Strong convergence rates for markovian representations of fractional processes

Harms, Philipp

doi:10.3934/dcdsb.2020367

Cited by 3 publications

(6 citation statements)

References 33 publications

(47 reference statements)

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“…Inspired by conventional techniques on linear time invariant systems, we introduce the "spectral" representation of the kernel K. This transformation is also proposed in Refs. [44,45].…”

Section: A Approximation Of Fractional Brownian Motionmentioning

confidence: 99%

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Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

et al. 2021

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“…Inspired by conventional techniques on linear time invariant systems, we introduce the "spectral" representation of the kernel K. This transformation is also proposed in Refs. [44,45].…”

Section: A Approximation Of Fractional Brownian Motionmentioning

confidence: 99%

“…Following the idea of [45], with an appropriate quadrature, the integral can be replaced by a system of finite number of Ornstein-Uhlenbeck processes. We call X ∞ t the process defined with the infinite sum and X N t the one obtained with N points of quadrature.…”

Section: Processmentioning

confidence: 99%

Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

et al. 2021

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“…For instance, in [49], it is shown that the fBM can be represented as a linear functional of an infinite-dimensional Markov process. Later in [37], the fBM was represented as an integral over a family of the Ornstein-Uhlenbeck (OU) processes. The author proposes numerical discretizations, which have strong convergence rates of an arbitrarily high polynomial order.…”

mentioning

confidence: 99%

“…The author concludes that his OU-OU model on timescales of days, weeks and months works well (at least, not worse than the rough volatility models), and is much easier to deal with being a bivariate Gaussian diffusion, amenable to the multiscale option pricing techniques. It is also worth mentioning that the approach of [51] can be considered as a simplistic approximation of the approach in [37], where only two OU processes out of the whole family are taken into account.…”

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confidence: 99%

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Short time behavior of the ATM implied skew in the ADO-Heston model

Itkin

2024

FMF

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In the current literature, there exists an opinion that Markovian approximations of rough volatility models are able to catch the behavior of the at-the-money implied volatility skew S(T ) at small maturities T for vanilla options, while for forward started options this is not the case. To analyze this, similar to [P. Carr, A. Itkin, 2019], we construct another Markovian approximation of the rough Heston-like volatility model -the ADO-Heston model. By using the Dobric-Ojeda process instead of the fractional Brownian motion in the rough variance process, we came up to an inhomogeneous Markovian model where the characteristic function (CF) solves an unsteady three-dimensional inhomogeneous partial differential equation (PDE) with some coefficients being function of the Hurst exponent H. It then turned out that the characteristic function of the log-price and S(T ) in this model can be represented in a semi-analytical form. Since we were basically interested in the behavior of the S(T ) at T → 0, instead of using numerical inverse transforms, we further simplified the representation of the CF for vanillas by making a particular choice of the market price of risk and looking at the asymptotic of the CF at T → 0. It was observed that the behavior of thus obtained S(T ) ∝ a(H)T b•(H−1/2) , b = const was not exactly the same as in rough volatility models since b ̸ = 1, but to be close enough for all practical values of T > 0. Thus, we concluded that the proposed Markovian model was able to replicate the behavior of the corresponding rough volatility model for the range of T inherent to the options' market. Similar analysis was provided for the forward starting options, where the S(t, s, T ) for the forward started options exhibited a similar behavior for any s > t when T → s.

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Approximation of Stochastic Volterra Equations with kernels of completely monotone type

Alfonsi,

Kebaier

2023

Math. Comp.

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In this work, we develop a multifactor approximation for d d -dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an L 2 L^2 -estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in ( 0 , 1 / 2 ) (0,1/2) , we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

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Strong convergence rates for markovian representations of fractional processes

Cited by 3 publications

References 33 publications

Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

Short time behavior of the ATM implied skew in the ADO-Heston model

Approximation of Stochastic Volterra Equations with kernels of completely monotone type

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