MALLIAVI随机分析和相关论题 (美)纽勒特 著作 文教 文轩网

Introduction
1 Analysis on the Wiener space
1.1 Wiener chaos and stochastic integrals
1.1.1 The Wiener chaos decomposition
1.1.2 The white noise case: Multiple Wiener-It5 integrals.
1.1.3 It5 stochastic calculus
1.2 The derivative operator
1.2.1 The derivative operator in the white nois case
1.3 The divergence operator
1.3.1 Properties of the divergence operator
1.3.2 The Skorohod integral
1.3.3 The It5 stochastic integral as a particular case of the Skorohod integral
1.3.4 Stochastic integral representation of Wiener functionals
1.3.5 Local properties
1.4 The Ornstein-Uhlenbeck semigroup
1.4.1 The semigroup of Ornstein-Uhlenbeck
1.4.2 The generator of the Ornstein-Uhlenbeck semigroup
1.4.3 Hypercontractivity property and the multiplier theorem
1.5 Sobolev spaces and the equivalence of norms
2 Regularity of probability laws
2.1 Regularity of densities and related topics
2.1.1 Computation and estimation of probability densities
2.1.2 A criterion for absolute continuity based on the integration-by-parts formula
2.1.3 Absolute continuity using Bouleau and Hirsch's ap- proach
2.1.4 Smoothness of densities
2.1.5 Composition of tempered distributions with nonde-generate random vectors
2.1.6 Properties of the support of the law
2.1.7 Regularity of the law of the maximum of continuous processes
2.2 Stochastic differential equations
2.2.1 Existence and uniqueness of solutions
2.2.2 Weak differentiability of the solution
2.3 Hypoe|lipticity and HSrmander's theorem
2.3.1 Absolute continuity in the case of Lipschitz coefficients
2.3.2 Absolute continuity under HSrmander's conditions .
2.3.3 Smoothness of the density under HSrmander's condition
2.4 Stochastic partial differential equations
2.4.1 Stochastic integral equations on the plane
2.4.2 Absolute continuity for solutions to the stochastic heat equation
3 Anticipating stochastic calculus
3.1 Approximation of stochastic integrals
3.1.1 Stochastic integrals defined by Riemann sums
3.1.2 The approach based on the L2 development of the process
3.2 Stochastic calculus for anticipating integrals
3.2.1 Skorohod integral processes
3.2.2 Continuity and quadratic variation of the Skorohod integral
3.2.3 It6's formula for the Skorohod and Stratonovich integrals
3.2.4 Substitution formulas
3.3 Anticipating stochastic differential equations
3.3.1 Stochastic differential equations in the Sratonovich sense
3.3.2 Stochastic differential equations with boundary con-ditions
3.3.3 Stochastic differential equations in the Skorohod sense
4 Transformations of the Wiener measure
4.1 Anticipating Girsanov theorems
4.1.1 The adapted case
4.1.2 General results on absolute continuity of transformations
4.1.3 Continuously differentiable variables in the direction of H1
4.1.4 Transformations induced by elementary processes.
4.1.5 Anticipating Cirsanov theorems
4.2 Markov random fields
4.2.1 Markov field property for stochastic differential equations with boundary conditions
4.2.2 Maxkov field property for solutions to stochastic partial differential equations
4.2.3 Conditional independence and factorization properties
5 Fractional Brownian motion
5.1 Definition, properties and construction of the fractional Brownian motion
5.1.1 Semimartingale property
5.1.2 Moving average representation
5.1.3 Representation of iBm on an interval
5.2 Stochastic calculus with respect to fBm
5.2.1 Malliavin Calculus with respect to the ibm
5.2.2 Stochastic calculus with respect to ibm. Case H
5.2.3 Stochastic integration with respect to iBm in the case H
5.3 Stochastic differential equations driven by a ibm
5.3.1 Generalized Stieltjes integrals
5.3.2 Deterministic differential equations ;
5.3.3 Stochastic differential equations with respect to ibm
5.4 Vortex filaments based on ibm
6 Malliavin Calculus in finance
6.1 Black-Scholes model
6.1.1 Arbitrage opportunities and martingale measures
6.1.2 Completeness and hedging
6.1.3 Black-Schol~s formula
6.2 Integration by parts formulas and computation of Greeks
6.2.1 Computation of Greeks for European options
6.2.2 Computation of Greeks for exotic options
6.3 Application of the Clark-Ocone formula in hedging
6.3.1 A generalized Clark-Ocone formula
6.3.2 Application to finance
6.4 Insider trading
A Appendix
A.1 A Gaussian formula
A.2 Martingale inequalities
A.3 Continuity criteria
A.4 Carleman-Fredholm determinant
A.5 Fractional integrals and derivatives
References
Index