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Affine processes and applications in finance. (English) Zbl 1048.60059
The regular affine processes are studied systematically. Recently these processes are frequently applied to model the price processes of some derivatives in the finance market. A regular homogeneous Markov process (not necessarily conservative) \((X, (P_x)_{x \in D})\) is called a regular affine process on \(D \subset \mathbb R_+^m \times \mathbb R^n\) if for any \(t \geq 0\) the logarithm of the characteristic function of the transition function \(P_t(x,\cdot)\) has an affine dependence on its initial state \(x\). The affine process has the advantage of computational tractability and flexibility in capturing many of the empirical features of financial time series.
Basic results are done fairly completely. These include: The explicit formula of its generator; semimartingale property and the relationship between the parameters of the generator and the characteristics of this semimartingale; describing the coefficients of this affine dependence of \(x\) by a pair of generalized Riccati equations; Feller property; infinitely decomposable property and this is also shown to be a sufficient condition of a regular homogeneous Markov process to be a regular affine process; Feynman-Kac formula; a regular affine process \(X=(Y,Z)\) on \( \mathbb R_+^m \times \mathbb R_n\) is exactly the same to say that the “component process” \(Y\) is a continuous-state branching process with immigration and the “component process” \(Z\) is an Ornstein-Uhlenbeck type process. Finance application in the term structure of interest rate, default risk and options pricing are briefly discussed.

MSC:
60J25 Continuous-time Markov processes on general state spaces
91G80 Financial applications of other theories
91G20 Derivative securities (option pricing, hedging, etc.)
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