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Extremal behavior of diffusion models in finance. (English) Zbl 0931.60036

The authors study the extremal behavior of diffusion processes \(X_{t}\) defined by SDE \(dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dw(t),\) with \(t>0\) and \(X_{0}=x,\) where \(w\) is a standard Brownian motion, \(\mu\) is the drift term and \(\sigma\) is the diffusion coefficient or volatility. Extremal behavior of a stochastic process \(X_{t}\) is for instance manifested in the asymptotic behavior of the running maxima \(M^{X}_{t}=\max_{0\leq s\leq t}X_{s},\;t>0,\) or running minima. The authors restrict themselves to the investigation of maxima, the mathematical treatment for minima being similar. The authors furthermore investigate the point process of \(\varepsilon\)-upcrossing of a high threshold \(u\) by \(X_{t}.\) For fixed \(\varepsilon>0\) the process has an \(\varepsilon\)-upcrossing at \(t\) if it has remained below \(u\) on the interval \((t-\varepsilon,t)\) and is equal to \(u\) at \(t.\) Under the weak conditions the point process of \(\varepsilon\)-upcrossing, properly scaled in time and space, converges in distribution to a homogeneous Poisson process. Two standard models like above-mentioned SDE in finance are the Black-Scholes model and the Vasicek model. The paper also aims at applications in finance: for the various models, namely Vasicek model, Cox-Ingersoll-Ross model and its generalized version, the generalized hyperbolic diffusion, the authors derive the distributional behavior of \(M^{X}_{t}\) as \(t\to+\infty\) which describes the extremal behavior of the whole process \(X_{t}.\) A new model, the generalized inverse Gaussian diffusion, is introduced. The results of this paper can be applied to study the risk measures of financial products as for instance the value of risk or related quantile risk measures.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60J60 Diffusion processes
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60G70 Extreme value theory; extremal stochastic processes
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