Lab | Stochastic processes environment

Chapter 35
Stochastic processes environment

Stochastic processes are collection of random variables ${X_{t}}_{t \in T}$ indexed by time. Therefore, they constitute the main quantitative tool to model randomness that evolves over time and make predictions.

Table 35.1 below, which we explain step by step in this section, summarizes the relevant macro-classes of processes and their field of application, in terms of

- how time is monitored: continuous versus discrete time;

- how the processes are identified: probabilistic versus mean-covariance / kernel, which in this framework we prefer naming kernel representation for reasons that will be clear soon.

Refer to the “Sequential decisions” row and “Mean-covariance” and “Probabilistic” columns of Figure 34.2 for a summary of this key elements which defines the probabilistic and the mean-covariance environment environment.

In analogy with the single variables framework Chapter 2-8, in the probabilistic environment, the evolution through time of randomness is modeled by the full distributions of all the possible subsets ${X_{t_{1}}, \dots, X_{t_{m}}}$ of instances over time. In the mean-covariance framework, the evolution through time of randomness is modelled only by the first two moments of of all the possible subsets ${X_{t_{1}}, \dots, X_{t_{m}}}$ of instances over time, discarding the rest of the information available from the full distributions.

Theory A stochastic process is a collection of random variables

{X_{t}}_{t \in T},

(35.1)

where time $t$ is either discrete ( $T \subset Z$ ) or continuous ( $T \subset R$ ) (35.9). To represent the randomness over time of the process, to each possible finite subset of the process ${X_{t}}_{t \in T}$ we assign:
- in the probabilistic representation, a distribution (35.18)-(35.23).
- in the kernel representation, a mean-covariance class (35.24), by means of an expectation $μ_{X} (t) = E {X_{t}}$ and an autocovariance function $k (u, t) =$ . $C v {X_{u}, X_{t}}$ . In each class, Gaussian processes (35.1.4) are privileged representative

We can add structure to the distributions (35.18)-(35.23) and mean-covariance classes (35.24) by imposing stationarity (35.48) and covariance stationarity (35.49)-(35.50) respectively.

In principle, all the structures summarized in Table 35.2 allow solve any prediction problem (35.73), that include forecasting, stress-test, smoothing, etc., via conditioning (35.74) or Krieging (linear projection) (35.75)-(35.76)-(35.77)

Stochastic processes are the main core of modern quantitative finance.

Practice Mean-covariance stochastic processes (35.24) are at the foundation of classic econometrics and time series analysis, see for instance [Brockwell and Davis, 2009]. Many mean-covariance models are extensively used to model financial quantities displaying trends, seasonal, and mean-reverting features.

We use extensively probabilistic processes (35.18) across the Lab, see Table 35.1. In discrete time (35.8) they play a dominant role across the “Checklist”, and in particular in the Quest for invariance Step 53. In continuous time (35.9), they play a dominant role in financial engineering, in particular for valuation purposes Section 51d.

The remainder of this chapter is organized as follows.

In Section 35.1 we provide with the main definitions and we describe both probabilistic and kernel representations of stochastic processes.

In Section 35.2 we discuss stationarity and ergodicity.

In Section 35.3 we introduce the prediction problem and its most relevant cases (forecast, smoothing etc.) to quantitative finance.

T = R,

X_{P} \sim f_{X} (x_{n_{1}, t_{1}}, \dots, x_{n_{k}, t_{k}}) .

f_{X} (x_{P}) = \int f_{X} (x_{P \cup G}) d x_{G},

{X_{t}}_{t \in T} m c \sim {μ_{X} (\cdot), k_{X} (\cdot, \cdot)}

{X_{t_{1}}, X_{t_{2}}, \dots, X_{t_{k}}} d = {X_{t_{1} + τ}, X_{t_{2} + τ}, \dots, X_{t_{k} + τ}} .

μ_{X} (t) \equiv E {X_{t}} \equiv μ_{X},

k_{X} (t + τ, t) \equiv C v {X_{t + τ}, X_{t}} \equiv κ_{X} (τ),

X_{P} | x_{G} \sim f (x_{P} | x_{G}) .

f (x_{P} | x_{G}) = \frac{f (x_{P \cup G})}{\int f (x_{P \cup G}) d x_{P}},

X_{P} ∥ x_{G} m c \sim {\begin{matrix} E {X_{P} ∥ x_{G}} C v {X_{P} ∥ x_{G}} . \end{matrix}

E {X_{P} ∥ x_{G}} = μ_{X} (P) + k_{X} (P, G) k_{X} (G, G))^{- 1} (x_{G} - μ_{X} (G)),

C v {X_{P} ∥ x_{G}} = k_{X} (P, P) - k_{X} (P, G) {(k_{X} (G, G))}_{X}^{- 1} k_{X} (P, G) .

T = Z,

	Discrete time	Continuous time

Probabilistic	Financial engineering ( $P$ world) Chapters 53a-53b	Financial engineering ( $Q$ world) Chapters 51d
Mean-covariance	Classic econometrics Section 37-39	Spectral analysis Section 38.1

(Probabilistic) processes	$\supset$	Gaussian processes	$\leftrightarrow$	Kernel representation
${X_{t}}_{t \in T} \sim f_{X} (\cdot)$	$\supset$	${X_{t}}_{t \in T} \sim G P (μ_{X} (\cdot), k_{X} (\cdot, \cdot))$	$\leftrightarrow$	${X_{t}}_{t \in T} m c \sim {μ_{X} (\cdot), k_{X} (\cdot, \cdot)}$

Chapter 35
Stochastic processes environment

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Chapter 35Stochastic processes environment

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Chapter 35
Stochastic processes environment