Affine process

An affine process is a stochastic process in continuous time, the Fourier transform of which has a special shape. Many of the processes in a wide variety of applications belong to this process class, and many of the functions relevant to applications can be calculated explicitly.

definition

The Fourier transform of the transition kernel of an affine process can be written in exponential-affine form.

Or more formally:
An affine process is a stochastically continuous, time-homogeneous Markov process on where the kumulantenerzeugende function is an affine function of the initial state: ${\ displaystyle (X_ {t}) _ {t> 0}}$ ${\ displaystyle \ mathbb {R} ^ {n} \ times \ mathbb {R} _ {+} ^ {m}}$

{\ displaystyle \ Phi _ {t} (u) = \ log (\ operatorname {E} \ left [\ mathrm {e} ^ {\ langle X_ {t}, u \ rangle} \ right]) = \ phi ( t, u) + \ langle X_ {0}, \ psi (t, u) \ rangle}

for all so that the expectation exists. ${\ displaystyle u \ in \ mathbb {C} ^ {(n + m)}}$

Important properties

Affine processes are Markov processes .
If it is an affine process, so too . ${\ displaystyle (X_ {t}) _ {t> 0}}$ ${\ displaystyle \ textstyle (\ int _ {0} ^ {t} {X_ {s} \, ds}, X_ {t})}$
The expected value of an expression that is often required can be written as follows:

{\ displaystyle \ operatorname {E} \ left [\ exp \ left (- \ int _ {t} ^ {t + x} X_ {s} \, ds \ right) \ right] = \ exp {(A (x ) + X_ {t} \ cdot B (x))}.}

Since this expected value (short-rate models) is of great importance in many interest-theoretical studies, for a long time all those processes where the expected value can be written in this way have been described as affine.

The functions A and B can be written as solutions to Riccati equations .

Related processes

The Wiener process and the Poisson process are affine processes, but the Ornstein-Uhlenbeck process (both Gaussian and non-Gaussian) is an affine process, as is the root diffusion process . Every Lévy process is affine. The geometric Brownian motion is not an affine process, but a very simple functional ( exponential function ) of an affine process.

Applications

In addition to the usual applications for all of the processes mentioned in the previous section, there are also models for stochastic volatility (e.g. Heston model , Barndorff-Nielsen Shepard model, etc.). There are many applications in financial mathematics (interest rate models, credit risk, option price models, etc.).

literature

D. Duffie, D. Filipovic, W. Schachermayer : Affine Processes and Applications in Finance. Annals of Applied Probability, Vol. 13 (2003), No. 3, pp. 984-1053. on [108 ]