# T-Stability of the Euler-Maruyama Algorithm for the Generalized Black-Scholes Model with Fractional Brownian Motion
##### Volume 14, Number 4, April 2018, pp. 815-820 DOI: 10.23940/ijpe.18.04.p23.815820
^{Hui Yu}
College of Science, Heilongjiang Bayi Agricultural University, Daqing, 163319, China
(Submitted on December 21, 2017; Revised on January 29, 2018; Accepted on March 5, 2018)
## Abstract:
On account of the fact that a fractional Brownian motion (fBm) with the Hurst parameter *H ∈(0,1/2)∪(1/2,1)* cannot follow the laws of the semimartingale and the Markov process, little work is presented about the T-stability for stochastic differential equations (SDEs) with fBm. Here, three results are obtained for the generalized Black-Scholes model (SDE) with *H∈(1/3,1/2)*. Firstly, the sufficient conditions of the stochastical and asymptotical stability in the large for such equation are presented by the aid of the Lyapunov exponent. Secondly, the Euler-Maruyama (EM) numerical algorithm with a given step-size for such model is constructed. Lastly, by taking advantage of the stable average function, the sufficient conditions of the T-stability that originated from the EM algorithm are presented. All the results show that on the basis of the stability of such equation, the T-stable region produced by the EM algorithm can be found. Moreover, one numerical example is afforded to the main conclusions.
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