Wednesday, January 28, 2015

simplest SDE (not PDE) - given by Greg

P91 of Greg Lawler's lecture notes states that the most basic, simple SDE

  dXt = At dBt     (1)

can be intuitively interpreted this way -- Xt is like a process that at time t evolves like a BM with zero drift and variance At2. 

In order to make sense of it, let's back track a bit. A regular BM with 0 drift and variance_parameter = 33 is a random walker. At any time like 64 days after the start (assuming days to be the unit of time), the walker still has 0 drift and variance_param=33. The position of this walker is a random variable ~ N(0, 64*33). However, If we look at the next interval from time 64 to 64.01, the BM's increment is a different random variable ~ N(0, 0.01*33).

This is a process with constant variance parameter. In contrast, our Xt process has a ... time-varying variance parameter! This random walker at time 64 is also a BM walker, with 0 drift, but variance_param= At2. If we look at the interval from time 64 to 64.01, (due to slow-changing At), the BM's increment is a random variable ~ N(0, 0.01At2).

Actually, the LHS "dXt" represents that signed increment. As such, it is a random variable ~ N(0, dt At2).

Formula (1) is another signal-noise formula, but without a signal. It precisely describes the distribution of the next increment. This is as precise as possible.

Note BS-E is a PDE not a SDE, because BS-E has no dB or dW term.

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