General conditions of weak convergence of discrete-time multiplicative scheme to asset price with memory

We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0,T ]. Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an exa...

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Bibliographic Details
Published in:Risks (Basel) 2020-03, Vol.8 (1), p.1-29
Main Authors: Mišura, Julija S, Ralchenko, Kostiantyn, Shklyar, S. V
Format: Article
Language:English
Subjects:
Approximation
Arbitrage
asset price model with memory
binary market model
Brownian motion
Cholesky decomposition
Decomposition
fractional Brownian motion
Random variables
Securities markets
Securities prices
Stochastic models
Theorems
Volatility
weak convergence
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Summary:We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space D [ 0,T ]. Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann-Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process.
ISSN:2227-9091
2227-9091
DOI:10.3390/risks8010011