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The main new feature of the fifth edition is the addition of a new chapter, Chapter 12, on applications to mathematical finance. I found it natural to include this material as another major application of stochastic analysis, in view of the amazing development in this field during the last 10-20 years. Moreover, the close contact between the theoretical achievements and the applications in this area is striking. For example, today very few firms (if any) trade with options without consulting the Black & Scholes formula! The first 11 chapters of the book are not much changed from the previous edition, but I have continued my efforts to improve the presentation through­ out and correct errors and misprints. Some new exercises have been added. Moreover, to facilitate the use of the book each chapter has been divided into subsections. If one doesn't want (or doesn't have time) to cover all the chapters, then one can compose a course by choosing subsections from the chapters. The chart below indicates what material depends on which sections. Chapter 6 Chapter IO Chapter 12 For example, to cover the first two sections of the new chapter 12 it is recom­ mended that one (at least) covers Chapters 1-5, Chapter 7 and Section 8.6. VIII Chapter 10, and hence Section 9.1, are necessary additional background for Section 12.3, in particular for the subsection on American options.



1. Introduction

To convince the reader that stochastic differential equations is an important subject let us mention some situations where such equations appear and can be used:
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2. Some Mathematical Preliminaries

Having stated the problems we would like to solve, we now proceed to find reasonable mathematical notions corresponding to the quantities mentioned and mathematical models for the problems. In short, here is a first list of the notions that need a mathematical interpretation:
A random quantity
Parametrized (discrete or continuous) families of random quantities
What is meant by a “best” estimate in the filtering problem (Problem 3)
What is meant by an estimate “based on” some observations (Problem 3)?
What is the mathematical interpretation of the “noise” terms?
What is the mathematical interpretation of the stochastic differential equations?
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3. Ito Integrals

We now turn to the question of finding a reasonable mathematical interpre- tation of the “noise” term in the equation of Problem 1 in the Introduction.
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4. The Ito Formula and the Martingale Representation Theorem

Example 3.1.9 illustrates that the basic definition of Ito integrals is not very useful when we try to evaluate a given integral. This is similar to the situation for ordinary Riemann integrals, where we do not use the basic definition but rather the fundamental theorem of calculus plus the chain rule in the explicit calculations.
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5. Stochastic Differential Equations

We now return to the possible solutions X t (w) of the stochastic differential equation
$$ \frac{{d{X_t}}}{{dt}} = b(t,{X_t}) + \sigma (t,Xt){W_t},b(t,x) \in R,\sigma (t,x) \in R$$
where W t is 1-dimensional “white noise”. As discussed in Chapter 3 the Ito interpretation of (5.1.1) is that X t satisfies the stochastic integral equation
$${X_t} = {X_0} + \int\limits_0^t {b(s,{X_s})} {d_s} + \int\limits_0^t {\sigma (s,{X_s})} d{B_s}$$
or in differential form
$$d{X_t} = b(t,{X_t})dt + \sigma (t,{X_t})d{B_t}$$
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6. The Filtering Problem

Problem 3 in the introduction is a special case of the following general filtering problem:
Suppose the state X t R n at time t of a system is given by a stochastic differential equation
$$\frac{{d{X_t}}}{{dt}} = b(t,{X_t}) + \sigma \left( {t,{X_t}} \right){W_t},t0,$$
where b: R n+1R n , σ : R n+1R n × p satisfy conditions (5.2.1), (5.2.2) and W t is p-dimensional white noise. As discussed earlier the Ito interpretation of this equation is (system)
$$d{X_t} = b\left( {t,{X_t}} \right)dt + \sigma \left( {t,{X_t}} \right)d{U_t},$$
where U t is p-dimensional Brownian motion. We also assume that the distribution of X 0 is known and independent of U t . Similarly to the 1-dimensional situation (3.3.6) there is an explicit several-dimensional formula which expresses the Stratonovich interpretation of (6.1.1):
$$d{X_t} = b\left( {t,{X_t}} \right)dt + \sigma \left( {t,{X_t}} \right) \circ d{B_t}$$
in terms of Ito integrals as follows:
$$d{X_t} = \tilde b\left( {t,{X_t}} \right)dt + \sigma \left( {t,{X_t}} \right)d{U_t},$$
$${\tilde b_i}\left( {t,x} \right) = {b_i}\left( {t,x} \right) + \frac{1}{2}\sum\limits_{j = 1}^p {\sum\limits_{k = 1}^n {\frac{{\partial {\sigma _{ij}}}}{{\partial {x_k}}}} } {\sigma _{kj}};1in.$$
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7. Diffusions: Basic Properties

Suppose we want to describe the motion of a small particle suspended in a moving liquid, subject to random molecular bombardments. If b(t, x) ∈ R 3 is the velocity of the fluid at the point x at time t, then a reasonable mathematical model for the position X t of the particle at time t would be a stochastic differential equation of the form
$$\frac{{dX}}{{dt}} = b\left( {t,{X_t}} \right) + \sigma \left( {t,{X_t}} \right){W_t},$$
where W t R 3 denotes “white noise” and σ(t, x) ∈ R 3×3. The Ito interpretation of this equation is
$$d{X_t} = b(t,{X_t})dt + \sigma (t,{X_t})d{B_t},$$
where B t is 3-dimensional Brownian motion, and similarly (with a correction term added to b) for the Stratonovich interpretation (see (6.1.3)).
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8. Other Topics in Diffusion Theory

In this chapter we study some other important topics in diffusion theory and related areas. Some of these topics are not strictly necessary for the remaining chapters, but they are all central in the theory of stochastic analysis and essential for further applications. The following topics will be treated:
Kolmogorov’s backward equation. The resolvent.
The Feynman-Kac formula. Killing.
The martingale problem.
When is an Ito process a diffusion?
Random time change.
The Girsanov formula.
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9. Applications to Boundary Value Problems

We now use results from the preceding chapters to solve the following generalization of the Dirichlet problem stated in the introduction.
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10. Application to Optimal Stopping

Problem 5 in the introduction is a special case of a problem of the following type:
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11. Application to Stochastic Control

Suppose that the state of a system at time t is described by an Ito process X t of the form
$$d{X_t} = dX_t^u = b\left( {t,{X_t}{u_t}} \right)dt + \sigma \left( {t,{X_t},{u_t}} \right)d{B_t},$$
where X t R n , b: R ×R n × UR n , σ: R ×R n × UR n ×m and B t is m-dimensional Brownian motion. Here u t UR k is a parameter whose value we can choose in the given Borel set U at any instant t in order to control the process X t . Thus u t = u(t, ω) is a stochastic process. Since our decision at time t must be based upon what has happened up to time t, the function ωu(t, ω) must (at least) be measurable w.r.t. F t (m) , i.e. the process u t must be F t (m) -adapted. Thus the right hand side of (11.1.1) is well-defined as a stochastic integral, under suitable assumptions on the functions b and σ. will not specify the conditions on b and σ further, but simply assume that the process X t satisfying (11.1.1) exists. See further comments on this in the end of this chapter.
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12. Application to Mathematical Finance

In this chapter we describe how the concepts, methods and results in the previous chapters can be applied to give a rigorous mathematical model of finance. We will concentrate on the most fundamental issues and those topics which are most closely related to the theory in this book. We emphasize that this chapter only intends to give a brief introduction to this exciting subject, which has developed very fast during the last years and shows no signs of slowing down. For a more comprehensive treatment we refer the reader to Duffle (1996), Karatzas (1996), Karatzas and Shreve (1997), Lamberton and Lapeyre (1996), Musiela and Rutkowski (1997), Kallianpur (1997) and the references therein.
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