Jump SDEs and the Study of Their Densities

Poisson processes are generalizations of the Poisson distribution which are often used to describe the random behavior of some counting random quantities such as the number of arrivals to a queue, the number of hits to a webpage etc.

In this chapter, we enlarge on the previous chapter by considering processes whose jumps may be random but independent between them. We also give some further definitions on general theory of stochastic processes and stochastic analysis which may be easier to understand given that they are used in this particular application.

In this chapter, we will generalize the previous construction of compound Poisson processes and allow the possibility of a infinite number of jumps on a fixed interval. The stochastic process constructed in this section will satisfy that the number of jumps whose absolute size is larger than any fixed positive value is finite in any fixed interval. Therefore the fact that there are infinite number of jumps is due to the fact that most of these jumps are small in size. The conditions imposed will also imply that the generated stochastic process has paths of bounded variation and therefore Stiltjes integration can be used to give a meaning to stochastic integrals. We also introduce the associated stochastic calculus.

In this chapter, we consider a class of Lévy processes which are not of bounded variation as in the preceding chapter but instead they are processes with paths of infinite variation. From the pedagogical point of view, this chapter provides the construction of the Lévy process, leaving for the reader most of the developments related to the construction of the stochastic integral, the Itô formula and the associated stochastic differential equations. This is done in the exercises in order to let you test your understanding of the subject. This is done on two levels. You will find the ideas written in words in the proofs. If you do not understand them you may try a further description that may be given in Chap. 14. It is a good exercise to try to link the words and the equations so that you understand the underlying meaning. This is also a chapter that may be used for promoting discussion between students and the guiding lecturer.

We briefly present in this chapter the definition and the regularity properties of the law of general Lévy processes in many dimensions. We could have taken the same approach as in previous chapters going slowly from Poisson processes to compound Poisson processes, finite variation and then infinite variation Lévy processes in many dimensions.

In this chapter, we will discuss how to obtain the flow properties for solutions of stochastic differential equations with jumps. This chapter is needed for the second part of this book and as the final goal is not to give a detailed account of the theory of stochastic differential equations driven by jump processes, we only give the main arguments, referring the reader to any specialized text on the subject. For example, see [2] (Sect. 6.6) or [48].

We learn early in any probability theory course that in order to compute any significant quantity we need to have information regarding the distribution function of random variables. This issue appears not only in applied problems where actual computation needs to be carried out but also in theoretical problems where qualitative information of the distribution function is needed.

In the previous chapters we started studying the density of some Lévy processes using some ad-hoc techniques (see e.g. Exercises 4.1.23 and 5.1.17). In order to study densities of random variables there are many different techniques. We will briefly describe some of them in this chapter. Most of these techniques are analytic in nature and they give a different range of results. We concentrate here on the multi-dimensional results, while in Chap. 1 some basic results were discussed in the one-dimensional case.

As explained in Sect. 8.2, the goal of this second part is to show how to obtain integration by parts (IBP) formulas for random variables which are obtained through systems based on an infinite sequence of independent random variables.

In many applied problems, one needs to compute expectations of a function of a random variable which are obtained through a certain theoretical development. This is the case of \( \mathbb {E}[G(Z_t)] \), where Z is a Lévy process with Lévy measure \( \nu \) which may depend on various parameters. Similarly, G is a real-valued bounded measurable function which may also depend on some parameters and is not necessarily smooth. For many stability reasons one may be interested in having explicit expressions for the partial derivatives of the previous expectation with respect to the parameters in the model. These quantities are called “Greeks” in finance but they may have different names in other fields.

In this chapter, we extend the method of analysis introduced in Chapter 11 to a general framework. This method was essentially introduced by Norris to obtain an integration by parts (IBP) formula for jump-driven stochastic differential equations. We focus our study on the directional derivative of the jump measure which respect to the direction of the Girsanov transformation. We first generalize the method in order to consider random variables on Poisson spaces and then show in various examples how the right choice of direction of integration is an important element of this formula.

The purpose of this chapter is to show an application of the concepts of integration by parts introduced so far in an explicit example. With this motivation in mind, we have chosen as a model the Boltzmann equation. The Boltzmann equation is a non-linear equation related to the density of particles within the gas. We consider an imaginary situation where particles collide in a medium and we observe a section of it, say \(\mathbb {R}^2\). These particles collide at a certain angle \(\theta \) and velocity v, which generates a certain force within a gas. The Boltzmann equation main quantity of interest, \(f_t(v)\), describes the density of particles traveling at speed v at time \(t>0\) supposing an initial distribution \(f_0\). We assume that these densities are equal all over the section and therefore independent of the position within the section. The feature of interest to be proven here is that even if \( f_0 \) is a degenerate law in the sense that it may be concentrated at some points, the noise in the corresponding equation will imply that for any \( t>0 \) \( f_t \) will be a well-defined function with some regularity. The presentation format in this section follows closely the one presented in [7] with some simplifications. This field of research is growing very quickly and therefore even at the present time the results presented here may be outdated. Still, our intention is to provide an explicit example of application of the method presented in the previous chapter. For the same reason, not all proofs are provided and some facts are relegated to exercises with some hints given in Chap. 14.

This chapter collects hints to solve various exercises. These solutions are not complete by any means, the given arguments can only be considered at best to be heuristic and need to be completed by the reader. These are a level above the hints given in each corresponding exercise which are given because we believe that some of the exercises may be difficult or even that some misunderstanding may occur.