Stochastic Processes Edited: February 2011 Page 1 Professor: Nina Kajiji Stochastic Processes Stochastic Process – Non Formal Definition: Non‐formal: A stochastic process (random process) is the opposite of a deterministic process such as one defined by a differential equation. A stochastic process deals with more than one possible reality of how a process might evolve. This means that even if the starting point (initial condition) is known there are many paths that a process may follow – some are more probable than others. For processes in time, a stochastic process is simply a process that develops in time according to probabilistic rules. Stochastic Process – Formal Definition Stochastic Process (X) is a family of random variables, dependent upon a parameter which usually denotes time (T) and defined on some sample space (Ω). Mathematically, {X t ,t є T} = { X t (ω), t є T, ωєΩ} Of course the parameter does not have to always denote time. It could be a vector representing location in space. In such a case the process will represent a random variable that varies across two‐dimensional space. In our case we will not delve into this level of details.
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Stochastic Processes Edited: February 2011 Page 1 Professor: Nina Kajiji
Stochastic Processes
Stochastic Process – Non Formal Definition:
Non‐formal: A stochastic process (random process) is the opposite of a deterministic
process such as one defined by a differential equation. A stochastic process deals with
more than one possible reality of how a process might evolve. This means that even if
the starting point (initial condition) is known there are many paths that a process may
follow – some are more probable than others.
For processes in time, a stochastic process is simply a process that develops in time
according to probabilistic rules.
Stochastic Process – Formal Definition
Stochastic Process (X) is a family of random variables, dependent upon a parameter
which usually denotes time (T) and defined on some sample space (Ω).
Mathematically,
{Xt, t є T} = { Xt(ω), t є T, ω є Ω}
Of course the parameter does not have to always denote time. It could be a vector
representing location in space. In such a case the process will represent a random
variable that varies across two‐dimensional space. In our case we will not delve into
this level of details.
Stochastic Processes Edited: February 2011 Page 2 Professor: Nina Kajiji
Stochastic Process – Discrete Time
When T is a set of integers, representing specific time points we have a stochastic
process in discrete time. In this case we generally define the random variable as Xn.
The random variable Xn will depend on earlier values of the process,
That is: Xn‐1, Xn‐2, …
Stochastic Processes Edited: February 2011 Page 3 Professor: Nina Kajiji
Stochastic Process – Continuous Time
When T is the real line (or some interval of the real line) we have a stochastic process in
continuous time. We will focus on this definition for our study. The random variable
X(t) will depend on values of X(u) for u < t.
Stochastic Processes Edited: February 2011 Page 4 Professor: Nina Kajiji
Examples of Stochastic Processes
Random Walk Models – such as exchange rate data. In a random walk model
changes in the “rate” are independent normal random variables with zero mean
and standard deviation of the actual data. In essence, the upward and
downward movement in the “rate” is equally likely and there is no scope for
profiteering by speculation except by luck.
Aside:
J.P. Morgan’s famous stock market prediction was that, “Prices will fluctuate.”
Bachelier’s Theory of Speculation in 1900 postulated that prices fluctuate
randomly.
These models make sense in a world where:
1. Most price changes result from temporary imbalances between buyers and
sellers.
2. Stronger price shocks are unpredictable.
3. Under efficient capital market hypothesis the current price of a stock reflects
all information about it.
Stochastic Processes Edited: February 2011 Page 5 Professor: Nina Kajiji
These models do not make sense if:
1. One believes in technical analysis
2. Random walk models assume that returns are normally or log‐normally
distributed. Thus, frequency of extreme events is underestimated.
NOTE: These models will be the focus of our study.
Poisson Processes – such as photon emissions. A photon is a minute particle of
light measured by a special machine. The problem – the machine has a dead
time period. That is, after it counts the photon it has to recharge before it can
count the next photon. Therefore, the counts are underestimated. A stochastic
model would need to be formulated that not only deals with the emissions but
also the dead time based on some probability of occurrence. A homogenous
poisson process satisfies:
o Starts at zero
o It is stationary, and has independent increments
o For every t > 0, X(t) has a poisson distribution
NOTE: Because poisson processes are counting processes they inherently have a
jump.
Epidemics – such as SARS. The disease is detected, spread, and eventually
controlled to eradicate it. The question arises, how soon will it spread. How
widespread will it be. How many should you vaccinate – because some will die
anyway so the ultimate goal of eradication is reached. The graphs generally have
a peak when the epidemic is at its peak.
Stochastic Processes Edited: February 2011 Page 6 Professor: Nina Kajiji
Point Processes – such as Earthquakes. Empirically we say that on average
there is one earthquake per year. However, in actuality there are years when
there have been no recorded earthquakes. Thus, there is a clustering effect in
the processes. The question would be, “is it related to magnitude?” “is it strictly