Abstract
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.
Highlights
The question of approximating prices in financial markets with continuous time using prices in markets with discrete time goes back to the approximation of Black–Scholes prices with prices changing in discrete time
For an initial acquaintance with the subject, we recommend a book, Föllmer and Schied (2011), that starts with the central limit theorem for approximation of the Black–Scholes model by the Cox–Ross–Rubinstein model. This area of research is immeasurably wider, since there are many more market models. They are functioning in discrete time, but their analytical research is easier to carry out in continuous time
Concerning finance, functional limit theorems allow us to investigate how the convergence of stock prices affects the convergence of option prices
Summary
The question of approximating prices in financial markets with continuous time using prices in markets with discrete time goes back to the approximation of Black–Scholes prices with prices changing in discrete time. The existence of arbitrage was first established in the paper Rogers (1997) and discussed in detail in the book Mishura (2008) Such an approach has the right to exist, if only because regardless of possible financial applications, it is reasonable to prove functional limit theorems in which the limit process is a fractional Brownian motion or some related process. Even fractional Black–Scholes model can be approximated by various discrete-time sequences, and the purpose of this article is to formulate and illustrate by examples the functional limit theorem and its multiplicative version, in which both the prelimit sequence of processes and the limiting process are quite general, but simple to consider. We explore the connection between this decomposition and time series prediction problems
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