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## Saturday, July 13, 2013

### Weiner process, better understood in discrete time

[[Hull]] presents a generalized Wiener process

dx = a dt + b dz

I now feel this equation/expression/definition is easier understood in discrete time. Specifically, x is a random variable, so its
Future value is unknown so we want to predict it with a pdf (super-granular histogram). Since x changes over time, we must clarify
our goal -- what's the probability distribution of x a a time t a short while later? I feel this question is best answered in
discrete time. So we throw out dt and dz. (As a Lazy guy U don't even need delta_t and delta_z).

Let's make some safe simplifying assumptions : a = 0; b = 1 and last observation is x = 0. These assumptions reduce x to a Weiner
variable (i.e. x follows a Weiner process). At at a (near) future t units[1] away, we predict x future value with a normal
distribution whose stdev=sqrt(t).

[1] time is measured in years by custom

Now, What if I want to estimate the rate of change ("slope" of the chart) i.e. dx/dt? I don't think we can, because this is stoch
calculus, not ordinary calculus. I am not sure if we can differentiate or integrate both sides.