\lambda=1/2$ (and hence $\tau_i=(t_{i-1}+t_i)/2$) and they don't even state under which assumptions on $\Phi$ this limit even exists!

So, the question is: How can we define the Stratonovich integral as rigorously as in the construction of the Itō integral that I've described above? Since I want to compare both integrals and derive a correction term for the conversion, it's clear that the corresponding classes of integrands must be somehow compatible (otherwise, we couldn't compare their integrals).

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  • $\begingroup$ The modern way to view stochastic integration is via Lyons's theory of rough paths, see e.g. arxiv:0306433. $\endgroup$ – pavel Oct 22 '16 at 19:35
  • $\begingroup$ What about the definition of the Stratonovich integral given in, e.g., Protter's book? This definition expresses the Stratonovich integral in terms of an Ito integral and a Lebesgue integral (the Ito-Stratonovich correction term). In other words, one can take the conversion formula to define the Stratonovich integral. Why isn't that satisfactory? $\endgroup$ – Nawaf Bou-Rabee Oct 22 '16 at 20:55
  • $\begingroup$ @NawafBou-Rabee You mean the definition on page 271? It's unsatisfactory, cause it's way too general for my application. It describes how to define the integral for integrators which are general semimartingales. I only need to declare it for a Brownian motion. Of course, everything presented in the book holds for the case of a Brownian motion too, but I'm not familiar with the theory of semimartingales and that's why I've hoped to find a rigorous definition for the case I'm looking for. $\endgroup$ – 0xbadf00d Oct 22 '16 at 23:25
  • $\begingroup$ When $\lambda=0$ then $\tau_i = t_{i-1}$ and $\Phi_{t_{i-1}} = \zeta_{i-2}$ which seems to disagree with the equation you write before (1). There is also a missing equality sign in the definition of $\mathcal{E}_0$ and I do not understand the limit statement given in (1). $\endgroup$ – Nawaf Bou-Rabee Oct 25 '16 at 12:18
  • $\begingroup$ @NawafBou-Rabee (a) Good point. It seems like I've made a slight mistake in one of the definitions. I will check that. $\endgroup$ – 0xbadf00d Oct 27 '16 at 9:34
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I think there might be a typo in your write up. Take a look at either chapter 8 from the book http://press.princeton.edu/titles/7566.html (http://www-math.mit.edu/~dws/ito/ito8.pdf). I will try to give a brief explanation, hope it helps.

Essentially the reason for defining Ito's integral of the process adapted process $h(s)$ with respect to Brownian Motion $W$ as the limit $$\label{eqn1} \tag 1 \int_0^t h(s)dW(s) := \lim_{n \to \infty } \sum_{i=0}^n h(t_{i}\wedge t)( W( t_{i+1}\wedge t ) - W( t_{i}\wedge t ) ) $$ for any sequence of random partitions tending towards the identity ucp(uniformly over compacts in probability) is the ability to use Ito's isometry. In other words, the fact that for a simple process, $\xi(t) = \sum_i \xi_i 1_{(t_{i+1},t_i]}(t)$ with $\xi_i$ $\mathcal{F}_{t_i}-$measurable, the identity \begin{equation} \mathbf{E} \left[ \int_0^t\xi(s)dW(s) \right]^2= \mathbf{E}\int_0^t\xi(s)^2ds \end{equation} holds, allows us to extend the class of processes we can integrate to a large class of processes: adapted with $\mathbf{P}\left( \int_0^t\xi(s)^2ds < \infty\right)=1.$ Having said this if you change the point in \eqref{eqn1} from $t_i$ to anything else bigger that $t_i$, Ito's isometry is not valid anymore. Stochastic integrals can still be defined through the Skorohod integral (https://en.wikipedia.org/wiki/Skorokhod_integral) for which I don't think there is a straightforward explanation.

Formally speaking when people refer to Stratonovich integral, I don't think the mean the limit in (1) with $h$ evaluated at the point $(t_i + t_{i+1})/2$, I think they mean the limit of sums of the form $$ \sum_{i=0}^n \left( \frac{ h(t_{i}\wedge t) + h(t_{i+1}\wedge t)}{2} \right )( W( t_{i+1}\wedge t ) - W( t_{i}\wedge t ) ), $$ see Section 8.1.2 from Strook's book attached, Of course Ito isometry does not apply straightforwardly in this case either. But not simple algebra implies $$ \tag 2 \left( h(t_{i}) + h(t_{i+1}) \right )( W( t_{i+1} ) - W( t_{i} ) )/2 = h(t_{i}) ( W( t_{i+1} ) - W( t_{i} ) )+\left( h(t_{i+1})-h(t_{i}) \right )( W( t_{i+1} ) - W( t_{i} ) )/2, $$ here I substitute $t_i \mapsto t_i\wedge t$ to avoid writing minimums everywhere. Now upon taking the sum and then the limit, the first term converged to the Ito integral while the second converges to the quadratic covariation $[h,W]$. With this intuition in hand, you arrive to the definition of the Stratonovich integral: $$ \int_0^t h(s)\circ dW(s) = \int_0^t h(s)dW(s) + \frac{1}{2} \left[ h,W \right](t). $$ And the appropriate set of $h$ are the ones for which both the Ito integral and the quadratic covariation are defined. This is how is defined in Protter's book that @NawafBou-Rabee posted on the comments.

Also @pavel mentioned rough paths which is a further (algebraic flavored) way of defining all sorts of integrals, but I let someone else with better expertise on that area comment.

Hope it helps!

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