Quadratic Variations and the Ito Isometry

As local martingales are semimartingales, they have a well-defined quadratic variation. These satisfy several useful and well known properties, such as the Ito isometry, which are the subject of this post. First, the covariation [X,Y] allows the product XY of local martingales to be decomposed into local martingale and FV terms. Consider, for example, a standard Brownian motion B. This has quadratic variation {[B]_t=t} and it is easily checked that {B^2_t-t} is a martingale.

Lemma 1 If X and Y are local martingales then XY-[X,Y] is a local martingale.

In particular, {X^2-[X]} is a local martingale for all local martingales X.

Proof: Integration by parts gives

\displaystyle  XY-[X,Y] = X_0Y_0+\int X_-\,dY+\int Y_-\,dX

which, by preservation of the local martingale property, is a local martingale. ⬜

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Integrating with respect to Brownian motion

In this post I attempt to give a rigorous definition of integration with respect to Brownian motion (as introduced by Itô in 1944), while keeping it as concise as possible. The stochastic integral can also be defined for a much more general class of processes called semimartingales. However, as Brownian motion is such an important special case which can be handled directly, I start with this as the subject of this post. If {\{X_s\}_{s\ge 0}} is a standard Brownian motion defined on a probability space {(\Omega,\mathcal{F},{\mathbb P})} and {\alpha_s} is a stochastic process, the aim is to define the integral

\displaystyle  \int_0^t\alpha_s\,dX_s. (1)

In ordinary calculus, this can be approximated by Riemann sums, which converge for continuous integrands whenever the integrator {X} is of finite variation. This leads to the Riemann-Stietjes integral and, generalizing to measurable integrands, the Lebesgue-Stieltjes integral. Unfortunately this method does not work for Brownian motion which, as discussed in my previous post, has infinite variation over all nontrivial compact intervals.

The standard approach is to start by writing out the integral explicitly for piecewise constant integrands. If there are times {0=t_0\le t_1\le\cdots\le t_n=t} such that {\alpha_s=\alpha_{t_k}} for each {s\in(t_{k-1},t_k)} then the integral is given by the summation,

\displaystyle  \int_0^t\alpha\,dX = \sum_{k=1}^n\alpha_{t_k}(X_{t_k}-X_{t_{k-1}}). (2)

We could try to extend to more general integrands by approximating by piecewise constant processes but, as mentioned above, Brownian motion has infinite variation paths and this will diverge in general.

Fortunately, when working with random processes, there are a couple of observations which improve the chances of being able to consistently define the integral. They are

  • The integral is not a single real number, but is instead a random variable defined on the probability space. It therefore only has to be defined up to a set of zero probability and not on every possible path of {X}.
  • Rather than requiring limits of integrals to converge for each path of {X} (e.g., dominated convergence), the much weaker convergence in probability can be used.

These observations are still not enough, and the main insight is to only look at integrands which are adapted. That is, the value of {\alpha_t} can only depend on {X} through its values at prior times. This condition is met in most situations where we need to use stochastic calculus, such as with (forward) stochastic differential equations. To make this rigorous, for each time {t\ge 0} let {\mathcal{F}_t} be the sigma-algebra generated by {X_s} for all {s\le t}. This is a filtration ({\mathcal{F}_s\subseteq\mathcal{F}_t} for {s\le t}), and {(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},{\mathbb P})} is referred to as a filtered probability space. Then, {\alpha} is adapted if {\alpha_t} is {\mathcal{F}_t}-measurable for all times {t}. Piecewise constant and left-continuous processes, such as {\alpha} in (2), which are also adapted are commonly referred to as simple processes.

However, as with standard Lebesgue integration, we must further impose a measurability property. A stochastic process {\alpha} can be viewed as a map from the product space {{\mathbb R}_+\times\Omega} to the real numbers, given by {(t,\omega)\mapsto\alpha_t(\omega)}. It is said to be jointly measurable if it is measurable with respect to the product sigma-algebra {\mathcal{B}({\mathbb R}_+)\otimes\mathcal{F}}, where {\mathcal{B}} refers to the Borel sigma-algebra. Finally, it is called progressively measurable, or just progressive, if its restriction to {[0,t]\times\Omega} is {\mathcal{B}([0,t])\otimes\mathcal{F}_t}-measurable for each positive time {t}. It is easily shown that progressively measurable processes are adapted, and the simple processes introduced above are progressive.

With these definitions, the stochastic integral of a progressively measurable process {\alpha} with respect to Brownian motion {X} is defined whenever {\int_0^t\alpha^2ds<\infty} almost surely (that is, with probability one). The integral (1) is a random variable, defined uniquely up to sets of zero probability by the following two properties.

  • The integral agrees with the explicit formula (2) for simple integrands.
  • If {\alpha^n} and {\alpha} are progressive processes such that {\int_0^t(\alpha^n-\alpha)^2\,ds} tends to zero in probability as {n\rightarrow\infty}, then
    \displaystyle  \int_0^t\alpha^n\,dX\rightarrow\int_0^t\alpha\,dX, (3)

    where, again, convergence is in probability.

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