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 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.
 A. Definition of change of measure.
 B. Most common application of change of measure.
 C. Transformation of SDE under 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.

Most common application of change of measure.

e are facing evaluation of the expectation under some probability measure and filtration where the and are -adapted processes. We are looking for a process that would deliver the property Hence, we would like to have The last expression may be regarded as the formula ( Main property of the change of measure ) with

 (Common application of change of measure)
The normalization by follows from the requirement . Note that such is a martingale. In addition, the must be positive. Thus, we also require that would not change sign.

Another way to arrive to the same conclusion would be to write

 Notation. Index. Contents.