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Mimicking an Itō process by a solution of a stochastic differential equation. (English) Zbl 1284.60109

The authors construct a process that mimics certain properties of a given Itō process, but is simpler in the sense that the mimicking process is a weak solution to a stochastic differential equation, while the Itō process may have drift and diffusion terms that are themselves stochastic processes. The work is motivated by the problem of model calibration in finance. The requirements to the Itō process are reduced to its integrability, without any conditions of nondegeneracy and boundedness of the covariance. It is shown that the mimicking process can preserve the joint distribution of certain functionals of the Itō process (e.g., running maximum and running average) at each fixed time. To prove the main result, a weakly relatively compact sequence of processes that mimic some initial target process is constructed. It is proved that the mimicking property is preserved under weak convergence and the semimartingale characteristics of the limiting process are computed.

MSC:

60G99 Stochastic processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
91G20 Derivative securities (option pricing, hedging, etc.)
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