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Akihiko Takahashi - One of the best experts on this subject based on the ideXlab platform.
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a weak approximation with Asymptotic Expansion and multidimensional malliavin weights
Annals of Applied Probability, 2016Co-Authors: Akihiko Takahashi, Toshihiro YamadaAbstract:This paper develops a new efficient scheme for approximations of expectations of the solutions to stochastic differential equations (SDEs). In particular, we present a method for connecting approximate operators based on an Asymptotic Expansion to compute a target expectation value precisely. The mathematical validity is given based on Watanabe and Kusuoka theories in Malliavin calculus. Moreover, a numerical experiment for option pricing under a local volatility model confirms the effectiveness of our scheme.
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Asymptotic Expansion for forward backward sdes with jumps
arXiv: Computational Finance, 2015Co-Authors: Masaaki Fujii, Akihiko TakahashiAbstract:This work provides a semi-analytic approximation method for decoupled forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an Asymptotic Expansion method for FBSDEs driven by the random Poisson measures with {\sigma}-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.
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a general computation scheme for a high order Asymptotic Expansion method
International Journal of Theoretical and Applied Finance, 2012Co-Authors: Akihiko Takahashi, Kohta Takehara, Masashi TodaAbstract:This paper presents a new computational scheme for an Asymptotic Expansion method of an arbitrary order. The Asymptotic Expansion method in finance initiated by Kunitomo and Takahashi (1992), Yoshida (1992b) and Takahashi (1995, 1999) is a widely applicable methodology for an analytic approximation of expectation of a certain functional of diffusion processes. Hence, not only academic researchers but also many practitioners have used the methodology for a variety of financial issues such as pricing or hedging complex derivatives under high-dimensional underlying stochastic environments. In practical applications of the Expansion, a crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. Takahashi (1995, 1999) and Takahashi and Takehara (2007) provided explicit formulas for those conditional expectations necessary for the Asymptotic Expansion up to the third order. This paper presents the new method for computing an arbitrary-order Expansion in a general diffusion-type stochastic environment, which is powerful especially for high-order Expansions: We develops a new calculation algorithm for computing coefficients of the Expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations directly. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for a λ-SABR model up to the fifth order.
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an Asymptotic Expansion with push down of malliavin weights
Siam Journal on Financial Mathematics, 2012Co-Authors: Akihiko Takahashi, Toshihiro YamadaAbstract:This paper derives Asymptotic Expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in multidimensional stochastic volatility models. In...
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a general computation scheme for a high order Asymptotic Expansion method
CARF F-Series, 2011Co-Authors: Akihiko Takahashi, Kohta Takehara, Masashi TodaAbstract:This paper presents a new computational scheme for an Asymptotic Expansion method of an arbitrary order. An Asymptotic Expansion method in finance initiated by Kunitomo and Takahashi[9], Yoshida[34] and Takahashi [20], [21] is a widely applicable methodology for an analytic approximation of the expectation of a certain functional of diffusion processes and not only academic researchers but also many practitioners have used the methodology for a variety of financial issues such as pricing or hedging complex derivatives under highdimensional underlying stochastic environments. In practical applications of the Expansion, the crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [20], [21] and Takahashi and Takehara [23] provided explicit formulas of conditional expectations necessary for the Asymptotic Expansion up to the third order. This paper presents the new method for computing an arbitrary-order Expansion in a general diffusion-type stochastic environment, which is powerful especially for a high-order Expansion: This develops a new calculation algorithm for computing coefficients of the Expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for the λ-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.
Nakahiro Yoshida - One of the best experts on this subject based on the ideXlab platform.
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High order Asymptotic Expansion for Wiener functionals
2019Co-Authors: Ciprian Tudor, Nakahiro YoshidaAbstract:By combining the Malliavin calculus with Fourier techniques, we develop a high-order Asymptotic Expansion theory for a sequence of vector-valued random variables. Our Asymptotic Expansion formulas give the development of the characteristic functional and of the local density of the random vectors up to an arbitrary order. We analyzed in details an example related to the wave equation with space-time white noise which also provides interesting facts on the correlation structure of the solution to this equation.
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Asymptotic Expansion of Skorohod integrals
arXiv: Probability, 2017Co-Authors: David Nualart, Nakahiro YoshidaAbstract:Asymptotic Expansion of the distribution of a perturbation $Z_n$ of a Skorohod integral jointly with a reference variable $X_n$ is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale Expansion. The second-order interpolation and Fourier inversion give Asymptotic Expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable functions $f$ and also measurable functions $f$. In the latter case, the interpolation method connects the two non-degeneracies of variables for finite $n$ and $\infty$. Random symbols are used for expressing the Asymptotic Expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. We identify these random symbols for a certain quadratic form of a fractional Brownian motion and for a quadratic from of a fractional Brownian motion with random weights. For a quadratic form of a Brownian motion with random weights, we observe that our formula reproduces the formula originally obtained by the martingale Expansion.
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Asymptotic Expansion for vector-valued sequences of random variables with focus on Wiener chaos
2017Co-Authors: Ciprian Tudor, Nakahiro YoshidaAbstract:We develop the Asymptotic Expansion theory for vector-valued sequences (F N) N ≥1 of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F N. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the Asymptotic Expansion of the density of F N and we illustrate our results by several examples. 2010 AMS Classification Numbers: 62M09, 60F05, 62H12
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Asymptotic Expansion for vector valued sequences of random variables with focus on wiener chaos
arXiv: Probability, 2017Co-Authors: Ciprian A Tudor, Nakahiro YoshidaAbstract:We develop the Asymptotic Expansion theory for vector-valued sequences (F N) N $\ge$1 of random variables in terms of the convergence of the Stein-Malliavin matrix associated to the sequence F N. Our approach combines the classical Fourier approach and the recent theory on Stein method and Malliavin calculus. We find the second order term of the Asymptotic Expansion of the density of F N and we illustrate our results by several examples. 2010 AMS Classification Numbers: 62M09, 60F05, 62H12
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Asymptotic Expansion formulas for functionals of e markov processes with a mixing property
Annals of the Institute of Statistical Mathematics, 2004Co-Authors: Yuji Sakamoto, Nakahiro YoshidaAbstract:The e-Markov process is a general model of stochastic processes which includes nonlinear time series models, diffusion processes with jumps, and many point processes. With a view to applications to the higher-order statistical inference, we will consider a functional of the e-Markov process admitting a stochastic Expansion. Arbitrary order Asymptotic Expansion of the distribution will be presented under a strong mixing condition. Applying these results, the third order Asymptotic Expansion of theM-estimator for a general stochastic process will be derived. The Malliavin calculus plays an essential role in this article. We illustrate how to make the Malliavin operator in several concrete examples. We will also show that the thirdorder Expansion formula (Sakamoto and Yoshida (1998, ISM Cooperative Research Report, No. 107, 53–60; 1999, unpublished)) of the maximum likelihood estimator for a diffusion process can be obtained as an example of our result.
Masashi Toda - One of the best experts on this subject based on the ideXlab platform.
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a general computation scheme for a high order Asymptotic Expansion method
International Journal of Theoretical and Applied Finance, 2012Co-Authors: Akihiko Takahashi, Kohta Takehara, Masashi TodaAbstract:This paper presents a new computational scheme for an Asymptotic Expansion method of an arbitrary order. The Asymptotic Expansion method in finance initiated by Kunitomo and Takahashi (1992), Yoshida (1992b) and Takahashi (1995, 1999) is a widely applicable methodology for an analytic approximation of expectation of a certain functional of diffusion processes. Hence, not only academic researchers but also many practitioners have used the methodology for a variety of financial issues such as pricing or hedging complex derivatives under high-dimensional underlying stochastic environments. In practical applications of the Expansion, a crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. Takahashi (1995, 1999) and Takahashi and Takehara (2007) provided explicit formulas for those conditional expectations necessary for the Asymptotic Expansion up to the third order. This paper presents the new method for computing an arbitrary-order Expansion in a general diffusion-type stochastic environment, which is powerful especially for high-order Expansions: We develops a new calculation algorithm for computing coefficients of the Expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations directly. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for a λ-SABR model up to the fifth order.
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a general computation scheme for a high order Asymptotic Expansion method
CARF F-Series, 2011Co-Authors: Akihiko Takahashi, Kohta Takehara, Masashi TodaAbstract:This paper presents a new computational scheme for an Asymptotic Expansion method of an arbitrary order. An Asymptotic Expansion method in finance initiated by Kunitomo and Takahashi[9], Yoshida[34] and Takahashi [20], [21] is a widely applicable methodology for an analytic approximation of the expectation of a certain functional of diffusion processes and not only academic researchers but also many practitioners have used the methodology for a variety of financial issues such as pricing or hedging complex derivatives under highdimensional underlying stochastic environments. In practical applications of the Expansion, the crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [20], [21] and Takahashi and Takehara [23] provided explicit formulas of conditional expectations necessary for the Asymptotic Expansion up to the third order. This paper presents the new method for computing an arbitrary-order Expansion in a general diffusion-type stochastic environment, which is powerful especially for a high-order Expansion: This develops a new calculation algorithm for computing coefficients of the Expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for the λ-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.
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computation in an Asymptotic Expansion method
CARF F-Series, 2009Co-Authors: Akihiko Takahashi, Kohta Takehara, Masashi TodaAbstract:An Asymptotic Expansion scheme in finance initiated by Kunitomo and Takahashi [15] and Yoshida[68] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. [46], [47] and [53] provide explicit formulas of conditional expectations necessary for the Asymptotic Expansion up to the third order. In general, the crucial step in practical applications of the Expansion is calculation of conditional expectations for a certain kind of Wiener functionals. This paper presents two methods for computing the conditional expectations that are powerful especially for high order Expansions: The first one, an extension of the method introduced by the preceding papers presents a general scheme for computation of the conditional expectations and show the formulas useful for Expansions up to the fourth order explicitly. The second one develops a new calculation algorithm for computing the coefficients of the Expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate their effectiveness, the paper gives numerical examples of the approximation for ƒE-SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.
Judd S. Gardner - One of the best experts on this subject based on the ideXlab platform.
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Uniform Asymptotic Expansion of the Associated Legendre Function to Leading Term for Complex Degree and Integral Order
IEEE Transactions on Antennas and Propagation, 2007Co-Authors: Judd S. GardnerAbstract:The associated Legendre function arises naturally in the study of spherical waves. Since in practical applications it is most often symbolically represented by Pn m(xi) for m les n and Pn m(xi) equiv 0 for m > n where m is the integer order and n is the integer degree, this form will be employed to develop the uniform Asymptotic Expansion. The considerable extent to which this function appears in literature substantiates its importance in engineering and science, and particularly to spherical harmonics. In his book, "Partial Differential Equations in Physics" Sommerfeld covers a variety of subjects including spherical harmonics, and gives a detailed account of obtaining an Expansion of the associated Legendre function, Pn m(cos(thetas)), by the method of steepest descents over the interval 0 les thetas les pi. The results he obtains are quite accurate for n Gt m except as thetas approaches the critical points, thetas rarr 0 or thetas rarr pi. Beginning with the same integral representation of the associated Legendre function with integer order and degree that Sommerfeld employed, a uniform Asymptotic Expansion is found that is applicable to the neighborhoods of thetas = 0 and thetas = pi and that becomes increasingly more accurate as n increases beyond m. Furthermore, the accuracy of the resulting uniform Asymptotic Expansion remains for real degree and complex degree as well. The results are plotted in order to assess the accuracy and the domain of validity of the uniform Asymptotic Expansion. The results of the uniform Asymptotic Expansion are also compared to the available approximation of the associated Legendre function given in terms of Bessel functions for small values of thetas.
Gergő Nemes - One of the best experts on this subject based on the ideXlab platform.
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Error bounds for the Asymptotic Expansion of the Hurwitz zeta function
Proceedings. Mathematical physical and engineering sciences, 2017Co-Authors: Gergő NemesAbstract:In this paper, we reconsider the large- a Asymptotic Expansion of the Hurwitz zeta function ζ ( s , a ). New representations for the remainder term of the Asymptotic Expansion are found and used to obtain sharp and realistic error bounds. Applications to the Asymptotic Expansions of the polygamma functions, the gamma function, the Barnes G -function and the s -derivative of the Hurwitz zeta function ζ ( s , a ) are provided. A detailed discussion on the sharpness of our error bounds is also given.
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error bounds for the Asymptotic Expansion of the hurwitz zeta function
arXiv: Classical Analysis and ODEs, 2017Co-Authors: Gergő NemesAbstract:In this paper, we reconsider the large-$a$ Asymptotic Expansion of the Hurwitz zeta function $\zeta(s,a)$. New representations for the remainder term of the Asymptotic Expansion are found and used to obtain sharp and realistic error bounds. Applications to the Asymptotic Expansions of the polygamma functions, the gamma function, the Barnes $G$-function and the $s$-derivative of the Hurwitz zeta function $\zeta(s,a)$ are provided. A detailed discussion on the sharpness of our error bounds is also given.
