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Math Finance I (Measure and Integration Theory (Conditional expectation…

- - - - Compliment belongs to F
      - Countable union belongs to F
      - Omega belongs to F
  - - - P(Omega)=1
      - P(countable union of disjoint sets in F) = sum of P(each individual set)
      - F->[0,1]
    - - Compliment is a null set
      - Null set: probability = 0
  - - - Partial averaging property holds
      - Measurable by the smaller sigma-algebra
    - - Radon-Nikodym's Theorem
        
        Absolutely continuous
        
        Equivalent measures
    - - Linear
      - for Y measurable by G and Y integrable w.r.t. P, E(XY|G)=YE(X|G)
      - sigma-algebras H<G<F, then E(E(X|G)|H)=E(X|H)
      - for X orthogonal to sigma-algebra G, E(X|G)=E(X)
      - Jensen's inequality: X, f(X) both integrable, f is convex, then E(f(X)|G)>f(E(X|G))
      - Monotone (dominate) convergence theorems hold
      - the independence lemma: let X orthogonal to Y, f(X,Y) integrable, then E(f(X,Y)|sigma(Y)) = E(f(X, y)|y=Y
  - - - P(X in A Intersect Y in B) = P(X in A)*P(Y in B)
      - Many equivalent statements
  - - - P_tilde(A) = \int_A ZdP=E[1_A Z]
      - Lemma for Y Fn measurable, Zn = E_n[Z], then E_tilde[Y] = E[Z_n Y]
- - - - Forwards
      - Options
  - - - Martingale
        
        adapted process
        
        E[|Mn|]<infinity
        
        E[M_(n+1)|F_n] = M_n
      - First theorem of asset pricing (part of it): exist P tilde s.t. S tilde is a martingale -> no arbitrage
      - Second theorem of asset pricing: exist only one P tilde s.t. S tilde is a martingale -> exist Delta_n n=1~N s.t. V_N can be replicated
      - S_n tilde is a martingale under P tilde
      - X_n tilde is a martingale under P tilde
      - X_0 = E tilde(V_N/(1+r)^N)
      - Levy martingales
        
        Give (Omega, F, P), sigma-algebras Fn < F(n+1) < F, random variable A measurable by F, A integrable, then E_n(A) is a martingale
        
        Radon-Nikogym process: Z > 0, E[Z] = 1, define Z_n = E_n[Z]
        
        Lemma: Z_n is a martingale
      - Markov process
        
        E[f(x_n+1)] = g(x_n)