scholarly journals Berry–Esséen bounds and almost sure CLT for the quadratic variation of the sub-fractional Brownian motion

2011 ◽  
Vol 375 (2) ◽  
pp. 667-676 ◽  
Author(s):  
Constantin Tudor
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Quadratic Variation

scholarly journals Limit theorems for a quadratic variation of Gaussian processes

2011 ◽  
Vol 16 (4) ◽  
pp. 435-452 ◽  
Author(s):  
Raimondas Malukas
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Fractional Brownian Motion ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Quadratic Variation

In the paper a weighted quadratic variation based on a sequence of partitions for a class of Gaussian processes is considered. Conditions on the sequence of partitions and the process are established for the quadratic variation to converge almost surely and for a central limit theorem to be true. Also applications to bifractional and sub-fractional Brownian motion and the estimation of their parameters are provided.

ARBITRAGE IN FRACTAL MODULATED BLACK–SCHOLES MODELS WHEN THE VOLATILITY IS STOCHASTIC

2005 ◽  
Vol 08 (03) ◽  
pp. 283-300 ◽  
Author(s):  
ERHAN BAYRAKTAR ◽  
H. VINCENT POOR
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Prices ◽  
Stock Price ◽  
Time Change ◽  
Quadratic Variation ◽  
Model Driven ◽  
Black Scholes ◽  
Arbitrage Strategy ◽  
Adapted Process

In this paper an arbitrage strategy is constructed for the modified Black–Scholes model driven by fractional Brownian motion or by a time changed fractional Brownian motion, when the volatility is stochastic. This latter property allows the heavy tailedness of the log returns of the stock prices to be also accounted for in addition to the long range dependence introduced by the fractional Brownian motion. Work has been done previously on this problem for the case with constant "volatility" and without a time change; here these results are extended to the case of stochastic volatility models when the modulator is fractional Brownian motion or a time change of it. (Volatility in fractional Black–Scholes models does not carry the same meaning as in the classic Black–Scholes framework, which is made clear in the text.) Since fractional Brownian motion is not a semi-martingale, the Black–Scholes differential equation is not well-defined sense for arbitrary predictable volatility processes. However, it is shown here that any almost surely continuous and adapted process having zero quadratic variation can act as an integrator over functions of the integrator and over the family of continuous adapted semi-martingales. Moreover it is shown that the integral also has zero quadratic variation, and therefore that the integral itself can be an integrator. This property of the integral is crucial in developing the arbitrage strategy. Since fractional Brownian motion and a time change of fractional Brownian motion have zero quadratic variation, these results are applicable to these cases in particular. The appropriateness of fractional Brownian motion as a means of modeling stock price returns is discussed as well.

scholarly journals Estimating the Hurst index of the solution of a stochastic integral equation

2009 ◽  
Vol 50 ◽  
Author(s):  
Kęstutis Kubilius ◽  
Dmitrij Melichov
Keyword(s):  
Integral Equation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Integral ◽  
Second Order ◽  
Quadratic Variation ◽  
Consistent Estimator ◽  
Hurst Index ◽  
Stochastic Integral Equation ◽  
Strongly Consistent

Let X(t) be a solution of a stochastic integral equation driven by fractional Brownian motion BH and let V2n (X, 2) = \sumn-1 k=1(\delta k2X)2 be the second order quadratic variation, where \delta k2X = X (k+1/N) − 2X (k/ n) +X (k−1/n). Conditions under which n2H−1Vn2(X, 2) converges almost surely as n → ∞ was obtained. This fact is used to get a strongly consistent estimator of the Hurst index H, 1/2 < H < 1. Also we show that this estimator retains its properties if we replace Vn2(X, 2) with Vn2(Y, 2), where Y (t) is the Milstein approximation of X(t).

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