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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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 MATH (see the section ( Definition of conditional probability )) and a filtration $\QTR{cal}{F}_{t}$ (see the section ( Filtration definition )). Let MATH be the expectation associated with the measure $P$ . We introduce another expectation

MATH (Definition of change of measure)
defined for any $X_{t}$ and a particular $a_{t}$ , where $a_{t}$ and $X_{t}$ are continuous processes adapted to the filtration $\QTR{cal}{F}_{t}$ and $a_{0}=1$ . The Girsanov's theorem ( Girsanov_theorem ) hints that $a_{t}$ should be a positive martingale.

We would like to calculate MATH in terms of MATH . 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 MATH for any smooth function $\phi$ . Because MATH is an $\QTR{cal}{F}_{t}$ -adapted random variable, we apply the formula ( Definition_of_change_of_measure ) on the left-hand side: MATH On the right-hand side we apply the formula ( Definition_of_change_of_measure ) directly MATH Hence, MATH The operation MATH is the original measure, hence, the formula ( Chain_rule ) holds and the last expression is MATH Left and right sides are equal: MATH for any $\phi$ . We conclude

MATH (Main property of change of measure)
for $t<T$ . 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 MATH converging to the delta function around such point and obtain a contradiction.

Notation. Index. Contents.

Copyright 2007