The fundamental object of probability theory is a set
that is divided into a collection,
, of measurable subsets.
A real-valued random variable a is a measurable function
on . This means, in particular, any open set on the real line
is the image of a member of
The probability measure
assigns a real number to each member of
. This framework is represented schematically as
With the probability measure defined on , it is possible to define, for a real-valued random variable a, the probability that a<x. This is an ordinary function of x given by
The density of (a ) (if it exists)
is the derivative of this function with respect to x:
This is sometimes put as: R_a(x)dx is the probability that a lies in (x,x+dx).
A notion of time is provided
by defining an increasing family
of sub sigma-algebras on
A stochastic process is a family of random variables indexed by t such that the random variable is -measurable for each t. Heuristically, X is a function of t and :
and each contains all events which can occur before or at time t.
The relationship between the formal construction of a
stochastic process and the SDE notation (1) is as follows.
The coefficient f(x,t) is the mean displacement
if =x then
The informal way to understand the second term in the SDE (1) is to write
More correctly, make the following construction. Define sets of times ( ; i=0,1,...,l) such that 0=t_0<t_1<...<t_l=t and - = t/l. Choose any path of X and let
Then [X], called the quadratic variation of X, satisfies the SDE
Note that if is constant and X_0=0 then [X]_t is proprtional to t. This is true of any path of with probability one.
The existence of the limit (10) is
a consequence of the roughness of paths and
the basis of stochastic calculus [1-6].
and Y are stochastic processes
obeying SDEs of the form (1), then
the Itô integral,
is itself a stochastic process given by the following limit:
Its quadratic variation is
As suggested by (9), second order infinitesimals are not always negligible in stochastic calculus. This is reflected in the Itô formula, which is the chain rule of stochastic calculus. The SDE for f( X), where f is a C^2 function, is related to that for X by [1-6]
d f(X) = f'(X)d X + 1/2f''(X)d[X].
An alternative definition of the stochastic integral is also used, in which t_i in (13) is replaced by t_s, where t_s=(t_i+t_i+1). If this alternative definition, called the Stratonovich integral, is chosen then changes of variables can be performed without the extra term that appears in (15). The extra term reappears, however, in the update formula or numerical algorithm. In the Stratonovich interpretation, the Euler algorithm corresponding to (1) is not (2), but
In any situation, the Itô and Stratonovich conventions can be used. One can change at will from one to the other using the following. The SDE (1) interpreted with the Stratonovich convention is equivalent to the Itô SDE
Having introduced the differential notation (1), it is natural to ask under what conditions is there a solution? When f and are ordinary functions, the conditions are like those for ordinary differential equations: a Lipschitz condition on f and is necessary for existence and a growth condition to ensure that trajectories don't go to infinity in a finite time. That a solution exists means, given a path of the Wiener process, a unique path is obtained. (For SPDEs a weaker definition than this is often useful.) The initial condition can be a random variable provided it has finite mean square. ( More generally, f and can be functionals of the whole path of up to time t.)
Forward References on Stochastic calculus