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
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