Stochastic Integrals
Published on 20. June 2014
154 pages
978-1-4832-5923-9 (ISBN)
Description
Stochastic Integrals discusses one area of diffusion processes: the differential and integral calculus based upon the Brownian motion. The book reviews Gaussian families, construction of the Brownian motion, the simplest properties of the Brownian motion, Martingale inequality, and the law of the iterated logarithm. It also discusses the definition of the stochastic integral by Wiener and by Ito, the simplest properties of the stochastic integral according to Ito, and the solution of the simplest stochastic differential equation. The book explains diffusion, Lamperti's method, forward equation, Feller's test for the explosions, Cameron-Martin's formula, the Brownian local time, and the solution of dx=e(x) db + f(x) dt for coefficients with bounded slope. It also tackles Weyl's lemma, diffusions on a manifold, Hasminski's test for explosions, covering Brownian motions, Brownian motions on a Lie group, and Brownian motion of symmetric matrices. The book gives as example of a diffusion on a manifold with boundary the Brownian motion with oblique reflection on the closed unit disk of R squared. The text is suitable for economists, scientists, or researchers involved in probabilistic models and applied mathematics.
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Content
PrefaceList of Notations1. Brownian Motion Introduction 1.1 Gaussian Families 1.2 Construction of the Brownian Motion 1.3 Simplest Properties of the Brownian Motion 1.4 A Martingale Inequality 1.5 The Law of the Iterated Logarithm 1.6 Lévy's Modulus 1.7 Several-Dimensional Brownian Motion2. Stochastic Integrals and Differentials 2.1 Wiener's Definition of the Stochastic Integral 2.2 Itô's Definition of the Stochastic Integral 2.3 Simplest Properties of the Stochastic Integral 2.4 Computation of a Stochastic Integral 2.5 A Time Substitution 2.6 Stochastic Differentials and Itô's Lemma 2.7 Solution of the Simplest Stochastic Differential Equation 2.8 Stochastic Differentials Under a Time Substitution 2.9 Stochastic Integrals and Differentials for Several-Dimensional Brownian Motion3. Stochastic Integral Equations (d = 1) 3.1 Diffusions 3.2 Solution of dX = e(X) db + f(X) dt for Coefficients with Bounded Slope 3.3 Solution of dX = e(X) db + f(X) dt for General Coefficients Belonging to C1(R1) 3.4 Lamperti's Method 3.5 Forward Equation 3.6 Feller's Test for Explosions 3.7 Cameron-Martin's Formula 3.8 Brownian Local Time 3.9 Reflecting Barriers 3.10 Some Singular Equations4. Stochastic Integral Equations (d = 2) 4.1 Manifolds and Elliptic Operators 4.2 Weyl's Lemma 4.3 Diffusions on a Manifold 4.4 Explosions and Harmonic Functions 4.5 Hasminskii's Test for Explosions 4.6 Covering Brownian Motions 4.7 Brownian Motions on a Lie Group 4.8 Injection 4.9 Brownian Motion of Symmetric Matrices 4.10 Brownian Motion with Oblique ReflectionReferencesSubject Index
