Index

abstract Wiener space, 45

anticipating stochastic integrals, 55

Black-Scholes model, 75

Brownian motion, 1

Brownian sheet, 42

Cameron-Martin space, 45

Cauchy semigroup, 25

central limit theorem, 67

chain rule, 5

chaotic central limit theorem, 70

Clark-Ocone formula, 17

contraction of order r, 67

covariant derivative, 49

Delta, 77

derivative operator, 3

difussion process, 45

divergence operator, 13

duality relationship, 13

ellipticity conditions, 48

European options, 77

forward stochastic integral, 58

fractional Brownian motion, 59

Gamma, 77

Gaussian white noise, 1

Greeks, 76

integration-by-parts formula, 3

Itˆ o stochastic integral, 16

Hermite Polynomials, 1

H¨ ormander’s conditions, 49

H¨ ormander’s theorem, 50

hypercontractivity, 20

Lie bracket, 49

local property of the derivative, 40

local property of the divergence, 56

local time, 18

Malliavin matrix, 33

mean return rate, 75

Mehler’s formula, 19

Meyer inequalities, 27

multiple stochastic integrals, 9

nondegenerate random vector, 35

Norris lemma, 50

Ornstein-Uhlenbeck process, 20

Ornstein-Uhlenbeck semigroup, 19

Picard approximations, 46

progressively measurable process, 16

reduced Malliavin matrix, 48

reproducing kernel Hilbert space, 42

Riemansn sums, 57

slef-financing portfolio, 75

self-similarity, 59

Skorohod integral, 55

smooth and cylindrical random variables, 3

Sobolev spaces, 29

Stein’s equation, 72

Stein’s method, 72

stochastic differential equation, 45

Stratonovich integral, 49

Stroock’s formula, 10

support of the law, 39

two-parameter Wiener process, 42

Vega, 77

volatility, 75

nth Wiener chaos, 3

Wiener chaos expansion, 6

Wiener measure, 45

85