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2 2015

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Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process

Mishura Y.

Opuscula Mathematica

2015

Vol. 35, no. 1

99--116

We adapt the general conditions of the weak convergence for the sequence of processes with discrete time to the diffusion process towards the weak convergence for the discrete-time models of a financial market to the continuous-time diffusion model. These results generalize a classical scheme of the weak convergence for discrete-time markets to the Black-Scholes model. We give an explicit and direct method of approximation by a recurrent scheme. As an example, an Ornstein-Uhlenbeck process is considered as a limit model.

A Weak-Type Inequality for Submartingales and Itô Processes

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 1

73--88

Let α∈[0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate [WZÓR]. Here W is the weak-L∞space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.