Brownian Motion and Stochastic Calculus

Abstract

I TAKEUTIIZARING . Introduction to Axiomatic Set Theory. 2nd ed . OXTOBY . Measure and Category. 2nd ed . SCHAEFFER. Topological Vector Spaces . HILTON/STAMMBACH . A Course in Hom ological Algebra. MACLANE. Categories for the Working Mathematician. HUGHES/PIPER . Projective Planes . SERRE. A Course in Arithmetic . TAKEUTI/ZARING . Axiomatic Set Theory . HUMPHREYS . Introduction to Lie Algebras and Representation Theory. COHEN . A Course in Simple Homotopy Theory. II CONWAY . Functions of One Complex Variable . 2nd ed . BEALS . Advanced Mathematical Analysis. ANDERSON/FuLLER . Rings and Categories of Modules . GOLUBITSKy/GUILLEMIN . Stable Mappings and Their Singularities . BERBERIAN . Lectures in Functional Analysis and Operator Theory. WINTER. The Structure of Fields. ROSENBLATT. Random Processes . 2nd ed . HALMos. Measure Theory . HALMos. A Hilbert Space Problem Book. 2nd ed . • revi sed . HUSEMOLLER . Fibre Bundles . 2nd ed . HUMPHREYS. Linear Algebraic Groups . BARNES/MACK . An Algebraic Introduction to Mathematical Logic . GREUB . Linear Algebra. 4th ed . HOLMES . Geometric Functional Analysis and its Applications . HEWITT/STROMBERG . Real and Abstract Analysis. MANES. Algebraic Theories. KELLEY . General Topology.

Spatial epidemics and local times for critical branching random walks in dimensions 2 and 3

“…Thus, to complete the proof of Theorem 2, it suffices to prove the assertion (18) when α = 2/(6 − d). For this, it suffices to show that the two sums in the exponential of equation (66) converge to the corresponding integrals in the exponential of equation (14). The convergence of the first sum follows from Proposition 3 and the representation (67), and can be proved by the same argument as in the proof of Corollary 4 of [16].…”

“…Since the law of the SIR-d epidemic with village size N is absolutely continuous relative to that of its branching envelope, and since the branching envelopes converge weakly, after renormalization, to super-Brownian motion, it suffices to prove that the likelihood ratios converge weakly to the likelihood ratio (14) of the appropriate Dawson-Watanabe process relative to super-Brownian motion. The one-and higher-dimensional cases differ only in the behavior of the occupation statistics that enter into the likelihood ratios.…”

“…This we will accomplish by verifying a form of the Kolmogorov-Centsov criterion (see, e.g., Theorem 2.8 and Problem 2.9 in [14]). According to this criterion, to prove tightness it suffices to prove that for each compact subset K of [0, ∞) × R d there exist constants C < ∞, α >0, and δ >d +1 such that for all k and all pairs (s, a),…”

Spectral Methods for Multiscale Stochastic Differential Equations

“…The second term can be bounded in a similar manner by using the Burkholder-Davis-Gundy inequality (see, for example, [35,Thm. 3.28]) and Itô isometry:…”

Optimal Investment Problem with Multiple Risky Assets under the Constant Elasticity of Variance (CEV) Model

“…Another method is the martingale approach which was adapted to the problem of utility maximization by Pliska [5], Karatzas, Lehoczky and Shreve [6] and Cox and Huang [7]. Much of this development appeared in [8,9]. Applying the martingale approach, Karatzas et al [10] investigated the utility maximization problem in an incomplete market and Zhang [11] considered a similar problem.…”