Dynamic Programming

  1. solve O-U Process

Questions

Easy
  1. Gambler's Ruin Problem
  2. Ant Go on the surface of a box, expected step of going back/ reaching the other end
  3. Drunk man standing at the 17m of a bridge
  4. Dice game - 1,2,3 paid, 4,5,6 roll again- expected payoff (Wald's Equality)
  5. Medium
  6. Hard

  7. Coin Triplets

    1. Probability
    2. THH before HHH/ HHH before THH
    3. Best Strategy

  8. Color Balls- n balls in a jar, paint two similar each time ( second after first), expected steps to be the same color

  9. Ticket line- 2n people waiting with 5 and 10 bills, what is the chance of no extra changes

Stochastic Process

  1. 1.6 Martingale & Markov Process

Martingale

Stopping Times (function that the event is in filtration)

Dobb’s Optional Sampling Theorem

Markov Process

Super martingale, Sub martingale

  1. Browian Motion

    Symmetric Random Walk

    Variance, Expectation,

    First Variation (use Mean-Value Theorem when it is differentiable)

    Quadratic Variation (Attention: it is a Sum)

    Brownian Motion

    continuous

    Stationary

    Independent increment,

    normal increments, variation

    Levy’s Theorem (Levy’s Criterion)

  2. Ito Integral

    Martingale Property

    Ito Isometry

    Quadratic Variation

    Ito-Doubelin formula

    Browian filtration

    F(s)< F(t), Browian motion measurable at time t.

    Martingles

    B,

    F(t,W) - \int_o^t (\diff_t f(s,W)- ½ \diff^2 f(s,W)) ds (Ito’s Lemma)

  3. Black-Scholes

  4. Multi-dimensional

Joint Quadratic Variation

Multi-dimensional Ito’s Formula

Multi-dimensional Browian Motion

Multi-Dimensional Levy’s Theorem

One-dimensional, multi-dimensional

  1. Risk-Neutral Pricing

Random-Nikoym Theorem

Girsanov Theorem

Martingale Representation Theorem

Futures and Forward

  1. Relationship with P.D.E

Transition Density

Markov Process / Markov Property

Feynman - Kac Theorem

Kolmogrov Forward Equation

Kolmogrov Backward Equation

Dupire’s Equation - Volatility

Delta Hedge

under different rates

with dividend (rate q)

  1. Fixed Income

Short Rate Models

Hoo-Lee, Vasicek, Hull-White, CIR

Yield, Yield Curve

Forward Interest Rate - instaneous interest rate

LIBOR

Whole Yield Models

Heath–Jarrow–Morton

  1. Change of Measure/ Change of Numeraire

    Multi-dimensionaly Levy’s Theorem

    Girsanov Theorem (Multi-dimensional)

    attention: used on independent Browian Motions

    eg. multi-factor interest models/ volatility models

Forward Measure

numeraire

* Quotient of Martingales is a martingale using the denominator as the numeraire

* The volatility ( vol - R-N Derivative) is the numerators’ vol vector - denominator’s vol vector

———————

做题技巧

  1. Tower Property

  2. For(t,t) = St , f(t,t) = Rt ( the underlying asset price)

  3. Change Measure A-B-C by Zt = *numeraire B/ Numeraire A (with normalization)

  4. 凑 exponential martingale，凑Black - Scholes Form/ Black’s Form

  5. MGF (构造

  6. 做差

  7. Conditional Expectation- Tower Property

  8. Ito’s Formula

Important Martingales

M^2- [M,M]t

MN - [M,N]

exp (M^2 - [M,M]t/2)

Ito f - \int_0^t ( df/dt +d^2f/dx^2)

Martingale stopped at stopping time

概率：构造partition， patrial averaging

正态： decomposition

martingale ：作差

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