Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance

Abstract

We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019). We discuss the applications of the GFBM and its mixtures in financial models, including stock price models, arbitrage and rough volatility. The GFBM is self-similar and has nonstationary increments, whose Hurst parameter H ∈ (0, 1) is determined by two parameters. We identify the region of these two parameter values in which the GFBM is a semimartingale. We also establish the p-variation results of the GFBM, which are used to provide an alternative proof of the non-semimartingale property when H < 1/2. We then study the semimartingale properties of the mixed process of an independent Brownian motion and a GFBM when the Hurst parameter H ∈ (1/2, 1), and derive the associated equivalent Brownian measure. INTRODUCTIONSemimartingale and non-semimartingale properties of the standard fractional Brownian motion (FBM) B H and its mixtures are well understood. These properties are important in modeling stock price [23,34], constructing arbitrage strategies and hedging policies [33,41,37,13], and modeling rough volatility [20,7, 44]. The standard FBM B H captures short/long-range dependence, and possesses the self-similar and stationary increment properties, as well as regular path properties. It may arise as the limit process of scaled random walks with long-range dependence or an integrated shot noise process [32].A generalized fractional Brownian motion (GFBM) X, introduced by Pang and Taqqu [31], is a selfsimilar Gaussian process, but does not have the stationary increments property. See the definition of the process in (2.1). The Hurst parameter H ∈ (0, 1) is determined by two-parameters (α, γ) in the range shown in Figure 1. The GFBM X is derived as the limit of integrated power-law shot noise processes in [31] (see a brief review in Section 6.1). We have studied in [21] some important path properties of the GFBM X, including the Hölder continuity property, the differentiability and non-differentiability properties, and functional and local law of the iterated logarithm, see Section 2 for a summary of its fundamental properties.

Lower functions and Chung's LILs of the generalized fractional Brownian motion

“…In [29], we studied some precise sample path properties of GFBM X, including the exact uniform modulus of continuity, small ball probabilities, and Chung's LIL at any fixed point t ą 0. In contrast to the theorems of Ichiba, Pang and Taqqu [14], our results show that the uniform modulus of continuity and Chung's LIL at any fixed point t ą 0 are determined mainly by the parameter α, while γ plays a less important role. Roughly speaking, for α ă 1{2, the results in [29] on uniform modulus of continuity and Chung's LIL at t ą 0 are analogous to the corresponding results for a fractional Brownian motion with index α `1{2.…”

“…There exists a positive constant κ 1 P p0, 8q such that lim inf Similarly to the theorems of Ichiba, Pang and Taqqu [14] mentioned above, the selfsimilarity index H plays an essential role in (1.7) and (1.8). The results in [14,29] and the present paper show that GFBM X is an interesting example of self-similar Gaussian processes which has richer sample path properties than the ordinary FBM and its close relatives such as the Riemann-Liouville FBM (cf. e.g., [5,10]), bifractional Brownian motion (cf.…”

“…Ichiba, Pang and Taqqu [13] established the Hölder continuity, the functional and local laws of the iterated logarithm of GFBM and showed that these properties are determined by the self-similarity index H " α´γ`1{2. More recently, Ichiba, Pang and Taqqu [14] studied the semimartingale properties of GFBM X and its mixtures and applied them to model the volatility processes in finance.…”

Exact uniform modulus of continuity and Chung's LIL for the generalized fractional Brownian motion

“…Ichiba, Pang and Taqqu [5] raised the interesting question: "how the parameter γ affects the sample path properties of GFBM?" They proved that, for any T ą 0 and ε ą 0, the sample paths of X are Hölder continuous in r0, T s of order H ´ε and the functional and local laws of the iterated logarithm of X are determined by the self-similarity index H. More recently, Ichiba, Pang and Taqqu [6] have studied the semimartingale properties of GFBM X and its mixtures and applied them to model the volatility processes in finance.…”