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Multiscale stochastic volatility asymptotics
 SIAM J. MULTISCALE MODELING AND SIMULATION
, 2003
"... In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the pri ..."
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Cited by 28 (11 self)
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In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the socalled term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the
Stability Of An Adaptive Regulator For Partially Known Nonlinear Stochastic Systems
 SIAM J. Control Optim
, 1999
"... . We investigate the properties of a fastidentification style of control algorithm applied to a class of stochastic dynamical systems in continuous time which are sampled at a constant rate. The algorithm does not assume that the system dynamics are known and estimates them using a simp ..."
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Cited by 2 (1 self)
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.<F3.835e+05> We investigate the properties of a fastidentification style of control algorithm applied to a class of stochastic dynamical systems in continuous time which are sampled at a constant rate. The algorithm does not assume that the system dynamics are known and estimates them using a simple filter. Under a mild smoothness condition on the system dynamics, we show that when the sampling rate is su#ciently fast, the control algorithm stabilizes the system in the sense that the sampled closedloop system becomes an ergodic Markov chain. Moreover, an explicit bound is given for the expected deviation of the system state from the origin. The result is also adapted for the case where statemeasurement is subject to random noise.<F4.005e+05> Key words.<F3.835e+05> adaptive, stochastic, control, nonlinear systems<F4.005e+05> AMS subject classifications.<F3.835e+05> 60H30, 60J25, 93C10, 93C90, 93E15, 93E35<F4.005e+05> PII.<F3.835e+05> S0363012997331512<F5.293e+05> 1. Introduction.<F4...
Deterministic and Random Dynamical Systems: Theory and Numerics
, 2001
"... The theory of (random) dynamical systems is a framework for the analysis of large time behaviour of timeevolving systems (driven by noise). These notes contain an elementary introduction to the theory of both dynamical and random dynamical systems. The subject matter is made accessible by means ..."
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Cited by 1 (0 self)
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The theory of (random) dynamical systems is a framework for the analysis of large time behaviour of timeevolving systems (driven by noise). These notes contain an elementary introduction to the theory of both dynamical and random dynamical systems. The subject matter is made accessible by means of very simple examples and highlights relationships between the deterministic and the random theories.
Two Deterministic Growth Models Related to DiffusionLimited Aggregation
, 1999
"... Growth models like DiffusionLimited Aggregation (DLA) have been actively studied in mathematics and physics the last twenty years. In this thesis we work on two deterministic models related to nonbranching conformal DLA, a model describing the growth, of a simply connected set of n >= 2 arcs in th ..."
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Cited by 1 (0 self)
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Growth models like DiffusionLimited Aggregation (DLA) have been actively studied in mathematics and physics the last twenty years. In this thesis we work on two deterministic models related to nonbranching conformal DLA, a model describing the growth, of a simply connected set of n >= 2 arcs in the plane with one common endpoint, along the external ray and in proportion to the harmonic measure of the tip of each arc. In its customary
vorgelegt von
"... from uncovered interest rates parity towards the identification of exchange rates risk premia ..."
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from uncovered interest rates parity towards the identification of exchange rates risk premia