A major difference between standard integral calculus and stochastic calculus is the existence of quadratic variations and covariations. Such terms show up, for example, in the stochastic version of the integration by parts formula.
For motivation, let us start by considering a standard argument for differentiable processes. The increment of a process over a time step can be written as . The following identity is easily verified,
Now, divide the time interval into equal parts. That is, set for . Then, using and summing equation (1) over these times,
If the processes are continuously differentiable, then the final term on the right hand side is a sum of terms, each of order , and therefore is of order . This vanishes in the limit , leading to the integration by parts formula
Now, suppose that are standard Brownian motions. Then, are normal random variables with standard deviation . It follows that the final term on the right hand side of (2) is a sum of terms each of which is, on average, of order . So, even in the limit as goes to infinity, it does not vanish. Consequently, in stochastic calculus, the integration by parts formula requires an additional term, which is called the quadratic covariation (or, just covariation) of and . Continue reading “Quadratic Variations and Integration by Parts”
In the previous two posts I gave a definition of stochastic integration. This was achieved via an explicit expression for elementary integrands, and extended to all bounded predictable integrands by bounded convergence in probability. The extension to unbounded integrands was done using dominated convergence in probability. Similarly, semimartingales were defined as those cadlag adapted processes for which such an integral exists.
The current post will show how the basic properties of stochastic integration follow from this definition. First, if is a cadlag process whose sample paths are almost surely of finite variation over an interval , then can be interpreted as a Lebesgue-Stieltjes integral on the sample paths. If the process is also adapted, then it will be a semimartingale and the stochastic integral can be used. Fortunately, these two definitions of integration do agree with each other. The term FV process is used to refer to such cadlag adapted processes which are almost surely of finite variation over all bounded time intervals. The notation represents the Lebesgue-Stieltjes integral of with respect to the variation of . Then, the condition for to be -integrable in the Lebesgue-Stieltjes sense is precisely that this integral is finite.
Lemma 1 Every FV process is a semimartingale. Furthermore, let be a predictable process satisfying
almost surely, for each . Then, and the stochastic integral agrees with the Lebesgue-Stieltjes integral, with probability one.
Continue reading “Properties of the Stochastic Integral”
Having covered the basics of continuous-time processes and filtrations in the previous posts, I now move on to stochastic integration. In standard calculus and ordinary differential equations, a central object of study is the derivative of a function . This does, however, require restricting attention to differentiable functions. By integrating, it is possible to generalize to bounded variation functions. If is such a function and is continuous, then the Riemann-Stieltjes integral is well defined. The Lebesgue-Stieltjes integral further generalizes this to measurable integrands.
However, the kinds of processes studied in stochastic calculus are much less well behaved. For example, with probability one, the sample paths of standard Brownian motion are nowhere differentiable. Furthermore, they have infinite variation over bounded time intervals. Consequently, if is such a process, then the integral is not defined using standard methods.
Stochastic integration with respect to standard Brownian motion was developed by Kiyoshi Ito. This required restricting the class of possible integrands to be adapted processes, and the integral can then be constructed using the Ito isometry. This method was later extended to more general square integrable martingales and, then, to the class of semimartingales. It can then be shown that, as with Lebesgue integration, a version of the bounded and dominated convergence theorems are satisfied.
In these notes, a more direct approach is taken. The idea is that we simply define the stochastic integral such that the required elementary properties are satisfied. That is, it should agree with the explicit expressions for certain simple integrands, and should satisfy the bounded and dominated convergence theorems. Much of the theory of stochastic calculus follows directly from these properties, and detailed constructions of the integral are not required for many practical applications. Continue reading “The Stochastic Integral”