Shot-Noise Processes in Finance

Schmidt, Thorsten

doi:10.1007/978-3-319-50986-0_18

Cited by 13 publications

(8 citation statements)

References 43 publications

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“…Stock price models with the (mixed) GFBM. Stock pricing models with a shot-noice component have been developed to study credit and insurance risks [2,38,43]. In particular, the stock price P (t) is modeled as…”

Section: Applications In Financial Modelsmentioning

confidence: 99%

“…µ ∈ R and σ > 0 are some real constants. Equivalent martingale measures for this price process are studied in [38,39]. In [38], it is also discussed when the shot-noise component is Markovian or a semimartingale.…”

Section: Applications In Financial Modelsmentioning

confidence: 99%

“…The price models we introduce in Section 6.2 generalize those using shot noise process and FBMs in the literature [23,2,38,43]. When the mixed BM and GFBM processes are semimartingales, we derive the Radon-Nikodym derivative for the equivalent martingale measure (Proposition 6.1), which can then be used for the price dynamics of various options.…”

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confidence: 99%

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

Ichiba¹,

Pang²,

Taqqu³

2020

Preprint

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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.

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Section: Applications In Financial Modelsmentioning

confidence: 99%

Section: Applications In Financial Modelsmentioning

confidence: 99%

See 1 more Smart Citation

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

Ichiba¹,

Pang²,

Taqqu³

2020

Preprint

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“…Applications of shot-noise processes are found in many areas besides queueing, covering, for example, the occurrence of earthquakes [82,103], water flows [104], financial models [90,93] and insurance risk [68]. In the latter case, some results can immediately be translated from the insurance to the queueing setting by using specific duality relations between the two classes of models.…”

Section: Introductionmentioning

confidence: 99%

Shot-noise queueing models

Boxma

Mandjes

2021

Queueing Syst

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We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process.

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“…We have mentioned the papers (Klüppelberg & Kühn, ; Klüppelberg & Mikosch, ; Klüppelberg et al., ). Earlier studies include (Samorodnitsky, ) and (Chobanov, ), and we also refer to the recent studies in financial models in Schmidt () and limit order books in Kaj and Caglar (). The shot noise processes can be used to model limit order books for very‐ or ultra‐high‐frequency data.…”

Section: Introductionmentioning

confidence: 99%

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

Pang

Taqqu

2019

High Frequency

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We study shot noise processes with Poisson arrivals and nonstationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: (a) the conditional variance functions of the noises have a power law and (b) the conditional noise distributions are piecewise. In both cases, the limit processes are self‐similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self‐similar Gaussian processes with nonstationary increments, via the time‐domain integral representation, which are natural generalizations of fractional Brownian motions.

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Shot-Noise Processes in Finance

Cited by 13 publications

References 43 publications

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

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

Shot-noise queueing models

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

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