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.

## Definition of change of measure.

We are given a probability space (see the section ( Definition of conditional probability )) and a filtration (see the section ( Filtration definition )). Let be the expectation associated with the measure . We introduce another expectation

 (Definition of change of measure)
defined for any and a particular , where and are continuous processes adapted to the filtration and . The Girsanov's theorem ( Girsanov_theorem ) hints that should be a positive martingale.

We would like to calculate in terms of . We noted in the section ( Filtration and conditional expectation ) that the formula ( Chain rule ) may regarded as a definition of conditional probability. Hence, we require for any smooth function . Because is an -adapted random variable, we apply the formula ( Definition_of_change_of_measure ) on the left-hand side: On the right-hand side we apply the formula ( Definition_of_change_of_measure ) directly Hence, The operation is the original measure, hence, the formula ( Chain_rule ) holds and the last expression is Left and right sides are equal: for any . We conclude

 (Main property of change of measure)
for . The proof of the last step is a standard analysis argument. We express the expectations in terms of integrals with respect to the corresponding distributions. If there is a point where the desired property is untrue then we take a sequence of converging to the delta function around such point and obtain a contradiction.

 Notation. Index. Contents.