Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 1 Conditional probability.
 2 Normal distribution.
 3 Brownian motion.
 4 Poisson process.
 5 Ito integral.
 6 Ito calculus.
 7 Change of measure.
 8 Girsanov's theorem.
 9 Forward Kolmogorov's equation.
 10 Backward Kolmogorov's equation.
 11 Optimal control, Bellman equation, Dynamic programming.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Ito integral. iven a filtration , Ito integral is initially defined for "simple" processes of the form where the variables are measurable for each and See the section ( Filtration_definition_section ) for the notations and . In other words, the value of remains constant during and it is known with certainty at . For such the stochastic (Ito) integral is defined as where is a standard Brownian motion adapted to the filtration . For each the is measurable while the is measurable and -independent. Hence, and are independent random variables for each . This observation is the key tool when doing anything with stochastic integral. In particular, we use such observation when proving the following properties.

Proposition

1. is an adapted martingale.

2. Ito isometry. .

Proof

By definition, We use the formula ( Chain_rule ):    Proposition

(Chebyshev's inequality). Proof

See the formula ( Chebyshev inequality ) with .

With help of the Ito isometry and Chebyshev's inequality we expand the notion of stochastic integral to the class of adapted processes with the property We approximate (in sense) any such process with a Cauchy sequence of simple processes. Then the stochastic integral of the process is a limit in probability of the Cauchy sequence of the simple integrals.

 Notation. Index. Contents.