scholarly journals Weak convergence to fractional Brownian motion in some anisotropic Besov space

2004 ◽  
Vol 11 (1) ◽  
pp. 1-17 ◽  
Author(s):  
M. Ait Ouahra
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Fractional Brownian Motion ◽  
Besov Space ◽  
Anisotropic Besov Space

scholarly journals General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory

Risks ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 11
Author(s):  
Yuliya Mishura ◽  
Kostiantyn Ralchenko ◽  
Sergiy Shklyar
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Fractional Brownian Motion ◽  
Covariance Matrix ◽  
Discrete Time ◽  
Asset Price ◽  
Cholesky Decomposition ◽  
Additive Scheme ◽  
With Memory ◽  
General Conditions

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.

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