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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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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 MATH under some probability measure and filtration $\QTR{cal}{F}_{t}$ where the $X_{t}$ and $Y_{t}$ are $\QTR{cal}{F}_{t}$ -adapted processes. We are looking for a process $a_{t}$ that would deliver the property MATH Hence, we would like to have MATH The last expression may be regarded as the formula ( Main property of the change of measure ) with

MATH (Common application of change of measure)
The normalization by MATH follows from the requirement $a_{0}=1$ . Note that such $a_{t}$ is a martingale. In addition, the $a_{t}$ must be positive. Thus, we also require that $X_{t}$ would not change sign.

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

Notation. Index. Contents.

Copyright 2007