Properties of Thursday, March 10, 2011 2:08 PM

Readings: Shreve Secs. 3.1-3.4

Homework 2 due Friday, March 11 at 5 PM

No class on Monday, March 21. No office hours on Tuesday, March 22.

How does one show that a unique well-defined Wiener process satisfying the conditions stated before exists? ○ Use the Kolmogorov extension theorem, using the fact that the properties assert a consistent set of finite-dimensional distributions. The assertion of continuity implies separability and the uniqueness of the extension. Actually one needs the Kolmogorov criterion which relates properties of the finite-dimensional distributions to the quality of smoothness of the . It tells us in this case that the paths of the Wiener process, just based on being a separable extension of the prescribed finite dimensional distributions, are continuous and moreover Holder continuous with any exponent  . see Oksendal, Ch.2 and Simon, Sec. 5 ○ Formally define the Wiener process as a limit of random walks (Shreve Secs. 3.2, 3.3). ○ Define the Wiener process via a probability measure on function space (Wiener measure) . see Simon, Sec. 5 . this mathematical approach is related to a formal physical approach of defining Feynman path integrals to represent the □ see Simon, Sec. 1 and 2

The Wiener process is not just an example of a continuous-time stochastic process, but it is a fundamental stochastic process in the following sense:

Levy-Khintchine theorem: Any stochastic process with independent increments is (more or less) a combination of a constant drift, a rescaled Wiener process, and some discontinuous (which can be represented as a Levy process (generalized compound Poisson processes).)

Some example calculations involving Wiener processes:

The basic trick for such calculations is just to decompose all quantities into sums of increments, which are independent of each other.

2011 Page 1 Another example:

2011 Page 2 Moreover, we can say that Z(t) is also Gaussian because as an integral of a Gaussian stochastic process, it can be written as a limit of a sum of Gaussian random variables, and the limit preserves the Gaussianity of the sum under the technical condition of mean-square convergence (Oksendal App A) and this condition is met for Riemann sums of integrals of continuous Gaussian functions. Extending the argument, one can in fact say that not only is Z(t) a Gaussian random variable for each value of t but it is also a Gaussian stochastic process in the sense that any finite-dimensional distribution will be Gaussian.

Note actually that we could also have derived the statement:

by extending the bilinearity of the covariance:

And then by taking limits of Riemann sums, we have for any integrable stochastic processes X(t), Y(t):

2011 Page 3 While all these calculations have been relatively easy, the Wiener process does have one challenging aspect, which is that its paths are too rough for ordinary integration theory. To show this, we will begin by computing a quantity of fundamental interest in and particularly in mathematical finance,

Quadratic variation of a stochastic process X: (Shreve Sec. 3.4)

If this limit doesn't exist, the quadratic variation is undefined. Note that in principle, the quadratic variation is random, but it will typically wind up being deterministic.

The quadratic variation is related to financial quantity of realized volatility.

We will show that the quadratic variation for the Wiener process is:

2011 Page 4 The derivation is in two steps. First we compute the expected value of the quadratic variation and show that it is T, then we show that the variance of the quadratic variation is 0.

First compute the expectation. Consider any partition:

Next we show the variance of the sum over a partition goes to zero as the norm of the partition goes to zero.

To do this calculation, one uses the fact that

2011 Page 5 But one can just look up formulas for moments of Gaussian random variables, and one special case is that for a mean zero Gaussian random variable, the fourth moment is just three times the square of the variance:

Note that this subresult indicates that even as you look at the squared- increments of the Wiener process on small time scales, they behave in a fundamentally random way on any given small time interval. (Standard deviation is comparable in size to the mean).

Continuing with the calculation:

So what this calculation shows, is that while the square of a Wiener process increment

2011 Page 6 So what this calculation shows, is that while the square of a Wiener process increment remains fundamentally random when looked at over short time intervals, the accumulation over a finite time interval T of squares of Wiener process increments behaves deterministically. This is just a manifestation of the central limit theorem (or even just the strong law of large numbers) that a sum of a large number of independent random variables (under some mild technical conditions) behaves deterministically in the limit where the sum is large.

This fact will be the reason why, when we develop stochastic calculus, the following rule works:

What does this tell us about the roughness of Wiener process paths? First, we note that any smooth function has quadratic variation of zero.

More importantly, the Wiener process has infinite absolute variation (first-order variation):

This is bad news for Riemann-Stieltjes integration theory because its most general implementation requires that what you integrate against have bounded variation, and the Wiener process doesn't!

2011 Page 7