Contents
Preface ix
General Notation xi
Contents of Volume 2 xiii
1 Stochastic Processes
1. The Kolmogorov construction of a stochastic process 1
2. Separable and continuous processes 6
3. Martingales and stopping times 9
Problems 15
2 Markov Processes
1. Construction of Markov processes 18
2. The Feller and the strong Markov properties 23
3. Time-homogeneous Markov processes 30
Problems 31
3 Brownian Motion
1. Existence of continuous Brownian motion 36
2. Nondifferentiability of Brownian motion 39
3. limit theorems 40
4. Brownian motion after a stopping time 44
5. Martingales and Brownian motion 46
6. Brownian motion in n dimensions 50
Problems 53
4 The Stochastic Integral
1. Approximation of functions by step functions 55
2. Definition of the stochastic integral 59
3. The indefinite integral 67
4. Stochastic integrals with stopping time 72
5. Ito's formula 78
6. Applications of Ito's formula 85
7. Stochastic integrals and differentials in n dimensions 89
Problems 93
5 Stochastic Differential Equations
1 Existence and uniqueness
2 Stronger uniqueness and existence theorems
3 The solution of a stochastic differential system
4 as a Markov process
5 Diffusion processes
6 Equations depending on a parameter
7 The Kolmogorov equation
Problems

6.Elliptic and Parabolic Partial Differential
Equations and Their Relations to
Stochastic Differential Equations
1. Square root of a nonnegative definite matrix 128
2. The maximum principle for elliptic equations 132
3. The maximum principle for parabolic equations 134
4. The Cauchy problem and fundamental solutions
for parabolic equations 139
5. Stochastic representation of solutions of partial
differential equations 144
Problems 150
7 The Cameron-Martln-Glrsanov Theorem
1. A class of absolutely continuous probabilities 152
2. Transformation of Brownian motion 156
3. Girsanov's formula 164
Problems 169
 

8 Asymptotic Estimates for Solutions
1. Unboundedness of solutions 172
2. Auxiliary estimates 174
3. Asymptotic estimates 180
4. Applications of the asymptotic estimates 185
5. The one-dimensional case 188
6. Counterexample 191
Problems 193
9 Recurrent and Transient Solutions

1 Transient solutions
2 Recurrent solutions 
3 Rate of wandering out to infinity
4 Obstacles
5 Transient solutions for degenerate diffusion
6 Recurrent solutions for degenerate diffusion
7 The one-dimensional case
Problems
Bibliographical Remarks
References
Index