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Quantitative Analysis
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Numerical Analysis
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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 $X_{t}$ is given by MATH 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 $dW_{t}$ is a column of standard Brownian motions, $\sigma_{t}$ is a column of $\QTR{cal}{F}_{t}$ -adapted stochastic processes and $\mu_{X,t}$ is a one-dimensional $\QTR{cal}{F}_{t}$ -adapted stochastic process. We introduce some process $a_{t}$ given by the equation MATH for some processes $\sigma_{a,t}$ . We would like to find SDE representation of $X_{t}$ with respect to the probability measure changed with $a_{t}$ 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 $\tilde{\mu}_{X,t}$ and MATH such that MATH where the $d\tilde{W}_{t}$ is an Ito process with the properties MATH where the $I$ is the unit matrix and the operation MATH is given by the formula ( Definition of change of measure ). Hence, we look for $\tilde{\mu}_{X,t}$ such that MATH The $dX_{t}$ is MATH -measurable: MATH . Hence MATH Note that MATH = MATH = MATH . MATH MATH We conclude that the "changed" drift is given by the recipe MATH We calculate the diffusion term using a similar approach: MATH MATH We keep only the terms of $dt$ -magnitude, see the section ( Ito calculus ). MATH Hence, MATH

We substitute the (*) and (**) into the $a_{t}$ -equation for MATH MATH MATH and conclude

MATH (General change of Brownian motion)

Summary

(Transformation of SDE under change of measure) Let $W_{t}$ be a column of standard Brownian motions adapted to $\QTR{cal}{F}_{t}$ and a positive-valued process $a_{t}$ is given by the SDE MATH where the $\sigma_{a,t}$ is a column of $\QTR{cal}{F}_{t}$ -adapted positive processes. Under the probability measure given by MATH , (see ( Definition_of_change_of_measure )), the process $dW_{t}$ has the SDE MATH for a column $\tilde{W}_{t}$ of some standard Brownian motions under the probability measure defined by MATH .





Notation. Index. Contents.


















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