prefaceacknowledgementabbreviatio
and some explanatio
�� stochastic differential equatio
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-dime
ional 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 equatio
3.1 strong solutio
to sde with jumps 3.1.1 notation 3.1.2 a priori estimate and uniqueness of solutio
3.1.3 existence of solutio
for the lipschitzian case 3.2 exponential solutio
to linear sde with jumps 3.3 gi
anov tra
formation and weak solutio
of sde with jumps 3.4 examples of weak solutio
4 some useful tools in stochastic differential equatio
4.1 yamada-watanabe type theorem 4.2 tanaka type formula and some applicatio
4.2.1 localization technique 4.2.2 tanaka type formula in d-dime
ional space 4.2.3 applicatio
to pathwise uniqueness and convergence of solutio
4.2.4 tanaka type formual in 1-dime
ional space 4.2.5 tanaka type formula in the component form 4.2.6 pathwise uniqueness of solutio
4.3 local time and occupation de
ity formula 4.4 krylov estimation 4.4.1 the case for 1-dime
ional space 4.4.2 the case for d-dime
ional space 4.4.3 applicatio
to convergence of solutio
to sde with jumps 5 stochastic differential equatio
with non-lipschitzian co efficients 5.1 strong solutio
. continuous coefficients with p- conditio
1 5.2 the skorohod weak convergence technique 5.3 weak solutio
. continuous coefficients 5.4 existence of strong solutio
and applicatio
to ode 5.5 weak solutio
. measurable coefficient case�� applicatio
6 how to use the stochastic calculus to solve sde 6.1 the foundation of applicatio
: ito's formula and gi
anov's theorem 6.2 more useful examples 7 linear and non-linear filtering 7.1 solutio
of sde with functional coefficients and gi
anov 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-dime
ional 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 solutio
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 solutio
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 solutio
for non-lipschitzian case 8.9 convergence of solutio
8.10 explanation of solutio
of bsdes to financial markets 8.11 comparison theorem for bsde with jumps 8.12 explanation of comparison theorem. arbitrage-free market 8.13 solutio
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 co
umption by h-j-b equation and lagrange method 9.1 optimal co
umption 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 applicatio
9.2.5 concluding remarks 10 comparison theorem and stochastic pathwise control ' 10.1 comparison for solutio
of stochastic differential equatio
10.1.1 1-dime
ional space case 10.1.2 component comparison in d-dime
ional space 10.1.3 applicatio
to existence of strong solutio
. weaker conditio
10.2 weak and pathwise uniqueness for 1-dime
ional sde with jumps 10.3 strong solutio
for 1-dime
ional 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-dime
ional 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 solutio
11.4 moment estimates and uniqueness of solutio
to
de 11.5 solutio
for
de with jumps and with continuous coef- ficients 11.6 solutio
for
de with jumps and with discontinuous co- etticients 11.7 solutio
to population sde and their properties 11.8 comparison of solutio
and stochastic population control 11.9 caculation of solutio
to population
de 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 vecto
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 integralsreferencesindex