Notation and Definitions.

The fundamental object of probability theory is a set equation1026 that is divided into a collection, equation1026 , of measurable subsets. A real-valued random variable a is a measurable function on equation1026. This means, in particular, any open set on the real line is the image of a member of equation1026. The probability measure equation1026 assigns a real number to each member of equation1026. This framework is represented schematically as follows [1]:

With the probability measure defined on equation1026, 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 tex2html_wrap_inline942. of sub sigma-algebras on tex2html_wrap_inline944.
A stochastic process tex2html_wrap_inline958 is a family of random variables indexed by t such that the random variable tex2html_wrap_inline950 is tex2html_wrap_inline954-measurable for each t. Heuristically, X is a function of t and tex2html_wrap_inline950:
and each tex2html_wrap_inline950 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 or `drift': if tex2html_wrap_inline950=x then  equation1045
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 tex2html_wrap_inline950 is constant and X_0=0 then [X]_t is proprtional to t. This is true of any path of tex2html_wrap_inline950 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]. If tex2html_wrap_inline950 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

where tex2html_wrap_inline1699.

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 tex2html_wrap_inline1703 are ordinary functions, the conditions are like those for ordinary differential equations: a Lipschitz condition on f and tex2html_wrap_inline1703 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 tex2html_wrap_inline954 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 tex2html_wrap_inline1703 can be functionals of the whole path of tex2html_wrap_inline1715 up to time t.)

Next Forward References on Stochastic calculus