SYLLABUS Previous: 5.1 Monte Carlo integration
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This section is intended as a very short introduction into the stochastic theory, which provides the mathematical foundation behind the Monte Carlo method; check Sczepessy , Kloeden  and Van Kampen  for complete courses on the subject.
where is the density distribution function of a stochastic variable . The notation refers to the set of Gaussian or normally distributed stochastic variable with a density distribution function
where is the mean and is the variance. Along the same lines, stands for the set of uniformly distributed random numbers in the interval .
It turns out that the distribution of Wiener increments is essentially the same function as the Green's function of the diffusion equation (1.3.2#eq.3).
The differential calculus of stochastic processes (or Itô calculus) involves new properties that are fundamentally different from the ordinary (or Riemann) calculus. The reason can be tracked down to the preferred direction of time in the Itô integral, which is defined as the limit of the explicit (forward Euler) discretization
for any sequence . Note that the circle here stands for Itô differential and states that is independent of ; also, an implicit discretization in (5.2#eq.5) would give a fundamentally different result.
Define a property that is a sum of a Riemann and an Itô integral
For an infinitely short time interval, this leads to the stochastic differential equation
which reduces to an ordinary differential equation in the absence of the stochastic component
Stochastic differential equations such as (5.2#eq.7) are used in the Monte-Carlo method to describe the trajectories or orbits of particles.
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