Skip to main content

A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation


This paper proposes a multidimensional Hilbert transform approach for pricing discretely monitored multi-asset barrier options and computing joint survival probability in multivariate exponential Lévy asset price models. We generalize the univariate Hilbert transform method of Feng and Linetsky (Math Financ 18(3), 337–384, 2008) for single-asset barrier options and the well-known Sinc approximation theory of Stenger (Numerical methods based on sinc and analytic functions. Springer, New York, 1993) for computing the one-dimensional Hilbert transform to any dimension. We prove that, for Lévy processes with joint characteristic functions having an exponentially decaying tail, the error of our method decays exponentially in some power of the number of terms used in the expansion for each dimension. Numerical experiments demonstrate the efficiency of our method in the two-dimensional and three-dimensional problems for some popular multivariate Lévy models.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2


  • Andricopoulos, A. D., Widdicks, M., Newton, D. P., & Duck, P. W. (2007). Extending quadrature methods to value multi-asset and complex path dependent options. Journal of Financial Economics, 83(2), 471–499.

    Article  Google Scholar 

  • Applebaum, D. (2009). Lévy processes and stochastic calculus. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Ballotta, L., & Bonfiglioli, E. (2016). Multivariate asset models using Lévy processes and applications. The European Journal of Finance, 22(13), 1320–1350.

    Article  Google Scholar 

  • Ballotta, L., Fusai, G., Loregian, A., & Perez, M. F. (2019). Estimation of multivariate asset models with jumps. Journal of Financial and Quantitative Analysis, 54(5), 2053–2083.

    Article  Google Scholar 

  • Barndorff-Nielsen, O. E. (1998). Processes of normal Inverse Gaussian type. Finance and Stochastics, 2, 41–68.

    Article  Google Scholar 

  • Bertoin, J. (1998). Lévy processes (Vol. 121). Cambridge: Cambridge University Press.

    Google Scholar 

  • Black, F., & Cox, J. C. (1976). Valuing corporate securities: Some effects of bond indenture provisions. The Journal of Finance, 31(2), 351–367.

    Article  Google Scholar 

  • Broadie, M., & Yamamoto, Y. (2005). A double-exponential fast Gauss transform algorithm for pricing discrete path-dependent options. Operations Research, 53(5), 764–779.

    Article  Google Scholar 

  • Cai, N., Li, C. & Shi, C. (2017). Correction from Black-Scholes: a new approach for pricing discretely monitored barrier options. Preprint.

  • Chen, Z., Feng, L., & Lin, X. (2012). Simulating Lévy processes from their characteristic functions and financial applications. ACM Transactions on Modeling and Computer Simulation, 22(3), 1–26.

    Article  Google Scholar 

  • Colldeforns-Papiol, G., Ortiz-Gracia, L., & Oosterlee, C. W. (2017). Two-dimensional Shannon wavelet inverse Fourier technique for pricing European options. Applied Numerical Mathematics, 117, 115–138.

    Article  Google Scholar 

  • Cont, R., & Tankov, P. (2004). Financial Modeling with Jump Processes. Cambridge: Chapman & Hall.

    Google Scholar 

  • Cont, R., & Voltchkova, E. (2005). A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM Journal on Numerical Analysis, 43(4), 1596–1626.

    Article  Google Scholar 

  • De Innocentis, M., & Levendorskiĭ, S. (2013). Pricing discrete barrier optins and credit default swaps under Lévy processes. Quantitative Finance.

  • Duffy, D. J. (2013). Finite difference methods in financial engineering: a partial differential equation approach. Chichester: Wiley.

    Google Scholar 

  • Fan, L. (2018). Some New Developments of Transform Methods for Financial Engineering Applications. Ph. D. thesis, The Chinese University of Hong Kong.

  • Fang, F., Jönsson, H., Oosterlee, C., & Schoutens, W. (2010). Fast valuation and calibration of credit default swaps under Lévy dynamics. The Journal of Computational Finance, 14, 57–86.

    Article  Google Scholar 

  • Fang, F., & Oosterlee, C. W. (2009). Pricing early-exercise and discrete barrier options by Fourier-cosine series expansions. Numerische Mathematik, 114(1), 27–62.

    Article  Google Scholar 

  • Fang, F., & Oosterlee, C. W. (2011). A Fourier-based valuation method for Bermudan and barrier options under Heston’s model. SIAM Journal on Financial Mathematics, 2(1), 439–463.

    Article  Google Scholar 

  • Feng, L., & Lin, X. (2013). Inverting analytic characteristic functions and financial applications. SIAM Journal on Financial Mathematics, 4(1), 372–398.

    Article  Google Scholar 

  • Feng, L., & Lin, X. (2013). Pricing Bermudan options in Lévy process models. SIAM Journal on Financial Mathematics, 4(1), 474–493.

    Article  Google Scholar 

  • Feng, L., & Linetsky, V. (2008). Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: A fast Hilbert transform approach. Mathematical Finance, 18(3), 337–384.

    Article  Google Scholar 

  • Feng, L., & Linetsky, V. (2008). Pricing options in jump-diffusion models: An extrapolation approach. Operations Research, 56(2), 304–325.

    Article  Google Scholar 

  • Feng, L., & Linetsky, V. (2009). Computing exponential moments of the discrete maximum of a Lévy process and lookback options. Finance and Stochastics, 13(4), 501–529.

    Article  Google Scholar 

  • Fusai, G., Abrahams, I. D., & Sgarra, C. (2006). An exact analytical solution for discrete barrier options. Finance and Stochastics, 10(1), 1–26.

    Article  Google Scholar 

  • Fusai, G., Germano, G., & Marazzina, D. (2016). Spitzer identity, Wiener-Hopf factorization and pricing of discretely monitored exotic options. European Journal of Operational Research, 251(1), 124–134.

    Article  Google Scholar 

  • Fusai, G., & Recchioni, M. C. (2007). Analysis of quadrature methods for pricing discrete barrier options. Journal of Economic Dynamics and Control, 31(3), 826–860.

    Article  Google Scholar 

  • Ge, Y., & Li, L. (2020). A Hilbert transform approach for controlled jump-diffusions with financial applications. International Journal of Financial Engineering, 7(4), 2050027.

    Article  Google Scholar 

  • Giles, M. B., & Xia, Y. (2017). Multilevel Monte Carlo for exponential Lévy models. Finance and Stochastics, 21(4), 995–1026.

    Article  Google Scholar 

  • Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stochastic Processes and Their Applications, 87(2), 167–197.

    Article  Google Scholar 

  • Gottlieb, D., & Shu, C. (1997). On the Gibbs phenomenon and its resolution. SIAM Review, 39(4), 644–668.

    Article  Google Scholar 

  • He, H., Keirstead, W. P., & Rebholz, J. (1998). Double lookbacks. Mathematical Finance, 8(3), 201–228.

    Article  Google Scholar 

  • King, F. W. (2009). Hilbert transforms (Vol. 2). Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Kirkby, J. L. (2017). Robust barrier option pricing by frame projection under exponential Lévy dynamics. Applied Mathematical Finance, 24(4), 337–386.

    Article  Google Scholar 

  • Kirkby, J. L. (2018). American and exotic option pricing with jump diffusions and other Lévy processes. Journal of Computational Finance 22(3).

  • Kou, S. (2008). Discrete barrier and lookback options. In J. R. Birge & V. Linetsky (Eds.), Handbook of Financial Engineering, Handbooks in Operations Research and Management Sciences, Chapter 8 (pp. 343–373). Elsevier.

  • Kou, S., & Zhong, H. (2016). First-passage times of two-dimensional Brownian motion. Advances in Applied Probability, 48(4), 1045–1060.

    Article  Google Scholar 

  • Lando, D. (2009). Credit risk modeling: Theory and applications. Princeton University Press.

  • Li, L., & Linetsky, V. (2015). Discretely monitored first passage problems: an eigenfunction expansion approach. Finance and Stochastics, 19(4), 941–977.

    Article  Google Scholar 

  • Li, L., & Zhang, G. (2018). Error analysis of finite difference and Markov chain approximations for option pricing. Mathematical Finance, 28(3), 877–919.

    Article  Google Scholar 

  • Lin, X. (2010). The Hilbert Transform and Its Applications in Computational Finance. PhD thesis, University of Illinois at Urbana-Champaign.

  • Lipton, A. & Savescu, I. (2013). CDSs, CVA and DVA-a structural approach. Risk 26(4), 56.

  • Lipton, A., & Savescu, I. (2014). Pricing credit default swaps with bilateral value adjustments. Quantitative Finance, 14(1), 171–188.

    Article  Google Scholar 

  • Littlejohn, R. G., & Cargo, M. (2002). Multidimensional discrete variable representation bases: Sinc functions and group theory. The Journal of Chemical Physics, 116(17), 7350–7361.

    Article  Google Scholar 

  • Luciano, E., Marena, M., & Semeraro, P. (2016). Dependence calibration and portfolio fit with factor-based subordinators. Quantitative Finance, 16(7), 1037–1052.

    Article  Google Scholar 

  • Luciano, E., & Schoutens, W. (2006). A multivariate jump-driven financial asset model. Quantitative Finance, 6(5), 385–402.

    Article  Google Scholar 

  • Madan, D. B., Carr, P. P., & Chang, E. C. (1998). The variance gamma process and option pricing. Review of Finance, 2(1), 79–105.

    Article  Google Scholar 

  • Mendoza-Arriaga, R., & Linetsky, V. (2016). Multivariate subordination of Markov processes with financial applications. Mathematical Finance, 26(4), 699–747.

    Article  Google Scholar 

  • Mijatovic, A., & Pistorius, M. (2013). Continuously monitored barrier options under Markov processes. Mathematical Finance, 23(1), 1–38.

    Article  Google Scholar 

  • Ortiz-Gracia, L., & Oosterlee, C. W. (2013). Robust pricing of European options with wavelets and the characteristic function. SIAM Journal on Scientific Computing, 35(5), B1055–B1084.

    Article  Google Scholar 

  • Ortiz-Gracia, L., & Oosterlee, C. W. (2016). A highly efficient Shannon wavelet inverse Fourier technique for pricing European options. SIAM Journal on Scientific Computing, 38(1), B118–B143.

    Article  Google Scholar 

  • Petrella, G., & Kou, S. G. (2004). Numerical pricing of discrete barrier and lookback options via Laplace transforms. Journal of Computational Finance, 8(1), 1–37.

    Article  Google Scholar 

  • Phelan, C. E., Marazzina, D., Fusai, G., & Germano, G. (2019). Hilbert transform, spectral filters and option pricing. Annals of Operations Research, 282(1), 273–298.

    Article  Google Scholar 

  • Ruijter, M. J., & Oosterlee, C. W. (2012). Two-dimensional Fourier cosine series expansion method for pricing financial options. SIAM Journal on Scientific Computing, 34(5), B642–B671.

    Article  Google Scholar 

  • Ruijter, M. J., Versteegh, M., & Oosterlee, C. W. (2015). On the application of spectral filters in a Fourier option pricing technique. Journal of Computational Finance, 19(1), 75–106.

    Article  Google Scholar 

  • Sato, K. (1999). Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press.

    Google Scholar 

  • Schoutens, W. (2003). Lévy processes in finance: Pricing financial derivatives. Chichester: Wiley.

    Book  Google Scholar 

  • Schoutens, W., & Cariboni, J. (2010). Lévy processes in credit risk. Chichester: Wiley.

    Google Scholar 

  • Stenger, F. (1993). Numerical methods based on sinc and analytic functions. New York: Springer.

    Book  Google Scholar 

  • Vandeven, H. (1991). Family of spectral lters for discontinuous problems. Journal of Scientific Computing, 6(2), 159–192.

    Article  Google Scholar 

  • Ye, W., & Entezari, A. (2011). A geometric construction of multivariate sinc functions. IEEE Transactions on Image Processing, 21(6), 2969–2979.

    Article  Google Scholar 

  • Zeng, P., & Kwok, Y. K. (2014). Pricing barrier and Bermudan style options under time-changed Lévy processes: fast Hilbert transform approach. SIAM Journal on Scientific Computing, 36(3), B450–B485.

    Article  Google Scholar 

  • Zhang, G., & Li, L. (2019). Analysis of Markov chain approximation for option pricing and hedging: grid design and convergence behavior. Operations Research, 67(2), 407–427.

    Google Scholar 

Download references


The research of Lingfei Li was supported by Hong Kong Research Grant Council General Research Fund Grant No. 14203418. The research of Gongqiu Zhang was supported by the National Natural Science Foundation of China Grant No. 12171408.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Lingfei Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The statements made and views expressed herein are solely of the author and do not necessarily represent policies, statements or views of JP Morgan.

A Proofs

A Proofs

To prove Theorem 2.1, we need the following lemma.

Lemma A.1

Assume (2.5) holds. For any \(\pmb {\beta }\in \{0,1\}^n\), \(\pmb {\beta }\ne \pmb 0\) and \(\pmb {\alpha }\in \{-1,1\}^n\),

$$\begin{aligned}&\left| \frac{\prod _{\{p:\beta _p=1\}}\sin (\pi z_p/h_p)}{(2\pi i)^{|\pmb {\beta }|_1}} \int _{{\mathbb {R}}^{|\pmb {\beta }|_1}} \frac{f(\pmb {\xi }_{\pmb {\beta }} + i\pmb {\alpha }_{\pmb {\beta }} \pmb {d}_{\pmb {\beta }}^- +\pmb {z}_{\pmb {1}-\pmb {\beta }})d\pmb {\xi }_{\pmb {\beta }}}{\prod _{\{j:\beta _j=1\}}(\xi _j+\alpha _jd_j^--z_j)\sin (\pi \zeta _j/h_j)} \right| \\&\quad \le C \prod _{\{j:\beta _j=1\}} \frac{e^{-\pi (d_j-|y_j|)/h_j} (1+ e^{-2\pi |y_j|/h_j}) }{(d_j-|y_j|)(1-e^{-2\pi d_j/h_j}) }. \end{aligned}$$

Proof of Lemma A.1

Applying the following two inequalities

$$\begin{aligned} \sinh (\pi |y|/h)&\le |\sin (\pi (x \pm iy)/h)| \le \cosh (\pi |y|/h), \end{aligned}$$
$$\begin{aligned} |t-z\pm id|&\ge d-|y|\ (z=x+iy,\ |y|<d,\ t\in {\mathbb {R}}), \end{aligned}$$

we obtain that

$$\begin{aligned}&\left| \frac{\prod _{\{p:\beta _p=1\}}\sin (\pi z_p/h_p)}{(2\pi i)^{|\pmb {\beta }|_1}} \int _{{\mathbb {R}}^{|\pmb {\beta }|_1}} \frac{f(\pmb {\xi }_{\pmb {\beta }} + i\pmb {\alpha }_{\pmb {\beta }} \pmb {d}_{\pmb {\beta }}^- +\pmb {z}_{\pmb {1}-\pmb {\beta }})d\pmb {\xi }_{\pmb {\beta }}}{\prod _{\{j:\beta _j=1\}}(\xi _j+\alpha _jd_j^--z_j)\sin (\pi \zeta _j/h_j)} \right| \\&\quad \le \prod _{\{j:\beta _j=1\}} \frac{1}{2\pi (d_j-|y_j|)} \cdot \frac{ \cosh (\pi |y_j|/h_j) }{ \sinh (\pi |d_j|/h_j)} \cdot \int _{{\mathbb {R}}^{|\pmb {\beta }|_1}} \left| f(\pmb {\xi }_{\pmb {\beta }} + i\pmb {\alpha }_{\pmb {\beta }} \pmb {d}_{\pmb {\beta }}^- +\pmb {z}_{\pmb {1}-\pmb {\beta }})\right| d\pmb {\xi }_{\pmb {\beta }}. \end{aligned}$$

Simplifying and using (2.5) give us the desired bound. \(\square \)

Theorem 2.1

(1) We prove the error bounds by induction. The case with \(n=1\) is proved in Stenger (1993). Suppose our error bounds hold for all dimensions that is less than n. For \(j=1,\cdots ,n\), let \(0< \delta _j < d_j\) and define

$$\begin{aligned} {\mathcal {D}}_j(m_j,\delta _j) = \left\{ z_j \in {\mathbb {C}}: |\text {Re}z_j|<\left( m_j+\frac{1}{2}\right) h_j, \, |\text {Im}z_j|<\delta _j \right\} . \end{aligned}$$

Fix \(z_j = x_j + i y_j\) in \({\mathcal {D}}_{d_j}\), and set \(\zeta _j = \xi _j + i \eta _j\) (\(j=1,\cdots ,n\)). If \(m_j\) is sufficiently large and \(\delta _j\) is sufficiently close to \(d_j\), then \(z_j \in {\mathcal {D}}_j(m_j,\delta _j)\). And let

$$\begin{aligned} E(\pmb {m}, \pmb \delta , f)(\pmb {z}) = f(\pmb {z}) - \sum _{k_1=-m_1}^{m_1}\cdots \sum _{k_n=-m_n}^{m_n} f(\pmb {kh}) \prod _{j=1}^{n}S(k_j,h_j)(z_j). \end{aligned}$$

We first show by induction that

$$\begin{aligned}&\frac{\prod _{p=1}^n\sin (\pi z_p/h_p)}{(2\pi i)^n} \int _{\partial {\mathcal {D}}_1(m_1,\delta _1)}\cdots \int _{\partial {\mathcal {D}}_n(m_n,\delta _n)}\frac{ f(\pmb \zeta )d\pmb \zeta }{\prod _{j=1}^n(\zeta _j-z_j)\sin (\pi \zeta _j/h_j)} \nonumber \\&\quad = \sum _{\pmb {\beta } \in \{0,1\}^n }(-1)^{|\pmb {\beta }|_1} \underset{-M_p\le k_p\le M_p}{\sum \textstyle ^{\pmb {\beta }} \textstyle } \left[ f(\pmb {k}_{\pmb {\beta }}\pmb {h}_{\pmb {\beta }} + \pmb {z}_{1-\pmb {\beta }}) \cdot \prod _{\{j: \beta _j=1 \}}S(k_j,h_j)(z_j) \right] . \end{aligned}$$

For \(n=1\), the result is given by Cauchy’s residue theorem. Now suppose the equality holds for the \(n-1\) case, then

$$\begin{aligned}&\frac{\prod _{p=1}^n\sin (\pi z_p/h_p)}{(2\pi i)^n} \int _{\partial {\mathcal {D}}_1(m_1,\delta _1)}\cdots \int _{\partial {\mathcal {D}}_n(m_n,\delta _n)}\frac{ f(\pmb \zeta )d\pmb \zeta }{\prod _{j=1}^n(\zeta _j-z_j)\sin (\pi \zeta _j/h_j)} \\&\quad = \frac{\prod _{p=1}^{n-1}\sin (\pi z_p/h_p)}{(2\pi i)^{n-1}} \int _{\partial {\mathcal {D}}_1(m_1,\delta _1)}\cdots \int _{\partial {\mathcal {D}}_{n-1}(m_{n-1},\delta _{n-1})} \frac{d\zeta _{1} \cdots d \zeta _{n-1} }{\prod _{j=1}^{n-1}(\zeta _j-z_j)\sin (\pi \zeta _j/h_j)} \\&\qquad \cdot \frac{\sin (\pi z_n/h_n)}{2\pi i}\int _{\partial {\mathcal {D}}_n(m_n,\delta _n)}\frac{f( \zeta _{1},\cdots ,\zeta _{n-1},\zeta _n) d\zeta _n}{(\zeta _n-z_n)\sin (\pi \zeta _n/h_n)} \\&\quad = \frac{\prod _{j=1}^{n-1}\sin (\pi z_j/h_j)}{(2\pi i)^{n-1}} \int _{\partial {\mathcal {D}}_1(m_1,\delta _1)}\cdots \int _{\partial {\mathcal {D}}_{n-1}(m_{n-1},\delta _{n-1})} \frac{d\zeta _{1} \cdots d \zeta _{n-1} }{\prod _{j=1}^{n-1}(\zeta _j-z_j)\sin (\pi \zeta _j/h_j)} \\&\qquad \cdot \left[ f(\zeta _{1},\cdots ,\zeta _{n-1}, z_n)- \sum _{k_n=-m_n}^{m_n}\frac{f(\zeta _{1},\cdots ,\zeta _{n-1},k_nh_n)(-1)^{k_n}}{\pi (z_n-k_nh_n)/h_n}\sin (\pi z_n/h_n)\right] \\&\quad = \sum _{\pmb {\beta }' \in \{0,1\}^{n-1} }(-1)^{|\pmb {\beta }'|_1} \underset{-M_p\le k_p'\le M_p}{\sum \textstyle ^{\pmb {\beta }' \textstyle }} \left[ f(\pmb {k}'_{\pmb {\beta }'} \pmb {h}'_{\pmb {\beta }'} + \pmb {z}'_{\pmb {1}-\pmb {\beta }'},z_n ) \cdot \prod _{\{j: \beta _j'=1 \}}S(k_j,h_j)(z_j) \right] \\&\qquad -\sum _{k_n=-m_n}^{m_n} S(k_n,h_n)(z_n) \sum _{\pmb {\beta }' \in \{0,1\}^{n-1} }(-1)^{|\pmb {\beta }'|_1} \underset{-M_p\le k_p'\le M_p}{\sum \textstyle ^{\pmb {\beta }' \textstyle }} \\&\qquad \quad \left[ f(\pmb {k}'_{\pmb {\beta }'} \pmb {h}'_{\pmb {\beta }'} + \pmb {z}'_{\pmb {1}-\pmb {\beta }'},k_nh_n )\prod _{\{j: \beta _j'=1 \}}S(k_j,h_j)(z_j) \right] \\&\quad = \sum _{\pmb {\beta } \in \{0,1\}^n }(-1)^{|\pmb {\beta }|_1} \left[ \underset{-M_p\le k_p\le M_p}{\sum \textstyle ^{\pmb {\beta } \textstyle }} \left( f(\pmb {k}_{\pmb {\beta }} \pmb {h}_{\pmb {\beta }} + \pmb {z}_{\pmb {1}-\pmb {\beta }} )\prod _{\{j: \beta _j=1 \}}S(k_j,h_j)(z_j) \right) \right] . \end{aligned}$$

The third equality follows from induction, the fourth equality is obtained from rewriting \(\pmb {z} = (\pmb {z}',z_n),\pmb {k} = (\pmb {k}',k_n),\pmb {h} = (\pmb {h}',h_n)\). Thus by induction, we obtain (A.4).

Noting that the number of positive and negative terms in \(\{(-1)^{|\pmb {\beta }|_1}:\pmb {\beta }\in \{0,1\}^n\}\) are the same, we can rewrite the right side of (A.4) as

$$\begin{aligned} \sum _{\pmb {\beta } \in \{0,1\}^n }(-1)^{|\pmb {\beta }|_1+1} \left[ f(\pmb {z}) - \underset{-M_p\le k_p\le M_p}{\sum \textstyle ^{\pmb {\beta } \textstyle }} \left( f(\pmb {k}_{\pmb {\beta }} \pmb {h}_{\pmb {\beta }} + \pmb {z}_{\pmb {1}-\pmb {\beta }} )\prod _{\{j: \beta _j=1 \}}S(k_j,h_j)(z_j) \right) \right] . \end{aligned}$$

Thus using (A.5), the term \(E(\pmb {m}, \pmb \delta , f)(\pmb {z})\) that is defined in (A.3) can be written as

$$\begin{aligned}&E(\pmb {m}, \pmb \delta , f)(\pmb {z}) =(-1)^{n+1} \frac{\prod _{p=1}^n\sin (\pi z_p/h_p)}{(2\pi i)^n} \nonumber \\&\quad \int _{\partial {\mathcal {D}}_1(m_1,\delta _1)}\cdots \int _{\partial {\mathcal {D}}_n(m_n,\delta _n)} \frac{ f(\pmb \zeta )d\pmb \zeta }{\prod _{j=1}^n(\zeta _j-z_j)\sin (\pi \zeta _j/h_j)} \nonumber \\&\quad + \sum _{\pmb {\beta } \in \{0,1\}^n, \pmb {\beta } \ne \pmb {1} }(-1)^{|\pmb {\beta }|_1+n+1} \left[ f(\pmb {z}) - \underset{-M_p\le k_p\le M_p}{\sum \textstyle ^{\pmb {\beta } \textstyle }} \nonumber \right. \\&\left. \qquad \left( f(\pmb {k}_{\pmb {\beta }} \pmb {h}_{\pmb {\beta }} + \pmb {z}_{\pmb {1}-\pmb {\beta }})\cdot \prod _{\{j: \beta _j=1 \}}S(k_j,h_j)(z_j) \right) \right] . \end{aligned}$$

We now derive the limit of the integral in (A.6) by taking \(m_j\) to \(\infty \) for all j. The integral can be split into contributions from the product of horizontal segments only, those from the product of vertical segments only and those from the mixed product of horizontal and vertical segments. We observe that whenever there is a vertical segment involved in the integration region, the integral vanishes in the limit. We illustrate this result by considering an integral of this type below, where a vertical segment is used for the n-th dimension and horizontal segments are used in all other dimensions. Here \(\zeta _j = \xi _j+id_j, 1\le j\le n-1, \zeta _n:= \left( m_n+\frac{1}{2} \right) h_n+i\eta _n\), and let \(\tilde{\pmb {\xi }} = (\xi _1,\cdots ,\xi _{n-1})' , \tilde{\pmb {d}} = (d_1,\cdots , d_{n-1} )'\).

$$\begin{aligned}&\left| \int _{-\delta _{n}}^{\delta _{n}} \int _{-\left( m_1+\frac{1}{2}\right) h_1}^{\left( m_1+\frac{1}{2}\right) h_1} \cdots \int _{-\left( m_{n-1}+\frac{1}{2}\right) h_{n-1}}^{\left( m_{n-1}+\frac{1}{2}\right) h_{n-1} } \frac{ f\left( \tilde{\pmb {\xi }} + i \tilde{\pmb {d}}^- , \left( m_n+\frac{1}{2} \right) h_n+i\eta _n \right) d\tilde{\pmb {\xi }}d\eta _n }{\prod _{j=1}^{n-1}(\zeta _j-z_j)\sin (\pi \zeta _j/h_j) \cdot (\zeta _n-z_n) \sin (\pi \zeta _n/h_n)} \right| \\&\le \int _{-\delta _{n}}^{\delta _{n}} \left| \int _{-\left( m_1+\frac{1}{2}\right) h_1}^{\left( m_1+\frac{1}{2}\right) h_1} \cdots \int _{-\left( m_{n-1}+\frac{1}{2}\right) h_{n-1}}^{\left( m_{n-1}+\frac{1}{2}\right) h_{n-1} } \frac{ f(\tilde{\pmb {\xi }} + i \tilde{\pmb {d}}^-, \left( m_n+\frac{1}{2} \right) h_n+i\eta _n ) d\tilde{\pmb {\xi }} }{\prod _{j=1}^{n-1}(\xi _j+id_j-z_j)\sin (\pi (\xi _j+id_j)/h_j) } \right| d\eta _n \\&\qquad \cdot \frac{1}{\left| \left( m_n+\frac{1}{2}\right) h_n-x_n \right| } \\&\rightarrow 0 \text { as } m_j \rightarrow \infty \ \text {for}\ j=1,\cdots ,n. \end{aligned}$$

The inequaltiy is obtained as follows. Along \(\{(m_n+\frac{1}{2})h_n+i\eta _n: -\delta _n\le \eta _n\le \delta _n\}\), we have

$$\begin{aligned}&|\zeta _n-z_n| \ge \left| \left( m_n+\frac{1}{2}\right) h_n-x_n \right| , \\&|\sin (\pi \zeta _n/h_n) | = | (-1)^n \cosh (\pi \eta _n/h_n)| \ge 1. \end{aligned}$$

In addition, the \((n-1)\)-dimensional integral can be bounded using Lemma A.1.

Putting these limiting results together, when \(m_j\rightarrow \infty \), \(\delta _j\rightarrow d_j\) for all j, the first part of the RHS of (A.6) becomes

$$\begin{aligned}&\frac{\prod _{p=1}^n\sin (\pi z_p/h_p)}{(2\pi i)^n} \int _{\partial {\mathcal {D}}_1(m_1,\delta _1)}\cdots \int _{\partial {\mathcal {D}}_n(m_n,\delta _n)}\frac{ f(\pmb \zeta )d\pmb \zeta }{\prod _{j=1}^n(\zeta _j-z_j)\sin (\pi \zeta _j/h_j)} \nonumber \\&\quad =\frac{\prod _{p=1}^n\sin (\pi z_p/h_p)}{(2\pi i)^n}\nonumber \\&\quad \sum _{\pmb {\alpha } \in \{-1,1\}^n } (-1)^{\sum _{\{j:\alpha _j=1\}}1}\int _{{\mathbb {R}}^n} \frac{f(\pmb {\xi } + i\pmb {\alpha } \pmb {d}^- )d\pmb {\xi }}{\prod _{j=1}^n(\xi _j+i\alpha _jd_j^--z_j)\sin (\pi \zeta _j/h_j)} . \end{aligned}$$

The RHS of (A.7) can be bounded as

$$\begin{aligned} C\prod _{ j=1}^n \frac{e^{-\pi (d_j-|y_j|)/h_j} (1+ e^{-2\pi |y_j|/h_j}) }{(d_j-|y_j|)(1-e^{-2\pi d_j/h_j})} \end{aligned}$$

using Lemma A.1. For the second part of the RHS of (A.6), applying the induction assumption together with (2.5) and (2.6), we obtain for each \(\pmb {\beta } \in \{0,1\}^n, \pmb {\beta } \ne \pmb 0, \pmb {1}\),

$$\begin{aligned}&\left| f(\pmb {z}) - \underset{-M_j\le k_j\le M_j}{\sum \textstyle ^{\pmb {\beta } \textstyle }} \left( f(\pmb {k}_{\pmb {\beta }} \pmb {h}_{\pmb {\beta }} + \pmb {z}_{\pmb {1}-\pmb {\beta }})\cdot \prod _{\{j: \beta _j=1 \}}S(k_j,h_j)(z_j) \right) \right| \nonumber \\&\quad \le \sum _{\pmb {\alpha } \in \{0,1\}^n, \pmb {\alpha } \le \pmb {\beta }, \pmb {\alpha } \ne 0} \left[ C_{\pmb {\alpha }} \prod _{\{p:\alpha _p =1 \}}\frac{e^{-\pi (d_p-|y_p|)/h_p}(1+e^{-2\pi |y_p|/h_p})}{(d_p-|y_p|)(1-e^{-2\pi d_p/h_p})} \right] . \end{aligned}$$

Here \( \pmb {\alpha } \le \pmb {\beta }\) means \(\alpha _j \le \beta _j\) for all \(1\le j \le n\). Adding (A.8) and (A.9) over \(\pmb {\beta }\ne \pmb 0,\pmb 1\) yields (2.7).

(2) We have \(E_H(f,\pmb {h})(\pmb {\xi })=\frac{1}{\pi ^n} \int _{{\mathbb {R}}^n}\frac{E(f,\pmb {h})(\pmb {x})}{\prod _{i=1}^{n}(\xi _i - x_i)}d\pmb {x}\). Thus (2.8) can be derived by using (A.7) with \(z_i=x_i\), interchanging the order of integration and applying the identity

$$\begin{aligned} \frac{1}{\pi }\int _{{\mathbb {R}}}\frac{\sin (\pi x/h)}{(\omega -x)(z-x)}dx=\frac{e^{i\pi \omega \text {sgn}(\text {Im}(\omega ))/h}-e^{i\pi z\text {sgn}(\text {Im}(z))/h}}{\omega -z} \end{aligned}$$

together with (A.1) and (A.2).

(3) We have \(E_I(f,\pmb {h})=\int _{{\mathbb {R}}^n}E(f,\pmb {h})(\pmb {x})d\pmb {x}\). Using (A.7) with \(z_i=x_i\), interchanging the order of integration and applying the identities

$$\begin{aligned} \frac{1}{2\pi i}\int _{{\mathbb {R}}}\frac{\sin (\pi x/h)}{t-x-id}dx=\frac{i}{2}e^{-\pi (d+it)/h},\ \frac{1}{2\pi i}\int _{{\mathbb {R}}}\frac{\sin (\pi x/h)}{x-t-id}dx=\frac{i}{2}e^{-\pi (d-it)/h}, \end{aligned}$$

and (A.1), we obtain (2.9).

\(\square \)

Proposition 3.2

It holds that

$$\begin{aligned} E_{\pmb {x}}[ f(\pmb {X}_t)] = E_{\pmb {x}}[ e^{-\pmb {\alpha }' \pmb {X}_t}f_{\pmb {\alpha }}(\pmb {X}_t) ]=e^{-\pmb {\alpha }'\pmb {x}-t\psi (i\pmb {\alpha })}E_{\pmb {x}}^{(\pmb {\alpha })}[ f_{\pmb {\alpha }}(\pmb {X}_t)]. \end{aligned}$$

Take the Fourier transform of \(E_{\pmb {x}}^{(\pmb {\alpha })}\left[ f_{\pmb {\alpha }}(\pmb {X}_t) \right] \) as a function of \(\pmb {x}\) and use Proposition 9 in Bertoin (1998), we obtain

$$\begin{aligned} {\mathcal {F}}_n\left( E_{\pmb {x}}^{(\pmb {\alpha })}\left[ f_{\pmb {\alpha }}(\pmb {X}_t) \right] \right) (\pmb {\xi })=\phi _t^{\pmb {\alpha }}(-\pmb {\xi }){\hat{f}}_{\pmb {\alpha }}(\pmb {\xi }). \end{aligned}$$

Due to the integrability condition (3.6), we can invert the Fourier transform and obtain (3.8). \(\square \)

Proposition 3.3

Given \(\pmb {\beta } \in \{0,1\}^n\), let \(k_{\pmb {\beta }}(\pmb {\xi }) = \prod _{\{j:\beta _j=1 \}}1/(\pi \xi _j)\). The partial Hilbert transform \({\mathcal {H}}_{\pmb {\beta }}\) of \({\hat{f}}\) is the convolution of \({\hat{f}}\) and \(k_{\pmb {\beta }}\), i.e. \({\mathcal {H}}_{\pmb {\beta }}{\hat{f}}(\pmb {\xi }) = ({\hat{f}} *k_{\pmb {\beta }})(\pmb {\xi })\) (see Eq.(15.39) in King 2009). The convolution theorem shows that

$$\begin{aligned}&{\mathcal {F}}^{-1}_{\pmb {\beta }}\left( {\mathcal {H}}_{\pmb {\beta }}{\hat{f}}(\pmb {\xi })\right) (\pmb {x}_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }})\nonumber \\&\quad = {\mathcal {F}}^{-1}_{\pmb {\beta }}({\hat{f}}(\pmb {\xi }))(\pmb {x}_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }}) \cdot (2\pi )^{|\pmb {\beta }|_1}{\mathcal {F}}^{-1}_{\pmb {\beta }}(k_{\pmb {\beta }}(\pmb {\xi }))(\pmb {x}_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }}), \end{aligned}$$


$$\begin{aligned}&{\mathcal {F}}^{-1}_{\pmb {\beta }}(k_{\pmb {\beta }}(\pmb {\xi }))(\pmb {x}_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }}) = \left( \frac{1}{2\pi } \right) ^{|\pmb {\beta }|_1}\\&\quad \int _{{\mathbb {R}}^{|\pmb {\beta }|_1}}\prod _{\{j:\beta _j=1 \}}\frac{e^{-i x_j\xi _j}d\xi _j}{\pi \xi _j}= \left( \frac{-i}{2\pi } \right) ^{|\pmb {\beta }|_1} \prod _{\{j:\beta _j=1 \}}\text {sgn}(x_j). \end{aligned}$$

The function \(k_{\pmb {\beta }}(\pmb {\xi })\) is not in \(L^1({\mathbb {R}}^n)\), so \({\mathcal {F}}^{-1}_{\pmb {\beta }}(k_{\pmb {\beta }}(\pmb {\xi }))(\pmb {x}_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }})\) must be interpreted as a Cauchy principal integral. We further apply Fourier inversion to those dimensions p with \(\beta _p=0\) on both sides of (A.10) and obtain that

$$\begin{aligned} {\mathcal {F}}^{-1}_n\left( {\mathcal {H}}_{\pmb {\beta }}{\hat{f}}(\pmb {\xi })\right) (\pmb {x}) = i^{|\pmb {\beta }|_1} f(\pmb {x})\prod _{\{j:\beta _j=1 \}}\text {sgn}(x_j). \end{aligned}$$

Here, we use the identity that \({\mathcal {F}}^{-1}_{\pmb {1}-\pmb {\beta }}({\mathcal {F}}^{-1}_{\pmb {\beta }}({\hat{f}}(\pmb {\xi }))(\pmb {x}_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }}))(\pmb {x})={\mathcal {F}}^{-1}_n{\hat{f}}(\pmb {x})=f(\pmb {x})\) as \({\hat{f}}\in L^1({\mathbb {R}}^n,{\mathbb {C}})\). Further applying \({\mathcal {F}}_n\) on both sides of the above equation gives us (3.10). \(\square \)

Proposition 3.4

Consider the translation operators

$$\begin{aligned} \left( {\mathcal {T}}_{l_j}^{(j)}f\right) (\pmb {x})&:= f(x_1,\cdots ,x_{j-1},x_j-l_j,x_{j+1},\cdots , x_n). \end{aligned}$$

We can write

$$\begin{aligned} \mathbbm {1}_{(\pmb {l},\infty )}(\pmb {x})f(\pmb {x})= & {} \frac{1}{2^n} \prod _{j=1}^{n} \left( 1+ {\mathcal {T}}_{l_j}^{(j)}\text {sgn}(x_j) \right) f(\pmb {x})\\= & {} \frac{1}{2^n} \sum _{\pmb {\beta } \in \{0,1\}^n } f(\pmb {x}) \cdot {\mathcal {T}}_{\pmb {l}}^{\pmb {\beta }}\left( \prod _{ \{ p:\beta _p = 1\}} \text {sgn}(x_p) \right) , \end{aligned}$$

where \({\mathcal {T}}_{\pmb {l}}^{\pmb {\beta }} = \prod _{ \{ p:\beta _p = 1\}} {\mathcal {T}}_{l_p}^{(p)}\). Furthermore, we have

$$\begin{aligned}f(\pmb {x}) \cdot {\mathcal {T}}_{\pmb {l}}^{\pmb {\beta }}\left( \prod _{ \{ p:\beta _p = 1\}} \text {sgn}(x_p) \right) = {\mathcal {T}}_{\pmb {l}}^{\pmb {\beta }} \left( \prod _{ \{ p:\beta _p = 1\}} \text {sgn}(x_p) \cdot {\mathcal {T}}_{-\pmb {l}}^{\pmb {\beta }}f(\pmb {x}) \right) . \end{aligned}$$

Then, by (3.10) and the property of the Fourier transform w.r.t. translation, we obtain

$$\begin{aligned} {\mathcal {F}}_n\left( {\mathcal {T}}_{\pmb {l}}^{\pmb {\beta }} \left( \prod _{ \{ p:\beta _p = 1\}} \text {sgn}(x_p) \cdot {\mathcal {T}}_{-\pmb {l}}^{\pmb {\beta }}f(\pmb {x}) \right) \right) (\pmb {\xi }) = i^{|\pmb {\beta }|} e^{i \pmb {\xi }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta } }} {\mathcal {H}}_{\pmb {\beta }} \left( e^{-i \pmb {\eta }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta } }}{\hat{f}}\left( \pmb {\eta }_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }}\right) \right) (\pmb {\xi }). \end{aligned}$$

Applying the \({\mathcal {F}}_n\) to \(\mathbbm {1}_{(\pmb {l},\infty )}(\pmb {x})f(\pmb {x})\) and using the above results gives us (3.11). \(\square \)

Theorem 3.1

First, notice that if (3.18) holds, then for any \(t\ge \chi \),

$$\begin{aligned} \int _{{\mathbb {R}}^n}\big |\phi _{t}^{\pmb {\alpha }}(\pmb {\xi }) \big |d\pmb {\xi }=\int _{{\mathbb {R}}^n}\big |\phi _{\chi }^{\pmb {\alpha }}(\pmb {\xi })\phi _{t-\chi }^{\pmb {\alpha }}(\pmb {\xi }) \big |d\pmb {\xi }\le \int _{{\mathbb {R}}^n}\big |\phi _{\chi }^{\pmb {\alpha }}(\pmb {\xi })\big |d\pmb {\xi }<\infty . \end{aligned}$$

We prove that (3.19) holds for every j. For \(j=N\), it is given by the assumption (3.17). Now assume that the claim (3.19) holds for j to N and we prove below that it also holds for \(j-1\). Using the recursion (3.15), we have

$$\begin{aligned} \int _{{\mathbb {R}}^n}\big |\phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) {\hat{v}}^{j-1}_{\pmb {\alpha }}(\pmb {\xi })\big |d\pmb {\xi } \le \frac{e^{-\Delta \psi (i\pmb {\alpha })}}{2^n}\sum _{\pmb {\beta } \in \{0,1\}^n } I_{\pmb {\beta }}, \end{aligned}$$


$$\begin{aligned} I_{\pmb {\beta }} := \int _{{\mathbb {R}}^n} \left| \phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) {\mathcal {H}}_{\pmb {\beta }} \left( e^{-i \pmb {\eta }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta } }} \phi _\Delta ^{\pmb {\alpha }} \big (-\pmb {\eta }_{\pmb {\beta }} -\pmb {\xi }_{\pmb {1}-\pmb {\beta }} \big ){\hat{v}}_{\pmb {\alpha }}^j\big (\pmb {\eta }_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }}\big ) \right) (\pmb {\xi }) \right| d\pmb {\xi }. \end{aligned}$$

As \(|\phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi })|\) is bounded over \(\pmb {\xi }\), \(I_{\pmb {0}}\) is finite by the induction assumption. Now pick \(p > 1\) such that \(p\Delta \ge \chi \) and set \(q=p/(p-1)\). Note that \((\phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }))^p = \phi _{p\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) \in L^1({\mathbb {R}}^n)\), and this implies \(\phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) \in L^p({\mathbb {R}}^n)\). The Calderón–Zygmund inequality (see Eq. (15.115) in King 2009) says that, if \(f \in L^p({\mathbb {R}}^n)\) with \(p>1\), then

$$\begin{aligned} \Vert {\mathcal {H}}_nf \Vert _p&= \left\| \prod _{k=1}^n {\mathcal {H}}_{(k)}f \right\| _p = \left\| {\mathcal {H}}_{(1)}\prod _{k=2}^n {\mathcal {H}}_{(k)}f \right\| _p\\&\le C_p^{(1)} \left\| \prod _{k=2}^n {\mathcal {H}}_{(k)}f \right\| _p \le \cdots \le C_p^{(n)} \left\| f\right\| _p, \end{aligned}$$

for some constants \(C_p^{(k)},1\le k\le n\). Here, we write \({\mathcal {H}}_{(k)}\) as the partial Hilbert transform with \(\pmb {\beta } = (0,\cdots ,0,1,0,\cdots ,0) \), where only \(\beta _k=1\). Thus for \(I_{\pmb {1}}\), by Hölder’s inequality,

$$\begin{aligned} I_{\pmb {1}}&\le \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\Vert _p \cdot \left\| {\mathcal {H}}_{n}\left( e^{-i\pmb {\eta }'\pmb {l}}\phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\eta }){\hat{v}}^j_{\pmb {\alpha }}(\pmb {\eta })\right) (\cdot )\right\| _q \le C \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\Vert _p \cdot \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ){\hat{v}}^j_{\pmb {\alpha }}(\cdot ) \Vert _q. \end{aligned}$$

Using the boundedness of \({\hat{v}}^j_{\pmb {\alpha }}\), we have

$$\begin{aligned} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ){\hat{v}}^j_{\pmb {\alpha }}(\cdot ) \Vert _q\le \Vert {\hat{v}}^j_{\pmb {\alpha }}(\cdot )\Vert _{\infty }^{q-1}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ){\hat{v}}^j_{\pmb {\alpha }}(\cdot ) \Vert _{1}, \end{aligned}$$

which is finite by the induction assumption. This shows \(I_{\pmb {1}}\) is finite. The finiteness of \(\{I_{\pmb {\beta }}: \pmb {\beta } \ne \pmb {0},\pmb {1} \}\) can be proved similarly. Finally, (3.20) follows from Proposition 3.2. \(\square \)

Theorem 3.2

(1) For \({\mathcal {P}}^\Delta g(\pmb {\xi })\), the total approximation error consists of the error for the terms involving several partial Hilbert transforms and the term with the n-dimensional Hilbert transform. We can directly apply Theorem 2.1 to estimate the approximation error for the partial Hilbert transforms under (3.24) and the boundedness of g. Below we only analyze the error for the n-dimensional Hilbert transform. We have

$$\begin{aligned} \left| {\mathcal {P}}^{\Delta }g(\pmb {\xi })-{\mathcal {P}}^{\Delta }_{h,M}g(\pmb {\xi })\right|&\le \left| {\mathcal {P}}^{\Delta }g(\pmb {\xi })-{\mathcal {P}}^{\Delta }_{h,\infty }g(\pmb {\xi })\right| +\left| {\mathcal {P}}^{\Delta }_{h,\infty }g(\pmb {\xi })-{\mathcal {P}}^{\Delta }_{h,M}g(\pmb {\xi })\right| . \end{aligned}$$

The first term can be estimated by applying (2.8) as our assumption on characteristic function (3.24) and the boundedness of g allows us to verify the conditions in Theorem 2.1. For the second term, it is bounded by

$$\begin{aligned}&\Vert g\Vert _{L^\infty ({\mathbb {R}}^n)}\sum _{|m_j|>M_j, 1\le j\le n}\left| \phi _{\Delta }(- \pmb {mh})\right| \nonumber \\&\quad \le a\cdot 2^n\Vert g\Vert _{L^\infty ({\mathbb {R}}^n)} \left( \prod _{j=1}^nh_j^{-1} \right) \sum _{m_j>M_j,1\le j\le n} \left( e^{-\Delta b \left( \sum _{j=1}^{n}m_j^2h_j^2 \right) ^{\nu /2} } \cdot \prod _{j=1}^nh_j\right) \nonumber \\&\quad \le a \cdot 2^n\Vert g\Vert _{L^\infty ({\mathbb {R}}^n)} \left( \prod _{j=1}^nh_j^{-1} \right) \sum _{m_j>M_j,1\le j\le n} e^{-\Delta b n^{\nu /2-1}\sum _{j=1}^{n}\left( m_jh_j \right) ^{\nu }}\cdot \prod _{j=1}^nh_j \nonumber \\&\quad \le a\cdot 2^n\Vert g\Vert _{L^\infty ({\mathbb {R}}^n)} \left( \prod _{j=1}^nh_j^{-1} \right) \cdot \prod _{j=1}^n \int _{M_jh_j}^\infty e^{-\Delta bn^{\nu /2-1}x_j^{\nu }}dx_j\nonumber \\&\quad \le a\cdot 2^n\Vert g\Vert _{L^\infty ({\mathbb {R}}^n)} \prod _{j=1}^{n} \Gamma \left( 1/\nu ,\Delta bn^{\nu /2-1}(M_jh_j)^\nu \right) /h_j, \end{aligned}$$

where we used \(|(1-\cos (x))/x|{\le } 1\), (3.23), \(\left( \sum _{j{=}1}^{n}m_j^2h_j^2\right) ^{\nu /2}\ge n^{\nu /2-1}\sum _{j{=}1}^{n}(m_jh_j)^\nu \) (due to the concavity of \(x^{\nu /2}\) as \(\nu \in (0,2]\)). Combining Eq.(6.20) in Feng and Linetsky (2008a), (2.8) and (A.11), we arrive at (3.26). Now we set \(h_j\) according to (3.27) and \(h_j\) is bounded as \(M_j\ge 1\). Therefore, for some constant \(C>0\),

$$\begin{aligned} \frac{e^{-\pi d_j/h_j}}{1-e^{-\pi d_j/h_j}}\le Ce^{-\pi d_j/h_j}, \end{aligned}$$

where the constant C is independent of j. As \(\Gamma (a,x)\sim x^{a-1}e^{-x}\) for x large, we can bound \(\Gamma (a,x)\) as a constant times \(x^{a-1}e^{-x}\) for all x bounded away from 0. Using this estimate, (3.27), (A.12) and that the term \(M_j^{\frac{2}{1+\nu }}\exp \left( -2cM_j^{\frac{\nu }{1+\nu }}\right) \le A M_j^{\frac{1}{1+\nu }}\exp \left( -cM_j^{\frac{\nu }{1+\nu }}\right) \) for some constant \(A>0\) when \(c>0\), we obtain (3.28).

(2) The proof for \({\mathcal {R}}^\Delta g(\pmb {x})\) is similar to the proof for \({\mathcal {P}}^\Delta g(\pmb {\xi })\), so the detail is omitted. \(\square \)

Theorem 3.3

(1) From the proof of Theorem 3.1, we have that for every \(q>1\),

$$\begin{aligned} \Vert {\mathcal {H}}_{\pmb {\beta }}f\Vert _{L^q}\le C_q^{\pmb {\beta }}\Vert f\Vert _{L^q}, \end{aligned}$$

for some constant \(C_q^{\pmb {\beta }}>0\). Now we use this result to bound \(\Vert {\mathcal {P}}^{\Delta } g \Vert _{L^q}\) for \(g\in L^q\). Using the expression of \({\mathcal {P}}^{\Delta } g\) yields

$$\begin{aligned} \left\| {\mathcal {P}}^{\Delta }g(\pmb {\xi }) \right\| _{L^q}&= \left\| \sum _{\pmb {\beta } \in \{0,1\}^n } i^{|\pmb {\beta }|_1} e^{i\pmb {\xi }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta }} }{\mathcal {H}}_{\pmb {\beta }} \left( e^{-i\pmb {\eta }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta }}}\phi _\Delta ^{\pmb {\alpha }} \big (-\pmb {\eta }_{\pmb {\beta }}-\pmb {\xi }_{\pmb {1}-\pmb {\beta }}\big )g\big (\pmb {\eta }_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }} \big ) \right) (\pmb {\xi }) \right\| _{L^q} \\&\le \sum _{\pmb {\beta } \in \{0,1\}^n } \left\| {\mathcal {H}}_{\pmb {\beta }} \left( e^{-i\pmb {\eta }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta }}}\phi _\Delta ^{\pmb {\alpha }} \big (-\pmb {\eta }_{\pmb {\beta }}-\pmb {\xi }_{\pmb {1}-\pmb {\beta }}\big )g\big (\pmb {\eta }_{\pmb {\beta }}+\pmb {\xi }_{\pmb {1}-\pmb {\beta }} \big ) \right) (\pmb {\xi }) \right\| _{L^q} \\&\le \sum _{\pmb {\beta } \in \{0,1\}^n } C_q^{\pmb {\beta } } \left\| e^{-i\pmb {\eta }_{\pmb {\beta }}' \pmb {l}_{\pmb {\beta }}} \phi _\Delta ^{\pmb {\alpha }} (\pmb {\xi })g (\pmb {\xi }) \right\| _{L^q} \\&\le \sum _{\pmb {\beta } \in \{0,1\}^n } C_q^{\pmb {\beta } } \left\| \phi _\Delta ^{\pmb {\alpha }} (\pmb {\xi })g (\pmb {\xi }) \right\| _{L^q} \\&\le C_q \left\| \phi _\Delta ^{\pmb {\alpha }} (\pmb {\xi })g (\pmb {\xi }) \right\| _{L^q}, \end{aligned}$$

where we set \(C_q = 2^n\max _{\pmb {\beta } \in \{0,1\}^n} C_q^{\pmb {\beta } }\).

(2) Set

$$\begin{aligned} q_1 =1, \quad \frac{1}{p_k}+\frac{1}{q_k}=\frac{1}{q_{k-1}},\ k=2,\cdots ,N. \end{aligned}$$


$$\begin{aligned}&C_{norm} = \prod _{i=1}^n2M_ih_i,\ C_3 = C_1\cdot 2^nC_{q_2},\\&C_k = \frac{C_3C_{norm}}{2^{n(k-2)}} \prod _{i=2}^{k}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_i}}, k \ge 4. \end{aligned}$$

For \(i=1,\cdots , N-1\), we have

$$\begin{aligned}&\left| {\tilde{v}}^{i}_{\pmb {\alpha }}(\pmb {\xi }) - {\hat{v}}^{i}_{\pmb {\alpha }} (\pmb {\xi }) \right| = \frac{e^{-\Delta \psi (i\pmb {\alpha })}}{2^n} \left| {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{i+1}_{\pmb {\alpha }}(\pmb {\xi }) - {\mathcal {P}}^{\Delta }{\hat{v}}^{i+1}_{\pmb {\alpha }} (\pmb {\xi })\right| \\&\quad \le \frac{e^{-\Delta \psi (i\pmb {\alpha })}}{2^n} \left( \left| {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{i+1}_{\pmb {\alpha }}(\pmb {\xi }) - {\mathcal {P}}^{\Delta }{\tilde{v}}^{i+1}_{\pmb {\alpha }} (\pmb {\xi }) \right| + \left| {\mathcal {P}}^{\Delta }\left( {\tilde{v}}^{i+1}_{\pmb {\alpha }} -{\hat{v}}^{i+1}_{\pmb {\alpha }}\right) (\pmb {\xi }) \right| \right) . \end{aligned}$$

Now we estimate \(I_1\) and \(I_2\).

$$\begin{aligned} I_1&= C_1 \int _{{\mathbb {R}}^n}\left| \phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) \right| \cdot \left| {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{2}_{\pmb {\alpha }}(\pmb {\xi }) - {\mathcal {P}}^{\Delta }{\tilde{v}}^{2}_{\pmb {\alpha }} (\pmb {\xi }) \right| d\pmb {\xi }\\&\le C_1 \int _{{\mathbb {R}}^n}\left| \phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) \right| d\pmb {\xi } \cdot \Vert {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{2}_{\pmb {\alpha }} - {\mathcal {P}}^{\Delta }{\tilde{v}}^{2}_{\pmb {\alpha }}\Vert _{L^{\infty }} \\&\le C_2 \Vert {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{2}_{\pmb {\alpha }} - {\mathcal {P}}^{\Delta }{\tilde{v}}^{2}_{\pmb {\alpha }} \Vert _{L^{\infty }}, \end{aligned}$$

where \(C_2 = C_1 \Vert \phi _{\Delta }^{\pmb {\alpha }}\Vert _{L^1}\) and

$$\begin{aligned} I_2&= C_1 \int _{{\mathbb {R}}^n}\left| \phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) \right| \cdot \left| {\mathcal {P}}^{\Delta }\left( {\tilde{v}}^{2}_{\pmb {\alpha }} -{\hat{v}}^{2}_{\pmb {\alpha }}\right) (\pmb {\xi }) \right| d\pmb {\xi } \\&\le C_1 \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_2}} \cdot \Vert {\mathcal {P}}^{\Delta }\left( {\tilde{v}}^{2}_{\pmb {\alpha }} -{\hat{v}}^{2}_{\pmb {\alpha }}\right) \Vert _{L^{q_2}} \\&\le C_3 \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_2}} \cdot \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{2}_{\pmb {\alpha }} -{\hat{v}}^{2}_{\pmb {\alpha }}\right) (\cdot ) \Vert _{L^{q_2}}, \end{aligned}$$

where \(C_3 = C_1\cdot 2^nC_{q_2}\). Applying the Holder inquality and the Minkowski inequality, we obtain

$$\begin{aligned}&\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{2}_{\pmb {\alpha }} -{\hat{v}}^{2}_{\pmb {\alpha }}\right) (\cdot ) \Vert _{L^{q_2}} \nonumber \\&\quad \le \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_3}} \cdot \Vert {\tilde{v}}^{2}_{\pmb {\alpha }} -{\hat{v}}^{2}_{\pmb {\alpha }}\Vert _{L^{q_3}} \nonumber \\&\quad \le \frac{e^{-\Delta \psi (i\pmb {\alpha })}}{2^n} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_3}} \cdot \Vert {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{3}_{\pmb {\alpha }} - {\mathcal {P}}^{\Delta }{\hat{v}}^{3}_{\pmb {\alpha }} \Vert _{L^{q_3}} \nonumber \\&\quad \le \frac{e^{-\Delta \psi (i\pmb {\alpha })}}{2^n} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_3}} \left( \Vert {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{3}_{\pmb {\alpha }} - {\mathcal {P}}^{\Delta }{\tilde{v}}^{3}_{\pmb {\alpha }} \Vert _{L^{q_3}} + \Vert {\mathcal {P}}^{\Delta }\left( {\tilde{v}}^{3}_{\pmb {\alpha }} -{\hat{v}}^{3}_{\pmb {\alpha }}\right) (\pmb {\xi })) \Vert _{L^{q_3}} \right) \nonumber \\&\quad \le \frac{\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_3}} }{2^n} \Vert {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{3}_{\pmb {\alpha }} - {\mathcal {P}}^{\Delta }{\tilde{v}}^{3}_{\pmb {\alpha }} \Vert _{L^{q_3}} + C_{q_3}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_3}} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{3}_{\pmb {\alpha }} -{\hat{v}}^{3}_{\pmb {\alpha }}\right) (\cdot ) \Vert _{L^{q_3}} \nonumber \\&\quad \le \cdots \le \sum _{k=3}^{N}\frac{1}{2^{n(k-2)}} \prod _{i=3}^{k}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_i}} \Vert {\mathcal {P}}^{\Delta }_{\pmb {h},\pmb {M}}{\tilde{v}}^{k}_{\pmb {\alpha }} - {\mathcal {P}}^{\Delta }{\tilde{v}}^{k}_{\pmb {\alpha }} \Vert _{L^{q_k}}\nonumber \\&\qquad + \prod _{k=3}^{N} C_{q_k}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_k}} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{N}_{\pmb {\alpha }} -{\hat{v}}^{N}_{\pmb {\alpha }}\right) (\cdot )\Vert _{L^{q_N}}. \end{aligned}$$

We point out the inequality

$$\begin{aligned} \Vert f\Vert _{L^{p}} \le C_{norm}\Vert f\Vert _{L^{\infty }}, \end{aligned}$$

which holds for \(p>1\) and f supported in the bounded hyperrectangle. We have

$$\begin{aligned}&\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{N}_{\pmb {\alpha }} -{\hat{v}}^{N}_{\pmb {\alpha }}\right) (\cdot )\Vert _{L^{q_N}}\nonumber \\&\quad \le \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{N}_{\pmb {\alpha }} -{\hat{v}}^{N}_{\pmb {\alpha }}\right) (\cdot )1_{\Omega _{\pmb {h},\pmb {M}}}(\cdot )\Vert _{L^{q_N}}+\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{N}_{\pmb {\alpha }} -{\hat{v}}^{N}_{\pmb {\alpha }}\right) (\cdot )1_{\Omega ^c_{\pmb {h},\pmb {M}}}(\cdot )\Vert _{L^{q_N}}\nonumber \\&\quad \le C_{norm}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{N}_{\pmb {\alpha }} -{\hat{v}}^{N}_{\pmb {\alpha }}\right) (\cdot )1_{\Omega _{\pmb {h},\pmb {M}}}(\cdot )\Vert _{L^{\infty }} + \Vert {\hat{g}}_{\pmb {\alpha }}\Vert _{L^{\infty }}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )1_{\Omega ^c_{\pmb {h},\pmb {M}}}(\cdot )\Vert _{L^{q_N}}\nonumber \\&\quad \le C_{norm}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot )\left( {\tilde{v}}^{N}_{\pmb {\alpha }} -{\hat{v}}^{N}_{\pmb {\alpha }}\right) (\cdot )1_{\Omega _{\pmb {h},\pmb {M}}}(\cdot )\Vert _{L^{\infty }} + \Vert {\hat{g}}_{\pmb {\alpha }}\Vert _{L^{\infty }}I_{\pmb {h},\pmb {M}}. \end{aligned}$$


$$\begin{aligned} C'_N = C_3\max \{C_{norm},\Vert {\hat{g}}_{\pmb {\alpha }}\Vert _{L^{\infty }}\}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_2}} \prod _{k=3}^{N} C_{q_k}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_k}}. \end{aligned}$$

Using (A.14) and (A.15), we can bound the RHS of (A.13) by the RHS of (3.43).

Next we provide estimates for the constants. First, we have

$$\begin{aligned} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p}}&= \left( \int _{{\mathbb {R}}^n} |\phi _{\Delta }^{\pmb {\alpha }}(-\pmb {\xi }) |^p d\pmb {\xi } \right) ^{\frac{1}{p}} \le \left( \int _{{\mathbb {R}}^n} a^pe^{-p\Delta b|\pmb {\xi }|^{\nu }} d\pmb {\xi } \right) ^{\frac{1}{p}}\\&\le (p\Delta b)^{-\frac{n}{\nu p}}\left( \int _{{\mathbb {R}}^n}a^pe^{-|\pmb {\eta }|^{\nu }}d\pmb {\eta } \right) ^{\frac{1}{p}},\\&= A^{\frac{1}{p}} p^{-\frac{n}{\nu p}}, \end{aligned}$$


$$\begin{aligned} \pmb {\eta } = (p\Delta b)^{\frac{1}{\nu }}\pmb {\xi },\ A = \frac{\int _{{\mathbb {R}}^n}a^pe^{-|\pmb {\eta }|^{\nu }}d\pmb {\eta } }{(\Delta b)^{\frac{n}{\nu }}}. \end{aligned}$$

The constant \(C_p\) is given by

$$\begin{aligned}C_p^{\frac{1}{n}} = {\left\{ \begin{array}{ll} \tan (\pi /2p), &{} 1<p\le 2, \\ \cot (\pi /2p), &{} 2 \le p <\infty . \end{array}\right. } \end{aligned}$$

Then, with \(q_k = 2-\frac{1}{2^k}\), we obtain (assume \( A > 1\), otherwise replace \(A^{\frac{1}{p}}\) with 1)

$$\begin{aligned}&p_k = \frac{q_{k-1}q_k}{q_k-q_{k-1}} = 2^k\cdot \left( 2-\frac{1}{2^k} \right) \cdot \left( 2-\frac{1}{2^{k-1}} \right) \ge 2^k , \quad p_k ^{-\frac{1}{ p_k}} \le 2^{-\frac{k}{2^{k}}},\\&\tan \left( \frac{\pi }{2q_k}\right) \approx 1+2\left( \frac{\pi }{2q_k}-\frac{\pi }{4}\right) = 1+\frac{\pi }{2^{k+1}\left( 2-\frac{1}{2^k} \right) } \le 1+\frac{\pi }{2^{k+1}}, \end{aligned}$$


$$\begin{aligned}&\prod _{i=2}^{k} \Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_i}} \le \prod _{i=2}^{k} \left( 2^{-\frac{i}{2^{i}}} \right) ^{\frac{n}{\nu }} A^{\frac{1}{2^i}} \le \sqrt{A }, \quad 3\le k \le N, \\&\prod _{k=3}^{N} C_{q_k}\Vert \phi _{\Delta }^{\pmb {\alpha }}(-\cdot ) \Vert _{L^{p_k}} \le \prod _{k=3}^{N} \left( 2^{-\frac{k}{2^{k}}} \right) ^{\frac{n}{\nu }} \cdot A^{\frac{1}{2^k}} \cdot \left( 1+\frac{\pi }{2^{k+1}} \right) ^n \le A^{\frac{1}{4}}e^n. \end{aligned}$$

Thus \(C_k \sim O\left( \frac{1}{2^{n(k-2)}} \right) \) and \(C'_N \sim O(1)\). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, J., Fan, L., Li, L. et al. A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation. Rev Deriv Res 25, 189–232 (2022).

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Hilbert transform
  • Sinc approximation
  • Lévy processes
  • Barrier options
  • Survival probability

Mathematics Subject Classification

  • 44A15
  • 65R10
  • 60G51