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## Über dieses Buch

This is a substantial expansion of the first edition. The last chapter on stochastic differential equations is entirely new, as is the longish section §9.4 on the Cameron-Martin-Girsanov formula. Illustrative examples in Chapter 10 include the warhorses attached to the names of L. S. Ornstein, Uhlenbeck and Bessel, but also a novelty named after Black and Scholes. The Feynman-Kac-Schrooinger development (§6.4) and the material on re­ flected Brownian motions (§8.5) have been updated. Needless to say, there are scattered over the text minor improvements and corrections to the first edition. A Russian translation of the latter, without changes, appeared in 1987. Stochastic integration has grown in both theoretical and applicable importance in the last decade, to the extent that this new tool is now sometimes employed without heed to its rigorous requirements. This is no more surprising than the way mathematical analysis was used historically. We hope this modest introduction to the theory and application of this new field may serve as a text at the beginning graduate level, much as certain standard texts in analysis do for the deterministic counterpart. No monograph is worthy of the name of a true textbook without exercises. We have compiled a collection of these, culled from our experiences in teaching such a course at Stanford University and the University of California at San Diego, respectively. We should like to hear from readers who can supply VI PREFACE more and better exercises.

## Inhaltsverzeichnis

### 1. Preliminaries

Abstract
For each interval I in IR = (−∞, ∞) let B(I) denote the σ-field of Borel subsets of I. For each tIR + = [0, ∞), let B t denote B([0, t]) and let B denote $$B(I{R_ + }) = {V_{t \in I{R_ + }}}$$ B t — the smallest σ-field containing B t for all t in IR+. Let $$overline {I{R_ + }} = [0,\infty ]$$ and $$overline B$$ denote the Borel σ-field of $$overline {I{R_ + }}$$ generated by B and the singleton {∞}. Let λ denote the Lebesgue measure on IR.
K. L. Chung, R. J. Williams

### 2. Definition of the Stochastic Integral

Abstract
In this chapter, we shall define stochastic integrals of the form $$int_{[0,t]}$$ X dM where M is a right continuous local L 2-martingale and X is a process satisfying certain measurability and integrability assumptions, such that the family of stochastic integrals $$\{ \int_{{[0,t]}} {X\,dM,t \in {{\mathbb{R}}_{ + }}} \}$$ is a right continuous local L 2-martingale. For certain M and X, the integral can be defined path-by-path. For instance, if M is a right continuous local L 2-martingale whose paths are locally of bounded variation, and X is a continuous adapted process, then $$int_{[0,t]}^{}{{X_s}d{M_s}}(\omega )$$ is well-defined as a Riemann-Stieltjes integral for each t and ω, namely by the limit as n → ∞ of
$$\sum\limits_{{k = 0}}^{{[{{2}^{n}}t]}} {{{X}_{{k{{2}^{{ - n}}}}}}} (\omega )({{M}_{{(k + 1){{2}^{{ - n}}}}}}(\omega ) - {{M}_{{k{{2}^{{ - n}}}}}}(\omega )).$$
K. L. Chung, R. J. Williams

### 3. Extension of the Predictable Integrands

Abstract
In this chapter, we show that the definition of the stochastic integral can be extended to a larger class of integrands than the predictable ones, when either a mild condition on the Doléans measure μM is satisfied or M is continuous.
K. L. Chung, R. J. Williams

Abstract
For the remainder of this book, we shall only consider integrators M which are continuous local martingales. By Proposition 1.9 these are automatically local L2-martingales. A more extensive treatment, encompassing right continuous integrators would require more elaborate considerations which are not suitable for inclusion in this short book.
K. L. Chung, R. J. Williams

### 5. The Ito Formula

Abstract
One of the most important results in the theory of stochastic integrals is the rule for change of variables known as the Itô formula, after Itô who first proved it for the special case of integration with respect to Brownian motion. The essential aspects of Itô’s formula are conveyed by the following. If M is a continuous local martingale and f is a twice continuously differentiable real-valued function on IR, then the Itô formula for f(M t ) is
$$f\left( {{M_t}} \right) - f\left( {{M_0}} \right) = \int\limits_0^t {f'\left( {{M_s}} \right)d{M_s} + \frac{1}{2}\int\limits_0^t {f''\left( {{M_s}} \right)d{{\left[ M \right]}_{s.}}} }$$
(5.1)
K. L. Chung, R. J. Williams

### 6. Applications of the Ito Formula

Abstract
A process $$M = \{ {M_t},t \in I{R_ + }\}$$ is a Brownian motion in $$IR$$ if and only if there is a standard filtration $${F_t}$$ such that $${M_t},{F_t},t \in I{R_ + }$$ is a continuous local martingale with quadratic variation [M] satisfying
$${[M]_t} = t{\text{ a}}{\text{.s}}{\text{. for all t}}{\text{.}}$$
(6.1)
K. L. Chung, R. J. Williams

### 7. Local Time and Tanaka’s Formula

Abstract
In this chapter B denotes a Brownian motion in IR. For each xIR we shall obtain a decomposition, known as Tanaka’s formula, of the positive submartingale |B − x| as the sum of another Brownian motion $$hat B$$ and a continuous increasing process L( · , x). The latter is called the local time of B at x, a fundamental notion invented by P. Lévy (see [54]). It may be expressed as follows:
$$L\left( {t,x} \right) = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{{2\varepsilon }}\int\limits_0^t {1\left( {x - \varepsilon ,x + \varepsilon } \right)} \left( {{B_s}} \right)ds{\text{ }} = \mathop {\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{{2\varepsilon }}\lambda \left\{ {s \in \left[ {0,t} \right]:{B_s} \in \left( {x - \varepsilon ,x + \varepsilon } \right)} \right\}{\text{ }}$$
(7.1)
where λ is the Lebesgue measure. Thus it measures the amount of time the Brownian motion spends in the neighborhood of x. It is well known that {t ∈ I R + : B t = x} is a perfect closed set of Lebesgue measure zero. The existence of a nonvanishing L defined in (7.1) is therefore far from obvious. In fact, the limit in (7.1) exists both in L2 and a.s., as we shall see. Moreover, L(t, x) may be defined to be a jointly continuous function of (t, x). This was first proved by H. F. Trotter, but our approach follows that of Stroock and Varadhan [73, p. 117]. The local time plays an important role in many refined developments of the theory of Brownian motion. One application, given at the end of Section 7.3, is a derivation of the exponential distribution of the local time accumulated up until the hitting time of a fixed level. Other applications of local time and Tanaka’s formula are discussed in the next two chapters.
K. L. Chung, R. J. Williams

### 8. Reflected Brownian Motions

Abstract
In this chapter, the processes L( · , 0) and $$\hat B$$( · , 0), defined by (7.7) and (7.16), will be denoted respectively by $$\hat B$$( · ) and L( · ).
K. L. Chung, R. J. Williams

### 9. Generalized Ito Formula, Change of Time and Measure

Abstract
In this chapter we shall first obtain a generalized Itô formula for convex functions of Brownian motion. Then we shall prove a result which shows that Brownian motion is truly the canonical example of a continuous local martingale. Namely, if M is a continuous local martingale with quadratic variation [M] t , then there is a random change of time τ t such that {M τt , tIR + } is a Brownian motion up to the (random) time [M] = sup t ≥0[M] t . An application of this result shows that for a one-dimensional Brownian motion B starting from x ≥ 0, there is a time change τ t such that {B τt , tIR + } is equivalent in law to |B|. Finally, we show how local martingales behave under mutually absolutely continuous changes of probability measure. Using this, we obtain a formula for transforming a local martingale into a local martingale plus a state-dependent drift. We illustrate how this can be applied to obtain weak solutions of some stochastic differential equations.
K. L. Chung, R. J. Williams

### 10. Stochastic Differential Equations

Abstract
In this chapter, we consider stochastic differential equations (SDE’s) of the form
$$dX\left( t \right) = \sigma \left( {X\left( t \right)} \right)dB\left( t \right) + b\left( {X\left( t \right)} \right)dt$$
(10.1)
, or equivalently in coordinate form
$$d{X_i}\left( t \right) = \sum\limits_{j = 1}^r {{\sigma _{ij}}} \left( {X\left( t \right)} \right)d{B_j}\left( t \right) + {b_i}\left( {X\left( t \right)} \right)dt, for i = 1, \ldots ,d,$$
(10.2)
where B = (B 1 , B r ) is an r-dimensional Brownian motion (r ≥ 1) starting from the origin, and σ : IR d IR d IR r and b: IR d IR d are Borel measurable functions. Here IR d IR r , d ≥ 1, r ≥ 1, denotes the space of d × r real-valued matrices with the norm
$$parallel A\parallel = {\left( {\sum\limits_{i = 1}^d {\sum\limits_{j = 1}^r {A_{ij}^2} } } \right)^{\frac{1}{2}}}$$
(10.3)
for AIR d IR r .
K. L. Chung, R. J. Williams

### Backmatter

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