# Hybrid scheme for Brownian semistationary processes

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## Abstract

We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel function by a power function near zero and by a step function elsewhere. The resulting approximation of the process is a combination of Wiener integrals of the power function and a Riemann sum, which is why we call this method a hybrid scheme. Our main theoretical result describes the asymptotics of the mean square error of the hybrid scheme, and we observe that the scheme leads to a substantial improvement of accuracy compared to the ordinary forward Riemann-sum scheme, while having the same computational complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we examine the finite-sample properties of an estimator of the roughness parameter of a Brownian semistationary process and study Monte Carlo option pricing in the rough Bergomi model of Bayer et al. (Quant. Finance 16:887–904, 2016), respectively.

## Keywords

Stochastic simulation Discretization Brownian semistationary process Stochastic volatility Regular variation Estimation Option pricing Rough volatility Volatility smile## Mathematics Subject Classification (2010)

60G12 60G22 65C20 91G60 62M09## JEL Classification

C22 G13 C13## 1 Introduction

*Brownian semistationary*(\(\mathcal {BSS}\)) processes, first introduced by Barndorff-Nielsen and Schmiegel [8, 9], which form a flexible class of stochastic processes that are able to capture some common features of empirical time series, such as stochastic volatility (intermittency), roughness, stationarity, and strong dependence. By now these processes have been applied in various contexts, most notably in the study of turbulence in physics [7, 16] and in finance as models of energy prices [4, 11]. A \(\mathcal {BSS}\) process \(X\) is defined via the integral representation

*moving average*of volatility-modulated Brownian noise, and setting \(\sigma_{s} = 1\), we see that stationary

*Brownian moving averages*are nested in this class of processes.

*informal*sense, anticipating a rigorous formulation of this relationship given in Sect. 2.2 using the theory of

*regular variation*[15], which plays a significant role in our subsequent arguments. The case \(\alpha= -\frac{1}{6}\) in (1.2) is important in statistical modeling of turbulence [16] as it gives rise to processes that are compatible with Kolmogorov’s scaling law for ideal turbulence. Moreover, processes of similar type with \(\alpha\approx- 0.4\) have been recently used in the context of option pricing as models of

*rough volatility*[1, 10, 18, 19]; see Sects. 2.5 and 3.3 below. The case \(\alpha=0\) would (roughly speaking) lead to a process that is a semimartingale, which is thus excluded.

Under (1.2), the trajectories of \(X\) behave locally like the trajectories of a *fractional Brownian motion* with Hurst index \(H = \alpha+ \frac{1}{2} \in(0,1) \setminus\{ \frac{1}{2} \}\). While the *local* behavior and roughness of \(X\), measured in terms of Hölder regularity, are determined by the parameter \(\alpha\), the *global* behavior of \(X\) (e.g., whether the process has long or short memory) depends on the behavior of \(g(x)\) as \(x \rightarrow\infty \), which can be specified independently of \(\alpha\). This should be contrasted with fractional Brownian motion and related *self-similar* models, which necessarily must conform to a restrictive affine relationship between their Hölder regularity (local behavior and roughness) and Hurst index (global behavior), as elucidated by Gneiting and Schlather [20]. Indeed, in the realm of \(\mathcal {BSS}\) processes, local and global behavior are conveniently decoupled, which underlines the flexibility of these processes as a modeling framework.

In connection with practical applications, it is important to be able to simulate the process \(X\). If the volatility process \(\sigma\) is deterministic and constant in time, then \(X\) will be strictly stationary and Gaussian. This makes \(X\) amenable to exact simulation using the Cholesky factorization or circulant embeddings; see e.g. [2, Chapter XI]. However, it seems difficult, if not impossible, to develop an exact method that is applicable with a stochastic \(\sigma\), as the process \(X\) is then neither Markovian nor Gaussian. Thus in the general case, one must resort to approximative methods. To this end, Benth et al. [13] have recently proposed a Fourier-based method of simulating \(\mathcal{BSS}\) processes, and more general *Lévy semistationary* (\(\mathcal{LSS}\)) processes, which relies on approximating the kernel function \(g\) in the frequency domain.

In this paper, we introduce a new discretization scheme for \(\mathcal{BSS}\) processes based on approximating the kernel function \(g\) in the time domain. Our starting point is the Riemann-sum discretization of (1.1). The Riemann-sum scheme builds on an approximation of \(g\) using step functions, which has the pitfall of failing to capture appropriately the steepness of \(g\) near zero. In particular, this becomes a serious defect under (1.2) when \(\alpha \in( -\frac{1}{2},0)\). In our new scheme, we mitigate this problem by approximating \(g\) using an appropriate power function near zero and a step function elsewhere. The resulting discretization scheme can be realized as a linear combination of Wiener integrals with respect to the driving Brownian motion \(W\) and a Riemann sum, which is why we call it a *hybrid scheme*. The hybrid scheme is only slightly more demanding to implement than the Riemann-sum scheme, and the schemes have the same computational complexity as the number of discretization cells tends to infinity.

Our main theoretical result describes the exact asymptotic behavior of the mean square error (MSE) of the hybrid scheme and, as a special case, that of the Riemann-sum scheme. We observe that switching from the Riemann-sum scheme to the hybrid scheme reduces the asymptotic root mean square error (RMSE) substantially. Using merely the simplest variant the of hybrid scheme, where a power function is used in a single discretization cell, the reduction is at least \(50\%\) for all \(\alpha\in( 0,\frac{1}{2})\) and at least \(80\%\) for all \(\alpha\in( -\frac{1}{2},0)\). The reduction in RMSE is close to \(100\%\) as \(\alpha\) approaches \(-\frac{1}{2}\), which indicates that the hybrid scheme indeed resolves the problem of poor precision that affects the Riemann-sum scheme.

To assess the accuracy of the hybrid scheme in practice, we perform two numerical experiments. Firstly, we examine the finite-sample performance of an estimator of the roughness index \(\alpha\), introduced by Barndorff-Nielsen et al. [6] and Corcuera et al. [16]. This experiment enables us to assess how faithfully the hybrid scheme approximates the fine properties of the \(\mathcal {BSS}\) process \(X\). Secondly, we study Monte Carlo option pricing in the rough Bergomi stochastic volatility model of Bayer et al. [10]. We use the hybrid scheme to simulate the volatility process in this model, and we find that the resulting implied volatility smiles are indistinguishable from those obtained using a method that involves exact simulation of the volatility process. Thus we are able to propose a solution to the problem of finding an efficient simulation scheme for the rough Bergomi model, left open in the paper [10].

The rest of this paper is organized as follows. In Sect. 2, we recall the rigorous definition of a \(\mathcal {BSS}\) process and introduce our assumptions. We also introduce the hybrid scheme, state our main theoretical result concerning the asymptotics of the mean square error and discuss an extension of the scheme to a class of *truncated* \(\mathcal{BSS}\) processes. Section 3 briefly discusses the implementation of the discretization scheme and presents the numerical experiments mentioned above. Finally, Sect. 4 contains the proofs of the theoretical and technical results given in the paper.

## 2 The model and theoretical results

### 2.1 Brownian semistationary process

**Throughout the paper**, we also assume that the process \(\sigma\) has finite second moments, \(\mathbb {E}[\sigma_{t}^{2}] < \infty\) for all \(t \in \mathbb {R}\), and that the process is covariance-stationary, namely,

*strictly*stationary as the dependence between the volatility process \(\sigma\) and the driving Brownian motion \(W\) may be time-varying.

### 2.2 Kernel function

As mentioned above, we consider a kernel function that satisfies \(g(x) \propto x^{\alpha}\) for some \(\alpha\in(-\frac{1}{2},\frac{1}{2})\setminus\{0 \}\) when \(x>0\) is near zero. To make this idea rigorous and to allow additional flexibility, we formulate our assumptions on \(g\) using the theory of regular variation [15] and, more specifically, slowly varying functions.

*slowly varying*at 0 if for any \(t>0\),

*regularly varying*at 0, with \(\beta\) being the

*index of regular variation*.

### Remark 2.1

Conventionally, slow and regular variation are defined at \(\infty\) [15, Sects. 1.2.1 and 1.4.1]. However, \(L\) is slowly varying (resp. regularly varying) at 0 if and only if \(x \mapsto L(1/x)\) is slowly varying (resp. regularly varying) at \(\infty\).

*Potter bounds*for slowly varying functions; see [15, Theorem 1.5.6(ii)] and (4.1) below. Making \(\delta\) very small, we see that slowly varying functions are asymptotically negligible in comparison with polynomially growing/decaying functions. Thus, by multiplying power functions and slowly varying functions, regular variation provides a flexible framework to construct functions that behave asymptotically like power functions.

- (A1)For some \(\alpha\in(-\frac{1}{2},\frac{1}{2}) \setminus \{0\}\),where \(L_{g} : (0,1] \to[0,\infty)\) is continuously differentiable, slowly varying at 0 and bounded away from 0. Moreover, there exists a constant \(C>0\) such that the derivative \(L'_{g}\) of \(L_{g}\) satisfies$$\begin{aligned} g(x) = x^{\alpha}L_{g}(x), \quad x \in(0,1], \end{aligned}$$$$\begin{aligned} |L_{g}'(x)| \leq C(1+x^{-1}), \quad x\in(0,1]. \end{aligned}$$
- (A2)
The function \(g\) is continuously differentiable on \((0,\infty)\), with derivative \(g'\) that is ultimately monotonic and also satisfies \(\int_{1}^{\infty}g'(x)^{2} \mathrm {d}x\).

- (A3)For some \(\beta\in(-\infty,-\frac{1}{2})\),$$ g(x) = \mathcal{O}(x^{\beta}), \quad x \rightarrow\infty. $$

*variogram*(also called

*second-order structure function*in the turbulence literature)

### Proposition 2.2

*Suppose that*(A1)

*–*(A3)

*hold*.

- (i)
*The variogram of*\(X\)*satisfies*$$ V_{X}(h) \sim \mathbb {E}[\sigma_{0}^{2}] \bigg(\frac{1}{2\alpha+ 1} + \int_{0}^{\infty}\big( (y+1)^{\alpha}- y^{\alpha}\big)^{2} \mathrm {d}y\bigg) h^{2\alpha+1} L_{g}(h)^{2}, \quad h \rightarrow0, $$*which implies that*\(V_{X}\)*is regularly varying at zero with index*\(2\alpha+1\). - (ii)
*The process*\(X\)*has a modification with locally*\(\phi\)-*Hölder*-*continuous trajectories for any*\(\phi\in(0,\alpha+ \frac{1}{2})\).

Motivated by Proposition 2.2, we call \(\alpha\) the *roughness index* of the process \(X\). Ignoring the slowly varying factor \(L_{g}(h)^{2}\) in (2.2), we see that the variogram \(V(h)\) behaves like \(h^{2\alpha+1}\) for small values of \(h\), which is reminiscent of the scaling property of the increments of a fractional Brownian motion (fBm) with Hurst index \(H = \alpha+ \frac{1}{2}\). Thus, the process \(X\) behaves locally like such an fBm, at least when it comes to second order structure and roughness. (Moreover, the factor \(\frac{1}{2\alpha + 1} + \int_{0}^{\infty}( (y+ 1)^{\alpha}- y^{\alpha})^{2} \mathrm {d}y\) coincides with the normalization coefficient that appears in the Mandelbrot–Van Ness representation [23, Theorem 1.3.1] of an fBm with \(H = \alpha+ \frac{1}{2}\).)

Let us now look at two examples of a kernel function \(g\) that satisfies our assumptions.

### Example 2.3

*gamma kernel*

### Example 2.4

*power-law kernel*function

### 2.3 Hybrid scheme

*hybrid scheme*, given by

- (A4)For some \(\gamma>0\),$$\begin{aligned} N_{n} \sim n^{\gamma+ 1},\quad n \rightarrow\infty. \end{aligned}$$

### 2.4 Asymptotic behavior of mean square error

We are now ready to state our main theoretical result, which gives a sharp description of the asymptotic behavior of the mean square error (MSE) of the hybrid scheme as \(n \rightarrow\infty\). We defer the proof of this result to Sect. 4.2.

### Theorem 2.5

*Suppose that*(A1)

*–*(A4)

*hold*,

*so that*

*and that for some*\(\delta>0\),

*Then for all*\(t\in\mathbb{R}\),

*where*

### Remark 2.6

When the hybrid scheme is used to simulate the \(\mathcal{BSS}\) process \(X\) on an equidistant grid \(\{0,\frac{1}{n},\frac{2}{n},\ldots,\frac {\lfloor nT \rfloor}{n} \}\) for some \(T>0\) (see Sect. 3.1 on the details of the implementation), the following consequence of Theorem 2.5 ensures that the covariance structure of the simulated process approximates that of the actual process \(X\).

### Corollary 2.7

*Suppose that the assumptions of Theorem*2.5

*hold*.

*Then for any*\(s\), \(t\in \mathbb {R}\)

*and*\(\varepsilon>0\),

### Proof

In Theorem 2.5, the asymptotics of the MSE (2.11) are determined by the behavior of the kernel function \(g\) near zero, as specified in (A1). The condition (2.9) ensures that the error from approximating \(g\) near zero is asymptotically larger than the error induced by the truncation of the stochastic integral (2.1) at \(t - \frac{N_{n}}{n}\). In fact, a different kind of asymptotics of the MSE, where truncation error becomes dominant, could be derived when (2.9) does not hold, under some additional assumptions, but we do not pursue this direction in the present paper.

While the rate of convergence in (2.11) is fully determined by the roughness index \(\alpha\), which may seem discouraging at first, it turns out that the quantity \(J(\alpha,\kappa,\mathbf{b})\), which we call the *asymptotic* MSE, can vary a lot, depending on how we choose \(\kappa\) and \(\mathbf{b}\), and can have a substantial impact on the precision of the approximation of \(X\). It is immediate from (2.12) that increasing \(\kappa\) will decrease \(J(\alpha,\kappa ,\mathbf{b})\). Moreover, for given \(\alpha\) and \(\kappa\), it is straightforward to choose \(\mathbf{b}\) so that \(J(\alpha,\kappa,\mathbf {b})\) is minimized, as shown in the following result.

### Proposition 2.8

*Let*\(\alpha\in(-\frac{1}{2},\frac{1}{2})\setminus\{ 0\}\)

*and*\(\kappa \geq0\).

*Among all sequences*\(\mathbf{b}=(b_{k})_{k=\kappa+1}^{\infty}\)

*with*\(b_{k} \in[k-1,k]\setminus\{0 \}\)

*for*\(k \geq\kappa+1\),

*the function*\(J(\alpha,\kappa,\mathbf{b})\),

*and consequently the asymptotic MSE induced by the discretization*,

*is minimized by the sequence*\(\mathbf {b}^{*}\)

*given by*

### Proof

*root mean square error*(RMSE) \(\sqrt{J(\alpha,\kappa,\mathbf{b})}\). In particular, we assess how much the asymptotic RMSE decreases relative to the RMSE of the forward Riemann-sum scheme (\(\kappa=0\) and \(\mathbf {b} = \mathbf{b}_{\mathrm{FWD}}\)) by using the quantity

### Remark 2.9

*Hurwitz zeta function*, which can be evaluated using accurate numerical algorithms.

### Remark 2.10

### 2.5 Extension to truncated Brownian semistationary processes

*truncated Brownian semistationary*(\(\mathcal{TBSS}\)) process, as \(Y\) is obtained from the \(\mathcal{BSS}\) process \(X\) by truncating the stochastic integral in (2.1) at 0. Of the preceding assumptions, only (A1) and (A2) are needed to ensure that the stochastic integral in (2.14) exists—in fact, of (A2), only the requirement that \(g\) is differentiable on \((0,\infty)\) comes into play.

### Proposition 2.11

*Suppose that*(A1)

*and*(A2)

*hold*.

- (i)
*The variogram of*\(Y\)*satisfies for any*\(t \geq0\)*that as*\(h \rightarrow0\),$$ V_{Y}(h,t) \sim \mathbb {E}[\sigma_{0}^{2}] \bigg(\frac{1}{2\alpha+ 1} + \mathbf{1}_{(0,\infty)}(t)\int_{0}^{\infty}\big( (y+1)^{\alpha}- y^{\alpha}\big)^{2} \mathrm {d}y\bigg) h^{2\alpha+1} L_{g}(h)^{2}, $$*which implies that*\(h \mapsto V_{Y}(h,t)\)*is regularly varying at zero with index*\(2\alpha+1\). - (ii)
*The process*\(Y\)*has a modification with locally*\(\phi\)-*Hölder*-*continuous trajectories for any*\(\phi\in(0,\alpha+ \frac{1}{2})\).

Note that while the increments of \(Y\) are not covariance-stationary, the asymptotic behavior of \(V_{Y}(h,t)\) is the same as that of \(V_{X}(h)\) as \(h \rightarrow0\) (cf. Proposition 2.2) for any \(t>0\). Thus, the increments of \(Y\) (apart from increments starting at time 0) are locally like the increments of \(X\).

The MSE of the hybrid scheme for the \(\mathcal{TBSS}\) process \(Y\) has the following asymptotic behavior as \(n \rightarrow\infty\), which is in fact identical to the asymptotic behavior of the MSE of the hybrid scheme for \(\mathcal{BSS}\) processes. We omit the proof of this result, which would be a simple modification of the proof of Theorem 2.5.

### Theorem 2.12

*Suppose that*(A1)

*and*(A2)

*hold*,

*and that for some*\(\delta>0\),

*Then for all*\(t>0\),

*where*\(J(\alpha,\kappa,\mathbf{b})\)

*is as in Theorem*2.5.

## 3 Implementation and numerical experiments

### 3.1 Practical implementation

*Gauss hypergeometric function*; see e.g. [17, Sect. 2.1.1] for the definition. (When \(k < j\), set \(\varSigma _{j,k} = \varSigma_{k,j}\).) For the convenience of the reader, we provide a proof of (3.5) in Sect. 4.3.

### Remark 3.1

*all*\(i = 0,1,\ldots,\lfloor nT \rfloor\) using an FFT is \(\mathcal{O}(N_{n} \log N_{n})\), see [22, Sect. 3.3.4], which under (A4) translates to \(\mathcal{O}(n^{\gamma+1} \log n)\). The computational complexity of the entire hybrid scheme is then \(\mathcal {O}(n^{\gamma+1} \log n)\), provided that \((\sigma ^{n}_{i})_{i=-N_{n}}^{\lfloor nT \rfloor-1}\) is generated using a scheme with complexity not exceeding \(\mathcal{O}(n^{\gamma+1} \log n)\). As a comparison, we mention that the complexity of an exact simulation of a stationary Gaussian process using circulant embeddings is \(\mathcal {O}(n \log n)\) [2, Chapter XI, Sect. 3], whereas the complexity of the Cholesky factorization is \(\mathcal{O}(n^{3})\) [2, Chapter XI, Sect. 2].

### Remark 3.2

With \(\mathcal{TBSS}\) processes, the computational complexity of the hybrid scheme via (3.7) is \(\mathcal {O}(n \log n)\).

### 3.2 Estimation of the roughness parameter

*change-of-frequency*(COF) statistics

To examine how well the hybrid scheme reproduces the fine properties of the \(\mathcal {BSS}\) process in terms of regularity/roughness, we apply the COF estimator to discretized trajectories of \(X\), where the kernel function \(g\) is again the gamma kernel (Example 2.3) with \(\lambda= 1\), generated using the hybrid scheme with \(\kappa= 1,2,3\) and \(\mathbf{b} = \mathbf{b}^{*}\). We consider the case where the volatility process satisfies \(\sigma_{t} = 1\), that is, the process \(X\) is Gaussian. This allows us to quantify and control the intrinsic bias and noisiness, measured in terms of standard deviation, of the estimation method itself, by initially applying the estimator to trajectories that have been simulated using an exact method based on the Cholesky factorization. We then study the behavior of the estimator when applied to a discretized trajectory, while decreasing the step size of the discretization scheme. More precisely, we simulate \(\hat{\alpha}(X^{n},m)\), where \(m=500\) and \(X_{n}\) is the hybrid scheme for \(X\) with \(n = ms\) and \(s \in\{1,2,5\}\). This means that we compute \(\hat{\alpha}(X^{n},m)\) using \(m\) observations obtained by subsampling every \(s\)th observation in the sequence \(X^{n}_{\frac{i}{n}}\), \(i = 0,1,\ldots,n\). As a comparison, we repeat these simulations substituting the hybrid scheme with the Riemann-sum scheme, using \(\kappa= 0\) with \(\mathbf {b}\in\{\mathbf{b}_{\mathrm{FWD}},\mathbf{b}^{*}\}\).

### 3.3 Option pricing under rough volatility

As another experiment, we study Monte Carlo option pricing in the *rough Bergomi* (rBergomi) model of Bayer et al. [10]. In the rBergomi model, the logarithmic spot variance of the price of the underlying is modeled by a rough Gaussian process, which is a special case of (2.14). By virtue of the rough volatility process, the model fits well to observed implied volatility smiles [10, Sect. 5].

*forward variance curve*[10, Sect. 3], which we assume here to be flat, \(\xi^{0}_{t} = \xi>0\) for all \(t \in[0,T]\).

Parameter values used in the rBergomi model

\(S_{0}\) | | | | |
---|---|---|---|---|

1 | 0.235 | 1.9 | −0.43 | −0.9 |

We observe that the Riemann-sum scheme (\(\kappa=0\), \(\mathbf{b} \in\{\mathbf{b}_{\mathrm{FWD}},\mathbf{b}^{*}\}\)) is able to capture the shape of the implied volatility smile, but not its level. But the method even breaks down with more extreme log-strikes (the prices are so low that the root-finding algorithm used to compute the implied volatility would return zero). In contrast, the hybrid scheme with \(\kappa= 1,2\) and \(\mathbf{b}=\mathbf{b}^{*}\) yields implied volatility smiles that are indistinguishable from the benchmark smiles obtained using exact simulation. Further, there is no discernible difference between the smiles obtained using \(\kappa=1\) and \(\kappa=2\). As in the previous section, we observe that the hybrid scheme is indeed capable of producing very accurate trajectories of \(\mathcal{TBSS}\) processes, in particular in the case \(\alpha\in(-\frac {1}{2},0)\), even when \(\kappa= 1\).

## 4 Proofs

### Lemma 4.1

*If*\(\alpha\in(-1,\infty)\)

*and*\(L : (0,1] \rightarrow[0,\infty)\)

*is slowly varying at*0,

*then*

### 4.1 Proof of Proposition 2.2

### 4.2 Proof of Theorem 2.5

As a preparation, we first establish an auxiliary result that deals with the asymptotic behavior of certain integrals of regularly varying functions.

### Lemma 4.2

*Suppose that*\(L : (0,1] \rightarrow[0,\infty)\)

*is bounded away from*0

*and*\(\infty\)

*on any set of the form*\((u,1]\), \(u\in(0,1)\),

*and slowly varying at*0.

*Moreover*,

*let*\(\alpha\in(-\frac{1}{2},\infty) \)

*and*\(k \geq1\).

*If*\(b \in[k-1,k] \setminus\{0\}\),

*then*

- (i)
\({\displaystyle\lim_{n \rightarrow\infty}\int _{k-1}^{k} \bigg( x^{\alpha}\frac{L(x/n)}{L(1/n)} - b^{\alpha}\frac {L(b/n)}{L(1/n)} \bigg)^{2} \mathrm {d}x = \int_{k-1}^{k} (x^{\alpha} - b^{\alpha })^{2} \mathrm {d}x<\infty}\).

- (ii)
\({\displaystyle\lim_{n \rightarrow\infty}\int _{k-1}^{k} x^{2\alpha}\bigg( \frac{L(x/n)}{L(1/n)} - \frac {L(b/n)}{L(1/n)} \bigg)^{2} \mathrm {d}x = 0}\).

### Proof

### Proof of Theorem 2.5

We now use the obtained asymptotic relations, (4.10)–(4.12) and (4.17), to complete the proof. To this end, it is convenient to introduce the relation \(x_{n} \gg y_{n}\) for any sequences \((x_{n})_{n=1}^{\infty}\) and \((y_{n})_{n=1}^{\infty}\) of positive real numbers satisfying \(\lim_{n \rightarrow\infty}\frac {x_{n}}{y_{n}} = \infty\). By (4.12), we have \(D'_{n} \gg D_{n}\). Since \(2\alpha +1 < 2\), we find that also \(D'_{n} \gg D''_{n}\), in view of (4.11). The assumption \(\gamma> -\frac{2\alpha+1}{2\beta+ 1}\) is equivalent to \(-(2\alpha+1) > \gamma(2\beta+1)\); so by the estimate (2.2) for slowly varying functions, we have \(D'_{n} \gg D'''_{n}\). It then follows that \(E_{n} \sim \mathbb {E}[\sigma_{0}^{2}] D'_{n}\) as \(n \rightarrow \infty\). Further, the condition (2.10) implies that \(E_{n} \gg E'_{n}\). In view of (4.9), we finally find that \(\mathbb{E}[| X^{n}_{t} - X_{t} |^{2}] \sim \mathbb {E}[\sigma_{0}^{2}] D'_{n}\) as \(n \rightarrow\infty\), which completes the proof. □

### 4.3 Proof of equation (3.5)

In order to prove (3.5), we rely on the following integral identity for the Gauss hypergeometric function \({}_{2} F_{1}\).

### Lemma 4.3

*For all*\(\alpha \in(-1,\infty)\)

*and*\(0 \leq a < b\),

### Proof

### Proof of equation (3.5)

## Notes

### Acknowledgements

We should like to thank Heidar Eyjolfsson and Emil Hedevang for useful discussions regarding the simulation of \(\mathcal{BSS}\) processes and Ulises Márquez for assistance with symbolic computation. Our research has been supported by CREATES (DNRF78), funded by the Danish National Research Foundation, by Aarhus University Research Foundation (project “Stochastic and Econometric Analysis of Commodity Markets”) and by the Academy of Finland (project 258042).

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