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**Introduction to stochastic integration.
2nd ed.**
*(English)*
Zbl 0725.60050

Probability and Its Applications. Boston, MA etc.: Birkhäuser. xv, 276 p. DM 68.00 (1990).

Since the first edition in 1983 (a Russian translation appeared in 1987) this introduction to stochastic integration was probably the basic textbook for innumerable courses on that subject. It is a well-written book giving a convincing access to the modern theory of stochastic integration. The selection and presentation of the material is especially well-done from the pedagogical point of view. The proofs are carefully elaborated, and a lot of motivations, examples and historical remarks is guiding through the general theory.

The second edition is a substantial expansion of the first one, filling the gap that the first edition did not explicitly treat the main application of stochastic integration, namely the theory of stochastic differential equations. For the first edition (essentially the Chapters 1 to 8 and the first part of Chapter 9) there is a detaled review of J. Walsh (Zbl 0527.60058), so I will describe this part rather shortly.

After the necessary prerequisites mainly on martingales in Chapter 1, the stochastic integral is developed in the next two chapters, first for right continuous, square integrable martingales and then for continuous local martingales in full generality. In Chapter 4 the quadratic variation process is introduced, which is needed for the Itô formula proved in Chapter 5. The following three chapters are devoted to diverse applications: Lévy’s characterization of Brownian motion, exponential processes, the Feynman-Kac functional and the Schrödinger equation (Chapter 6), local time and Tanaka’s formula (Chapter 7), and reflected Brownian motions (Chapter 8).

In Chapter 9 first the Itô formula is extended for convex functions and then it is proved that any continuous local martingale can be turned into a Brownian motion by a random change of time. The rest of the chapter is entirely new and contains a well-motivated proof of the Cameron-Martin- Girsanov formula.

The last and newly added chapter on stochastic differential equations first gives a detailed proof on the existence and uniqueness of solutions of stochastic differential equations of the form \[ dX(t)=\sigma (X(t))dB(t)+b(X(t))dt, \] where B is an r-dimensional Brownian motion, \(\sigma\) is (d\(\times r)\)-matrix valued, and b is an \({\mathbb{R}}^ d\)- valued function. \(\sigma\) and b are assumed to fulfill local Lipschitz and boundedness conditions. A careful proof of the strong Markov property of the solution is included, and the notions of strong and weak solutions are discussed in some detail. The general theory is applied to the Ornstein-Uhlenbeck process, the Bessel process, and - especially exhaustive - to an example from the financial market, the Black-Scholes option pricing formula, with an interesting application of the Cameron- Martin-Girsanov formula.

The second edition is a substantial expansion of the first one, filling the gap that the first edition did not explicitly treat the main application of stochastic integration, namely the theory of stochastic differential equations. For the first edition (essentially the Chapters 1 to 8 and the first part of Chapter 9) there is a detaled review of J. Walsh (Zbl 0527.60058), so I will describe this part rather shortly.

After the necessary prerequisites mainly on martingales in Chapter 1, the stochastic integral is developed in the next two chapters, first for right continuous, square integrable martingales and then for continuous local martingales in full generality. In Chapter 4 the quadratic variation process is introduced, which is needed for the Itô formula proved in Chapter 5. The following three chapters are devoted to diverse applications: Lévy’s characterization of Brownian motion, exponential processes, the Feynman-Kac functional and the Schrödinger equation (Chapter 6), local time and Tanaka’s formula (Chapter 7), and reflected Brownian motions (Chapter 8).

In Chapter 9 first the Itô formula is extended for convex functions and then it is proved that any continuous local martingale can be turned into a Brownian motion by a random change of time. The rest of the chapter is entirely new and contains a well-motivated proof of the Cameron-Martin- Girsanov formula.

The last and newly added chapter on stochastic differential equations first gives a detailed proof on the existence and uniqueness of solutions of stochastic differential equations of the form \[ dX(t)=\sigma (X(t))dB(t)+b(X(t))dt, \] where B is an r-dimensional Brownian motion, \(\sigma\) is (d\(\times r)\)-matrix valued, and b is an \({\mathbb{R}}^ d\)- valued function. \(\sigma\) and b are assumed to fulfill local Lipschitz and boundedness conditions. A careful proof of the strong Markov property of the solution is included, and the notions of strong and weak solutions are discussed in some detail. The general theory is applied to the Ornstein-Uhlenbeck process, the Bessel process, and - especially exhaustive - to an example from the financial market, the Black-Scholes option pricing formula, with an interesting application of the Cameron- Martin-Girsanov formula.

Reviewer: E.Dettweiler (Reutlingen)

### MSC:

60H05 | Stochastic integrals |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J65 | Brownian motion |