司徒荣编著的《带跳的随机微分方程理论及其应用（英文影印版）》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。

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 mar tingales

 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 tion by parts

2.9 independence of bm and poisson point processes

2.10 some examples

2.11 strong markov property of bm and poisson point processes

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 with jumps

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 of solutions

 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 with jumps

 5 stochastic differential equations with non-lipschitzian co efficients

5.1 strong solutions. continuous coefficients with p- conditions 1

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's theorem

6.2 more useful examples

 7 linear and non-linear filtering

7.1 solutions of sde with functional coefficients and girsanov theorems

7.2 martingale representation theorems (functional coefficient case)

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 option pricing

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-free market

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. bsde case

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 lagrange method

 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 differential equations

 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. weaker conditions

10.2 weak and pathwise uniqueness for 1-dimensional sde with jumps

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-linear system

 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 population control

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 with jumps

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