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## Über dieses Buch

This book provides a comprehensive introduction to the theory of stochastic calculus and some of its applications. It is the only textbook on the subject to include more than two hundred exercises with complete solutions.

After explaining the basic elements of probability, the author introduces more advanced topics such as Brownian motion, martingales and Markov processes. The core of the book covers stochastic calculus, including stochastic differential equations, the relationship to partial differential equations, numerical methods and simulation, as well as applications of stochastic processes to finance. The final chapter provides detailed solutions to all exercises, in some cases presenting various solution techniques together with a discussion of advantages and drawbacks of the methods used.

Stochastic Calculus will be particularly useful to advanced undergraduate and graduate students wishing to acquire a solid understanding of the subject through the theory and exercises. Including full mathematical statements and rigorous proofs, this book is completely self-contained and suitable for lecture courses as well as self-study.

## Inhaltsverzeichnis

### Chapter 1. Elements of Probability

Abstract
In this chapter we recall the basic facts in probability that are required for the investigation of the stochastic processes that are the object of the subsequent chapters.
Paolo Baldi

### Chapter 2. Stochastic Processes

Abstract
A stochastic process is a mathematical object that is intended to model the evolution in time of a random phenomenon. As will become clear in the sequel the appropriate setting is the following.
Paolo Baldi

### Chapter 3. Brownian Motion

Abstract
Brownian motion is a particular stochastic process which is the prototype of the class of processes which will be our main concern. Its investigation is the object of this chapter.
Paolo Baldi

### Chapter 4. Conditional Probability

Abstract
Let $$(\varOmega,\mathcal{F},\mathrm{P})$$ be a probability space and $$A \in \mathcal{ F}$$ an event having strictly positive probability. Recall that the conditional probability of P with respect to A is the probability P A on $$(\varOmega,\mathcal{F})$$, which is defined as
$$\displaystyle{\mathrm{P}_{A}(B) ={ \mathrm{P}(A \cap B) \over \mathrm{P}(A)} \quad \text{for every }B \in \mathcal{ F}\ .}$$
Intuitively the situation is the following: initially we know that every event $$B \in \mathcal{ F}$$ can appear with probability P(B). If, later, we acquire the information that the event A has occurred or will certainly occur, we replace the law P with P A , in order to keep into account the new information.
Paolo Baldi

### Chapter 5. Martingales

Abstract
Martingales are stochastic processes that enjoy many important, sometimes surprising, properties. When studying a process X, it is always a good idea to look for martingales “associated” to X, in order to take advantage of these properties.
Paolo Baldi

### Chapter 6. Markov Processes

Abstract
In this chapter we introduce an important family of stochastic processes. Diffusions, which are the object of our investigation in the subsequent chapters, are instances of Markov processes.
Paolo Baldi

### Chapter 7. The Stochastic Integral

Abstract
Let $$B = (\varOmega,\mathcal{F},(\mathcal{F}_{t})_{t},(B_{t})_{t},\mathrm{P})$$ be a (continuous) standard Brownian motion fixed once and for all: the aim of this chapter is to give a meaning to expressions of the form $$\displaystyle{ \int _{0}^{T}X_{ s}(\omega )\,dB_{s}(\omega ) }$$ where the integrand (X s )0 ≤ sT is a process enjoying certain properties to be specified. As already remarked in Sect. 3.​3, this cannot be done path by path as the function tB t (ω) does not have finite variation a.s. The r.v. (7.1) is a stochastic integral and it will be a basic tool for the construction and the investigation of new processes.
Paolo Baldi

### Chapter 8. Stochastic Calculus

Abstract
A process admitting a stochastic differential is called an Ito process. An Ito process is therefore the sum of a process with finite variation and of a local martingale.
Paolo Baldi

### Chapter 9. Stochastic Differential Equations

Abstract
In this chapter we introduce the notion of a Stochastic Differential Equation. In Sects. 9.4, 9.5, 9.6 we investigate existence and uniqueness. In Sect. 9.8 we obtain some L p estimates that will allow us to specify the regularity of the paths and the dependence from the initial conditions. In the last sections we shall see that the solution of a stochastic differential equation is a Markov process and even a diffusion associated to a differential operator that we shall specify.
Paolo Baldi

### Chapter 10. PDE Problems and Diffusions

Abstract
In this chapter we see that the solutions of some PDE problems can be represented as expectations of functionals of diffusion process. These formulas are very useful from two points of view. First of all, for the investigation and a better understanding of the properties of the solutions of these PDEs. Moreover, in some situations, they allow to compute the solution of the PDE (through the explicit computation of the expectation of the corresponding functional) or the expectation of the functional (by solving the PDE explicitly). The exercises of this chapter and Exercise 12.8 provide some instances of this way of reasoning.
Paolo Baldi

### Chapter 11. ∗Simulation

Abstract
Applications often require the computation of the expectation of a functional of a diffusion process. But for a few situations there is no closed formula in order to do this and one must recourse to approximations and numerical methods. We have seen in the previous chapter that sometimes such an expectation can be obtained by solving a PDE problem so that specific numerical methods for PDEs, such as finite elements, can be employed. Simulation of diffusion processes is another option which is explored in this chapter.
Paolo Baldi

### Chapter 12. Back to Stochastic Calculus

Abstract
The example above introduces a subtle way of obtaining new processes from old: just change the underlying probability. In this section we develop this idea in full generality, but we shall see that the main ideas are more or less the same as in this example.
Paolo Baldi

### Chapter 13. An Application: Finance

Abstract
Stochastic processes are useful as models of random phenomena, among which a particularly interesting instance is given by the evolution of the values of financial (stocks, bonds, …) and monetary assets listed on the Stock Exchange.
Paolo Baldi

### Backmatter

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