3
8.2
isalmostsurelyfinite................................ 97
8.3 The moment generating function for
........................ 99
8.4 Expectation of
.................................... 100
8.5 TheStrongMarkovProperty............................. 101
8.6 GeneralFirstPassageTimes ............................. 101
8.7 Example:PerpetualAmericanPut .......................... 102
8.8 DifferenceEquation.................................. 106
8.9 DistributionofFirstPassageTimes.......................... 107
8.10TheReflectionPrinciple ............................... 109
9 Pricing in terms of Market Probabilities: The Radon-Nikodym Theorem. 111
9.1 Radon-Nikodym Theorem .............................. 111
9.2 Radon-Nikodym Martingales . . . .......................... 112
9.3 TheStatePriceDensityProcess ........................... 113
9.4 Stochastic Volatility Binomial Model ......................... 116
9.5 Another Applicaton of the Radon-Nikodym Theorem . ............... 118
10 Capital Asset Pricing 119
10.1AnOptimizationProblem............................... 119
11 General Random Variables 123
11.1 Law of a Random Variable .............................. 123
11.2 Density of a Random Variable . . .......................... 123
11.3Expectation...................................... 124
11.4 Two random variables ................................. 125
11.5MarginalDensity ................................... 126
11.6ConditionalExpectation ............................... 126
11.7ConditionalDensity.................................. 127
11.8MultivariateNormalDistribution........................... 129
11.9Bivariatenormaldistribution............................. 130
11.10MGF of jointly normal random variables ....................... 130
12 Semi-Continuous Models 131
12.1Discrete-timeBrownianMotion ........................... 131
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