A reading guide for last passage times with financial applications in view

Abstract

In this survey on last passage times, we propose a new viewpoint which provides a unified approach to many different results which appear in the mathematical finance literature and in the theory of stochastic processes. In particular, we are able to improve the assumptions under which some well-known results are usually stated. Moreover we give some new and detailed calculations for the computation of the distribution of some large classes of last passage times. We have kept in this survey only the aspects of the theory which we expect potentially to be relevant for financial applications.

Appendix

In this appendix, we recall from [24] a few facts about the natural conditions and the Parathasarathy conditions (P). For more details see [24].

Most of the properties which generally hold under the usual conditions remain valid under the natural conditions (for example, existence of càdlàg versions of martingales, the Doob–Meyer decomposition, the début theorem, etc.). Let us recall here the definition.

Definition A.1

A filtered probability space $(\varOmega,\mathcal{F}, (\mathcal {F}_{t})_{t \geq0}, \mathbb{P})$ satisfies the natural conditions if the following two assumptions hold:

• The filtration $(\mathcal{F}_{t})_{t \geq0}$ is right-continuous.

• For all t≥0, and for every $\mathbb{P}$-negligible set $A \in\mathcal{F}_{t}$, all subsets of A are contained in $\mathcal{F}_{0}$.

This definition is slightly different from the definitions given in [5] and [24], but one can easily check that it is equivalent. The natural enlargement of a filtered probability space can be defined by using the following result.

Proposition A.2

([24])

Let $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq0}, \mathbb{P})$ be a filtered probability space. There exists a unique filtered probability space $(\varOmega, \widetilde {\mathcal{F}}, (\widetilde{\mathcal{F}}_{t})_{t \geq0}, \widetilde{\mathbb{P}})$ (with the same set Ω) such that:

• For all t≥0, $\widetilde{\mathcal{F}}_{t}$ contains $\mathcal{F}_{t}$, $\widetilde{\mathcal{F}}$ contains $\mathcal{F}$, and $\widetilde{\mathbb{P}}$ is an extension of $\mathbb{P}$.

• The space $(\varOmega, \widetilde{\mathcal{F}}, (\widetilde{\mathcal{F}}_{t})_{t \geq0}, \widetilde{\mathbb{P}})$ satisfies the natural conditions.

• For any filtered probability space $(\varOmega, \mathcal{F}', (\mathcal{F}'_{t})_{t \geq0}, \mathbb{P}')$ satisfying the two items above, $\mathcal{F}'_{t}$ contains $\widetilde {\mathcal{F}}_{t}$ for all t≥0, $\mathcal{F}'$ contains $\widetilde{\mathcal{F}}$, and $\mathbb {P}'$ is an extension of  $\widetilde{\mathbb{P}}$.

The space $(\varOmega, \widetilde{\mathcal{F}}, (\widetilde{\mathcal{F}}_{t})_{t \geq0}, \widetilde{\mathbb{P}})$ is called the natural enlargement of $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq0}, \mathbb{P})$.

Intuitively, the natural enlargement of a filtered probability space is its smallest extension which satisfies the natural conditions. We also introduce a class of filtered measurable spaces $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq0})$ such that any consistent family $(\mathbb {Q}_{t})_{t \geq0}$ of probability measures, with $\mathbb{Q}_{t}$ defined on $\mathcal {F}_{t}$, can be extended to a probability measure $\mathbb{Q}$ defined on $\mathcal{F}$.

Definition A.3

Let $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq0})$ be a filtered measurable space such that $\mathcal{F}$ is the σ-algebra generated by $\mathcal{F}_{t}$, t≥0, i.e., $\mathcal{F}=\bigvee_{t\geq0}\mathcal{F}_{t}$. We say that the property ¹ (P) holds if $(\mathcal{F}_{t})_{t \geq0}$ enjoys the following properties:

• For all t≥0, $\mathcal{F}_{t}$ is generated by a countable number of sets.

• For all t≥0, there exist a Polish space Ω _t and a surjective map π _t from Ω to Ω _t such that $\mathcal{F}_{t}$ is the σ-algebra of the inverse images, by π _t , of Borel sets in Ω _t , and such that for all $B \in\mathcal{F}_{t}$, ω∈Ω, π _t (ω)∈π _t (B) implies ω∈B.

• If (ω _n ) _n≥0 is a sequence of elements of Ω such that for all N≥0,

  $$\bigcap_{n = 0}^{N} A_n (\omega_n) \neq\emptyset, $$

  where A _n (ω _n ) is the intersection of the sets in $\mathcal {F}_{n}$ containing ω _n , then

  $$\bigcap_{n = 0}^{\infty} A_n (\omega_n) \neq\emptyset. $$

A fundamental example of a filtered measurable space $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq0})$ satisfying the property (P) can be constructed as follows. We take Ω to be equal to $\mathcal{C}(\mathbb{R}_{+},\mathbb{R}^{d})$, the space of continuous functions from $\mathbb{R}_{+}$ to $\mathbb{R}^{d}$, or $\mathcal{D}(\mathbb {R}_{+},\mathbb{R}^{d})$, the space of càdlàg functions from $\mathbb{R}_{+}$ to $\mathbb{R}^{d}$ (for some d≥1), and for t≥0, we define $(\mathcal{F}_{t})_{t \geq0}$ as the natural filtration of the canonical process, and we set

$$\mathcal{F} := \bigvee_{t\geq0}\mathcal{F}_t. $$

The combination of the property (P) and the natural conditions gives the following notion.

Definition A.4

Let $(\varOmega, \mathcal{F},(\mathcal{F}_{t})_{t \geq0}, \mathbb{P})$ be a filtered probability space. We say that it satisfies the property (NP) if it is the natural enlargement of a filtered probability space $(\varOmega, \mathcal{F}^{0},(\mathcal {F}^{0}_{t})_{t \geq0}, \mathbb{P}^{0})$ such that the filtered measurable space $(\varOmega, \mathcal {F}^{0},(\mathcal{F}^{0}_{t})_{t \geq0})$ enjoys property (P).

In [24], the following result about extensions of probability measures is proved (in a slightly more general form).

Proposition A.5

Let $(\varOmega, \mathcal{F}, (\mathcal{F}_{t})_{t \geq0}, \mathbb{P})$ be a filtered probability space satisfying property (NP). Then the σ-algebra $\mathcal{F}$ is the σ-algebra generated by $(\mathcal{F}_{t})_{t \geq0}$, and for all consistent families of probability measures $(\mathbb{Q}_{t})_{t \geq0}$ such that $\mathbb{Q}_{t}$ is defined on $\mathcal{F}_{t}$ and is absolutely continuous with respect to the restriction of $\mathbb{P}$ to $\mathcal{F}_{t}$, there exists a unique probability measure $\mathbb{Q}$ on $\mathcal{F}$ which coincides with $\mathbb{Q}_{t}$ on $\mathcal{F}_{t}$ for all t≥0.