Stochastic Processes and Financial Mathematics
(part two)

Chapter 10 The transition to continuous time

Up until now, we have always indexed time by the integers, $n = 1, 2, 3, \dots$ . For the remainder of the course, we will move into continuous time, meaning that our time will be indexed as $t \in [0, \infty)$ . We need to update some of our terminology to match.

As before, we work over some probability space $(Ω, F, P)$ .

Definition 10.0.1 A stochastic process (in continuous time) is a family of random variables $(X_{t})_{t = 0}^{\infty}$ . We think of $t \in [0, \infty)$ as time.

As in discrete time, we will usual write $(X_{t}) = (X_{t})_{t = 0}^{\infty}$ . We will also sometimes write simply $X$ instead of $(X_{t})$ .

Definition 10.0.2 We say that a stochastic process $(X_{t})$ is continuous (or, equivalently, has continuous paths) if, for almost all $ω \in Ω$ , the function $t \mapsto X_{t} (ω)$ is continuous.

In words, we should think of a continuous stochastic process as a random continuous function. For example, if $A, B$ and $C$ are i.i.d. $N (0, 1)$ random variables, $X_{t} = A t^{2} + B t + C$ for $t \in [0, \infty)$ is a random continuous (quadratic) function. In this course, we will usually be more interested in situations where, in some sense, randomness appears and causes the stochastic process to change value as time passes. To do so, we need to think about filtrations.

Definition 10.0.3 We say that a family $(F_{t})$ of $σ$ -fields is a (continuous time) filtration if $F_{u} \subseteq F_{t}$ whenever $u \leq t$ .

A stochastic process $(X_{t})$ is adapted to the filtration $(F_{t})$ if $M_{t} \in m F_{t}$ for all $t \geq 0$ .

In continuous time, our standard setup is that we will work over a filtered space $(Ω, F, (F_{t}), P)$ where $(Ω, F, P)$ is a probability space and $(F_{t})$ is a filtration. If we are given a stochastic process $(X_{t})$ , implicitly over some probability space, the generated (or natural) filtration of the stochastic process is $F_{t} = σ (X_{u}); u \leq t)$ .

Lastly, we upgrade our definition of a martingale into continuous time.

Definition 10.0.4 A (continuous time) stochastic process $(M_{t})$ is a martingale if
- 1. $(M_{t})$ is adapted,
- 2. $M_{t} \in L^{1}$ for all $t$ ,
- 3. $E [M_{t} | F_{u}] = M_{u}$ for all $0 \leq u \leq t$ .
We say that $(M_{t})$ is a submartingale if, instead of 3, we have $E [M_{t} | F_{u}] \geq M_{u}$ almost surely. We say that $(M_{t})$ is a supermartingale if, instead of 3, we have $E [M_{t} | F_{u}] \leq M_{u}$ almost surely.

There are continuous time equivalents of the results (e.g. the martingale convergence theorem) that we proved for discrete time martingales, but they are outside of the scope of this course.

Organization

Whilst studying these notes you will also start thinking about revision for the summer exam. Some advice on how to structure your revision for this course is included in Appendix D. We’ll discuss this in lectures at a suitable point during the semester.

Those of you taking MAS61023 have a chapter of independent reading in Chapter 19, marked with a (). This material relies only on the previous semester and is already already accessible to you. You can study this chapter whenever you like, during semester two. Section 13.3 is also independent reading for you, but you’ll have to wait until we reach up to that point within the semester two lectures to study it.

You can find general information on the organization of this course in Section 0.1, within part one of these lecture notes.

Stochastic Processes and Financial Mathematics (part two)

Chapter 10 The transition to continuous time

Organization

Stochastic Processes and Financial Mathematics
(part two)