Wednesday, September 18, 2013

Stoch Lesson 33 canonical Wiener variable ^ Gaussian variable

See Lesson 05 for the backgrounder on Level, Stepsize, time-varying random variable...
See Lesson 15 about TVRV
See Lesson 19 about N@T

In many formulas in this blog (and probably in the literature), W denotes not just some Wiener variable, but THE canonical TVRV random variable following a Wiener process a.k.a BM. Before we proceed it's good (perhaps necessary) to pick a concrete unit of time. Say 1 sec. Now I am ready to pin down THE canonical Wiener variable W in discrete-time --

   Over any time interval h seconds, the positive or negative increment in W's Level is generated from a Gaussian noisegen, with mean 0 and variance equal to h. This makes W THE canonical Wiener variable. [1]

Special case - If the interval is from last observation, when Level is 0, to 55 sec later, then dW = W(t=55) - 0 = W(t=55), and therefore W@55sec, as a N@T, also has a Gaussian distribution with variance = 55.

[1] I think this is the discrete version of Standard Brownian Motion or SBM, defined by Lawler on P42 with 2+1 defining properties -- 1) iid random increments 2) no-jump, which __implies__ 3) Gaussian random increments


Now let's look at the standard normal distro or canonical Gaussian distro or Gaussian noisegen -- If something epsilon follows a canonical Gaussian distribution, it's often a N@T, which is not a time-varying random variable. Also the variance and stdev are both 1.0.

I believe the canonical Wiener variable can be expressed in terms of the canonical Gaussian variable --

  deltaW = epsilon * sqrt(deltaT)  //  in discrete time
  dW = epsilon * sqrt(dT)            //  in continuous time

Let's be concrete and suppose deltaT is 0.3 yoctosecond (more brief than any price movement). In English, this says "over a brief 0.3 yoctosecond, step_size is generated from a Gaussian noisegen with variance equal to 0.3 * 10^-24". If we simulate this step 9999 times, we would get 9999 deltaW (stesp_size) realization values. These realizations would follow a bell-shaped histogram.

Given dW can be expressed this way, many authors including Hull uses it all the time.

Both the canonical Wiener variable and the canonical Gaussian distribution have their symbols -- W vs epsilon(ϵ), or sometimes Z. They show up frequently in formulas. Don't confuse them.

The Wiener var is always a TVRV; the Gaussian var is often a N@T.

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