- Martingale Representation Theorem
- Lévy Theorem
- Girsanov Theorem
- 1st Fundamental Theorem of Asset Pricing
- 2nd Fundamental Theorem of Asset Pricing
Notations and Ito Formula
Multiplication table
ㅤ | ||
0 | 0 | |
0 |
Ito Formula
Denote , , and is a real function of real variable, then
where .
For example,
Martingale Representation Theorem
Informally, the sources of uncertainty is from Brownian motions (). Then any martingale w.r.t. filtration of is sum of deterministic and stochastic integrals w.r.t. .
Suppose are independent BMs on the space . is generated by on . Denote as a martingale w.r.t. , that is
then
Lévy Theorem
Same settings as above. Denote are two martingales s.t. . Also assume , , , and are independent standard BMs.
Proof
If , then for any . And thus
if , then
if , then
Besides, since
and are independent (because their Generation Function are independent).
Girsanov Theorem
The theorem explains how stochastic processes change under changes in measure.
For example, consider 3 tosses of a coin. Denote as the probability of getting a H on toss i, which are independent for 3 tosses and .
The the probability for each output is as below
HHH | ||
HHT | … | |
HTH | … | |
HTT | … | |
THT | … | |
TTH | … | |
TTT |
Denote as the number of H’s in the first 2 tosses, then
Radon–Nikodym Random Variable () is defined as, for example,
Radon-Nikodym Martingale:
transformation
- Expectation:
- Condition Expectation and Bayes Rule: suppose is -measurable, the
where is -measurable.
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