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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.

## Transformation of SDE under change of measure.

uppose a one-dimensional process is given by with respect to some probability measure that we call "the original measure" in light of the considerations of the section ( Definition of change of measure ). Here and everywhere below is a column of standard Brownian motions, is a column of -adapted stochastic processes and is a one-dimensional -adapted stochastic process. We introduce some process given by the equation for some processes . We would like to find SDE representation of with respect to the probability measure changed with in the sense of formula ( Definition of change of measure ). To obtain the recipe for such calculation we rephrase the requirement as follows. We wish to find the processes and such that where the is an Ito process with the properties where the is the unit matrix and the operation is given by the formula ( Definition of change of measure ). Hence, we look for such that The is -measurable: . Hence Note that = = . We conclude that the "changed" drift is given by the recipe We calculate the diffusion term using a similar approach: We keep only the terms of -magnitude, see the section ( Ito calculus ). Hence,

We substitute the (*) and (**) into the -equation for and conclude

 (General change of Brownian motion)

Summary

(Transformation of SDE under change of measure) Let be a column of standard Brownian motions adapted to and a positive-valued process is given by the SDE where the is a column of -adapted positive processes. Under the probability measure given by , (see ( Definition_of_change_of_measure )), the process has the SDE for a column of some standard Brownian motions under the probability measure defined by .

 Notation. Index. Contents.