Brownian Motion and Stochastic Calculus
 List Price: $64.95
 Binding: Paperback
 Edition: 2
 Publisher: Springer Verlag
 Publish date: 09/01/1991
Description:
1 Martingales, Stopping Times, and Filtrations. 1.1. Stochastic Processes and ?Fields. 1.2. Stopping Times. 1.3. ContinuousTime Martingales. A. Fundamental inequalities. B. Convergence results. C. The optional sampling theorem. 1.4. The DoobMeyer Decomposition. 1.5. Continuous, SquareIntegrable Martingales. 1.6. Solutions to Selected Problems. 1.7. Notes. 2 Brownian Motion. 2.1. Introduction. 2.2. First Construction of Brownian Motion. A. The consistency theorem. B. The Kolmogorov?entsov theorem. 2.3. Second Construction of Brownian Motion. 2.4. The SpaceC[0, ?), Weak Convergence, and Wiener Measure. A. Weak convergence. B. Tightness. C. Convergence of finitedimensional distributions. D. The invariance principle and the Wiener measure. 2.5. The Markov Property. A. Brownian motion in several dimensions. B. Markov processes and Markov families. C. Equivalent formulations of the Markov property. 2.6. The Strong Markov Property and the Reflection Principle. A. The reflection principle. B. Strong Markov processes and families. C. The strong Markov property for Brownian motion. 2.7. Brownian Filtrations. A. Rightcontinuity of the augmented filtration for a strong Markov process. B. A "universal" filtration. C. The Blumenthal zeroone law. 2.8. Computations Based on Passage Times. A. Brownian motion and its running maximum. B. Brownian motion on a halfline. C. Brownian motion on a finite interval. D. Distributions involving last exit times. 2.9. The Brownian Sample Paths. A. Elementary properties. B. The zero set and the quadratic variation. C. Local maxima and points of increase. D. Nowhere differentiability. E. Law of the iterated logarithm. F. Modulus of continuity. 2.10. Solutions to Selected Problems. 2.11. Notes. 3 Stochastic Integration. 3.1. Introduction. 3.2. Construction of the Stochastic Integral. A. Simple processes and approximations. B. Construction and elementary properties of the integral. C. A characterization of the integral. D. Integration with respect to continuous, local martingales. 3.3. The ChangeofVariable Formula. A. The It rule. B. Martingale characterization of Brownian motion. C. Bessel processes, questions of recurrence. D. Martingale moment inequalities. E. Supplementary exercises. 3.4. Representations of Continuous Martingales in Terms of Brownian Motion. A. Continuous local martingales as stochastic integrals with respect to Brownian motion. B. Continuous local martingales as timechanged Brownian motions. C. A theorem of F. B. Knight. D. Brownian martingales as stochastic integrals. E. Brownian functionals as stochastic integrals. 3.5. The Girsanov Theorem. A. The basic result. B. Proof and ramifications. C. Brownian motion with drift. D. The Novikov condition. 3.6. Local Time and a Generalized It Rule for Brownian Motion. A. Definition of local time and the Tanaka formula. B. The Trotter existence theorem. C. Reflected Brownian motion and the Skorohod equation. D. A generalized It rule for convex functions. E. The EngelbertSchmidt zeroone law. 3.7. Local Time for Continuous Semimartingales. 3.8. Solutions to Selected Problems. 3.9. Notes. 4 Brownian Motion and Partial Differential Equations. 4.1. Introduction. 4.2. Harmonic Functions and the Dirichlet Problem. A. The meanvalue property. B. The Dirichlet problem. C. Conditions for regularity. D. Integral formulas of Poisson. E. Supplementary exercises. 4.3. The OneDimensional Heat Equation. A. The Tychonoff uniqueness theorem. B. Nonnegative solutions of the heat equation. C. Boundary crossing probabilities for Brownian motion. D. Mixed initial/boundary value problems. 4.4. The Formulas of Feynman and Kac. A. The multidimensional formula. B. The onedimensional formula. 4.5. Solutions to selected problems. 4.6. Notes. 5 Stochastic Differential Equations. 5.1. Introduction. 5.2. Strong Solutions. A. Definitions. B. The It theory. C. Comparison results and other refinements. D. Approximations of stochastic differential equations. E. Supplementary exercises. 5.3. Weak Solutions. A. Two notions of uniqueness. B. Weak solutions by means of the Girsanov theorem. C. A digression on regular conditional probabilities. D. Results of Yamada and Watanabe on weak and strong solutions. 5.4. The Martingale Problem of Stroock and Varadhan. A. Some fundamental martingales. B. Weak solutions and martingale problems. C. Wellposedness and the strong Markov property. D. Questions of existence. E. Questions of uniqueness. F. Supplementary exercises. 5.5. A Study of the OneDimensional Case. A. The method of time change. B. The method of removal of drift. C. Feller''s test for explosions. D. Supplementary exercises. 5.6. Linear Equations. A. GaussMarkov processes. B. Brownian bridge. C. The general, onedimensional, linear equation. D. Supplementary exercises. 5.7. Connections with Partial Differential Equations. A. The Dirichlet problem. B. The Cauchy problem and a FeynmanKac representation. C. Supplementary exercises. 5.8. Applications to Economics. A. Portfolio and consumption processes. B. Option pricing. C. Optimal consumption and investment (general theory). D. Optimal consumption and investment (constant coefficients). 5.9. Solutions to Selected Problems. 5.10. Notes. 6 P. Lvy''s Theory of Brownian Local Time. 6.1. Introduction. 6.2. Alternate Representations of Brownian Local Time. A. The process of passage times. B. Poisson random measures. C. Subordinators. D. The process of passage times revisited. E. The excursion and downcrossing representations of local time. 6.3. Two Independent Reflected Brownian Motions. A. The positive and negative parts of a Brownian motion. B. The first formula of D. Williams. C. The joint density of (W(t), L(t), ?+(t)). 6.4. Elastic Brownian Motion. A. The FeynmanKac formulas for elastic Brownian motion. B. The RayKnight description of local time. C. The second formula of D. Williams. 6.5. An Application: Transition Probabilities of Brownian Motion with TwoValued Drift. 6.6. Solutions to Selected Problems. 6.7. Notes.
Expand description
Seller  Condition  Comments  Price 

SRJCLIB1
Good 
$16.86


Solr Books
Very Good

$38.57


Broad Street Books
Like New

$44.95


MERCHACCESS
Acceptable 
$44.98


SurplusTextSeller
Good 
$49.26

Please Wait