带跳的随机微分方程理论及其应用（英文版）
出版时间：2012年版
内容简介
《带跳的随机微分方程理论及其应用（英文版）》是一部讲述随机微分方程及其应用的教程。内容全面，讲述如何很好地引入和理解ito积分，确定了ito微分规则，解决了求解sde的方法，阐述了girsanov定理，并且获得了sde的弱解。书中也讲述了如何解决滤波问题、鞅表示定理，解决了金融市场的期权定价问题以及著名的black-scholes公式和其他重要结果。特别地，书中提供了研究市场中金融问题的倒向随机技巧和反射sed技巧，以便更好地研究优化随机样本控制问题。这两个技巧十分高效有力，还可以应用于解决自然和科学中的其他问题。
目录
preface
acknowledgement
abbreviations and some explanations
Ⅰ stochastic differential equations with jumps inrd
1 martingale theory and the stochastic integral for point
processes
1.1 concept of a martingale
1.2 stopping times. predictable process
1.3 martingales with discrete time
1.4 uniform integrability and martingales
1.5 martingales with continuous time
1.6 doob-meyer decomposition theorem
1.7 poisson random measure and its existence
1.8 poisson point process and its existence
1.9 stochastic integral for point process. square integrable martingales
2 brownian motion, stochastic integral and ito's formula
2.1 brownian motion and its nowhere differentiability
2.2 spaces ~0 and z?
2.3 ito's integrals on l2
2.4 ito's integrals on l2,loc
2.5 stochastic integrals with respect to martingales
2.6 ito's formula for continuous semi-martingales
2.7 ito's formula for semi-martingales with jumps
2.8 ito's formula for d-dimensional semi-martingales. integra tionby parts
2.9 independence of bm and poisson point processes
2.10 some examples
2.11 strong markov property of bm and poisson pointprocesses
2.12 martingale representation theorem
3 stochastic differential equations
3.1 strong solutions to sde with jumps
3.1.1 notation
3.1.2 a priori estimate and uniqueness of solutions
3.1.3 existence of solutions for the lipschitzian case
3.2 exponential solutions to linear sde with jumps
3.3 girsanov transformation and weak solutions of sde withjumps
3.4 examples of weak solutions
4 some useful tools in stochastic differential equations
4.1 yamada-watanabe type theorem
4.2 tanaka type formula and some applications
4.2.1 localization technique
4.2.2 tanaka type formula in d-dimensional space
4.2.3 applications to pathwise uniqueness and convergence ofsolutions
4.2.4 tanaka type formual in 1-dimensional space
4.2.5 tanaka type formula in the component form
4.2.6 pathwise uniqueness of solutions
4.3 local time and occupation density formula
4.4 krylov estimation
4.4.1 the case for 1-dimensional space
4.4.2 the case for d-dimensional space
4.4.3 applications to convergence of solutions to sde withjumps
5 stochastic differential equations with non-lipschitzian coefficients
5.1 strong solutions. continuous coefficients with p- conditions1
5.2 the skorohod weak convergence technique
5.3 weak solutions. continuous coefficients
5.4 existence of strong solutions and applications to ode
5.5 weak solutions. measurable coefficient case
Ⅱ applications
6 how to use the stochastic calculus to solve sde
6.1 the foundation of applications: ito's formula and girsanov'stheorem
6.2 more useful examples
7 linear and non-linear filtering
7.1 solutions of sde with functional coefficients and girsanovtheorems
7.2 martingale representation theorems (functional coefficientcase)
7.3 non-linear filtering equation
7.4 optimal linear filtering
7.5 continuous linear filtering. kalman-bucy equation
7.6 kalman-bucy equation in multi-dimensional case
7.7 more general continuous linear filtering
7.8 zakai equation
7.9 examples on linear filtering
8 option pricing in a financial market and bsde
8.1 introduction
8.2 a more detailed derivation of the bsde for optionpricing
8.3 existence of solutions with bounded stopping times
8.3.1 the general model and its explanation
8.3.2 a priori estimate and uniqueness of a solution
8.3.3 existence of solutions for the lipschitzian case
8.4 explanation of the solution of bsde to option pricing
8.4.1 continuous case
8.4.2 discontinuous case
8.5 black-scholes formula for option pricing. two approaches
8.6 black-scholes formula for markets with jumps
8.7 more general wealth processes and bsdes
8.8 existence of solutions for non-lipschitzian case
8.9 convergence of solutions
8.10 explanation of solutions of bsdes to financial markets
8.11 comparison theorem for bsde with jumps
8.12 explanation of comparison theorem. arbitrage-freemarket
8.13 solutions for unbounded (terminal) stopping times
8.14 minimal solution for bsde with discontinuous drift
8.15 existence of non-lipschitzian optimal control. bsdecase
8.16 existence of discontinuous optimal control. bsdes in rl
8.17 application to pde. feynman-kac formula
9 optimal consumption by h-j-b equation and lagrange method
9.1 optimal consumption
9.2 optimization for a financial market with jumps by the lagrangemethod
9.2.1 introduction
9.2.2 models
9.2.3 main theorem and proof
9.2.4 applications
9.2.5 concluding remarks
10 comparison theorem and stochastic pathwise control '
10.1 comparison for solutions of stochastic differentialequations
10.1.1 1-dimensional space case
10.1.2 component comparison in d-dimensional space
10.1.3 applications to existence of strong solutions. weakerconditions
10.2 weak and pathwise uniqueness for 1-dimensional sde withjumps
10.3 strong solutions for 1-dimensional sde with jumps
10.3.1 non-degenerate case
10.3.2 degenerate and partially-degenerate case
10.4 stochastic pathwise bang-bang control for a non-linearsystem
10.4.1 non-degenerate case
10.4.2 partially-degenerate case
10.5 bang-bang control for d-dimensional non-linear systems
10.5.1 non-degenerate case
10.5.2 partially-degenerate case
11 stochastic population conttrol and reflecting sde
11.1 introduction
11.2 notation
11.3 skorohod's problem and its solutions
11.4 moment estimates and uniqueness of solutions to rsde
11.5 solutions for rsde with jumps and with continuous coef-ficients
11.6 solutions for rsde with jumps and with discontinuous co-etticients
11.7 solutions to population sde and their properties
11.8 comparison of solutions and stochastic populationcontrol
11.9 caculation of solutions to population rsde
12 maximum principle for stochastic systems with jumps
12.1 introduction
12.2 basic assumption and notation
12.3 maximum principle and adjoint equation as bsde withjumps
12.4 a simple example
12.5 intuitive thinking on the maximum principle
12.6 some lemmas
12.7 proof of theorem 354
a a short review on basic probability theory
a.1 probability space, random variable and mathematical ex-pectation
a.2 gaussian vectors and poisson random variables
a.3 conditional mathematical expectation and its properties
a.4 random processes and the kolmogorov theorem
b space d and skorohod's metric
c monotone class theorems. convergence of random processes41
c.1 monotone class theorems
c.2 convergence of random variables
c.3 convergence of random processes and stochastic integrals
references
index
