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SCFII-1 Big Theorems

  • 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
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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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