## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

#### thesis presentations

11:00-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)

A group action on higher Chow cycles on a family of Kummer surfaces

(あるクンマー曲面族の上の高次チャウサイクルへの群作用について)

**SATO Ken**(Graduate School of Mathematical Sciences University of Tokyo)A group action on higher Chow cycles on a family of Kummer surfaces

(あるクンマー曲面族の上の高次チャウサイクルへの群作用について)

#### thesis presentations

11:00-12:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Manifolds Graded by an Arbitrary Abelian Group

(任意のアーベル群で次数付けられた多様体)

**HARAKO Shuichi**(Graduate School of Mathematical Sciences University of Tokyo)Manifolds Graded by an Arbitrary Abelian Group

(任意のアーベル群で次数付けられた多様体)

#### thesis presentations

11:00-12:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Various problems for properties of solutions to fractional partial differential equations

(非整数階偏微分方程式の解の性質に関する諸問題)

**SATO Shoichi**(Graduate School of Mathematical Sciences University of Tokyo)Various problems for properties of solutions to fractional partial differential equations

(非整数階偏微分方程式の解の性質に関する諸問題)

#### thesis presentations

13:00-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Integral Structures in the Local Algebra of a Singularity

(特異点の局所代数の整構造について)

**ZHA Chenghan**(Graduate School of Mathematical Sciences University of Tokyo)Integral Structures in the Local Algebra of a Singularity

(特異点の局所代数の整構造について)

#### thesis presentations

13:00-14:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Refinement of Young’s convolution inequality on locally compact groups and generalizations of related inequalities

(局所コンパクト群上のYoung の畳み込み不等式の精密化と関連の不等式の拡張)

**SATOMI Takashi**(Graduate School of Mathematical Sciences University of Tokyo)Refinement of Young’s convolution inequality on locally compact groups and generalizations of related inequalities

(局所コンパクト群上のYoung の畳み込み不等式の精密化と関連の不等式の拡張)

### 2023/01/26

#### thesis presentations

9:15-10:30 Room #122号室 (Graduate School of Math. Sci. Bldg.)

Fourier-Mukai transforms for non-commutative complex tori

(非可換複素トーラスのフーリエ・向井変換)

**OKUDA Nobuki**(Graduate School of Mathematical Sciences University of Tokyo)Fourier-Mukai transforms for non-commutative complex tori

(非可換複素トーラスのフーリエ・向井変換)

#### thesis presentations

11:00-12:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Earthquake theorem and cluster algebras

(地震定理とクラスター代数)

**ASAKA Takeru**(Graduate School of Mathematical Sciences University of Tokyo)Earthquake theorem and cluster algebras

(地震定理とクラスター代数)

#### thesis presentations

11:00-12:15 Room #126 (Graduate School of Math. Sci. Bldg.)

On universality of the Kardar-Parisi-Zhang equation in high temperature regime

(高温相におけるKardar-Parisi-Zhang 方程式の普遍性について)

**HAYASHI Kohei**(Graduate School of Mathematical Sciences University of Tokyo)On universality of the Kardar-Parisi-Zhang equation in high temperature regime

(高温相におけるKardar-Parisi-Zhang 方程式の普遍性について)

#### thesis presentations

13:00-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Studies on the Cone Conjecture, Automorphisms, and Arithmetic Degrees

(錐予想, 自己同型と算術次数の研究)

**WANG LONG**(Graduate School of Mathematical Sciences University of Tokyo)Studies on the Cone Conjecture, Automorphisms, and Arithmetic Degrees

(錐予想, 自己同型と算術次数の研究)

#### thesis presentations

13:00-14:15 Room #122 (Graduate School of Math. Sci. Bldg.)

Finite path integral model and toric code based on homological algebra

(ホモロジー代数に基づく有限経路積分モデルとトーリックコード)

**KIM Minkyu**(Graduate School of Mathematical Sciences University of Tokyo)Finite path integral model and toric code based on homological algebra

(ホモロジー代数に基づく有限経路積分モデルとトーリックコード)

#### thesis presentations

14:45-16:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by sl2 subalgebras which are generalizations of principal sl2 subalgebras

(主sl2部分代数の一般化であるsl2部分代数によるrank2対称双曲型Kac-Moody Lie 代数の既約分解)

**TSURUSAKI Hisanori**(Graduate School of Mathematical Sciences University of Tokyo)Irreducible module decompositions of rank 2 symmetric hyperbolic Kac-Moody Lie algebras by sl2 subalgebras which are generalizations of principal sl2 subalgebras

(主sl2部分代数の一般化であるsl2部分代数によるrank2対称双曲型Kac-Moody Lie 代数の既約分解)

### 2023/01/20

#### Colloquium

15:30-16:30 Hybrid

If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please take part online [Reference URL].

Kyiv formula and its applications (ENGLISH)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUrduioqjouG9wBfhl35VPxN_K92oa1wB4P

If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please take part online [Reference URL].

**Mikhail Bershtein**(HSE University, Skoltech)Kyiv formula and its applications (ENGLISH)

[ Abstract ]

The Kyiv formula gives the generic tau function of Painleve' equation (and more generally isomonodromy deformation equations) in terms of conformal blocks or Nekrasov partition function. I will explain the statement, examples and different approaches to the proof. If time permits, I will discuss some applications of this formula.

[ Reference URL ]The Kyiv formula gives the generic tau function of Painleve' equation (and more generally isomonodromy deformation equations) in terms of conformal blocks or Nekrasov partition function. I will explain the statement, examples and different approaches to the proof. If time permits, I will discuss some applications of this formula.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tZUrduioqjouG9wBfhl35VPxN_K92oa1wB4P

#### Tokyo-Nagoya Algebra Seminar

10:30-12:00 Online

Please see the reference URL for details on the online seminar.

Tropical cluster transformations and train track splittings (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the reference URL for details on the online seminar.

**Shunsuke Kano**(Tohoku University)Tropical cluster transformations and train track splittings (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2023/01/19

#### Information Mathematics Seminar

16:50-18:35 Room #123 (Graduate School of Math. Sci. Bldg.)

Code-based cryptography and its decoding algorithm (Japanese)

**Shintaro Narisada**(KDDI Research, Inc.)Code-based cryptography and its decoding algorithm (Japanese)

[ Abstract ]

This talk overviews code-based cryptography and its decoding algorithm called Information Set Decoding (ISD). All lectures will be given in Japanese.

This talk overviews code-based cryptography and its decoding algorithm called Information Set Decoding (ISD). All lectures will be given in Japanese.

### 2023/01/18

#### Number Theory Seminar

17:00-18:00 Hybrid

The affine Grassmannian as a presheaf quotient (English)

**Kestutis Cesnavicius**(Paris-Saclay University)The affine Grassmannian as a presheaf quotient (English)

[ Abstract ]

The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.

The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.

### 2023/01/17

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Integral structures in the local algebra of a singularity (ENGLISH)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Chenghan Zha**(The Univesity of Tokyo)Integral structures in the local algebra of a singularity (ENGLISH)

[ Abstract ]

We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. Furthermore, we figure out the image of the Milnor lattice of the singularity of an invertible polynomial of chain type using the basis of middle homology constructed by Otani-Takahashi. We calculated the Seifert form of the basis as well.

[ Reference URL ]We compute the image of the Milnor lattice of an ADE singularity under a period map. We also prove that the Milnor lattice can be identified with an appropriate relative K-group defined through the Berglund-Huebsch dual of the corresponding singularity. Furthermore, we figure out the image of the Milnor lattice of the singularity of an invertible polynomial of chain type using the basis of middle homology constructed by Otani-Takahashi. We calculated the Seifert form of the basis as well.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

### 2023/01/16

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Holomorphic foliation associated with a semi-positive class of numerical dimension one (Japanese)

https://forms.gle/hYT2hVhDE3q1wDSh6

**Takayuki Koike**(Osaka Metropolitan University)Holomorphic foliation associated with a semi-positive class of numerical dimension one (Japanese)

[ Abstract ]

Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bidegree $(1,1)$ on $X$.

When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

[ Reference URL ]Let $X$ be a compact Kähler manifold and $\alpha$ be a Dolbeault cohomology class of bidegree $(1,1)$ on $X$.

When the numerical dimension of $\alpha$ is one and $\alpha$ admits at least two smooth semi-positive representatives, we show the existence of a family of real analytic Levi-flat hypersurfaces in $X$ and a holomorphic foliation on a suitable domain of $X$ along whose leaves any semi-positive representative of $\alpha$ is zero.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

As an application, we give the affirmative answer to a conjecture on the relation between the semi-positivity of the line bundle $[Y]$ and the analytic structure of a neighborhood of $Y$ for a smooth connected hypersurface $Y$ of $X$.

https://forms.gle/hYT2hVhDE3q1wDSh6

### 2023/01/13

#### Discrete mathematical modelling seminar

13:15-14:45 Room #126 (Graduate School of Math. Sci. Bldg.)

An infinite sequence of Heron triangles with two rational medians (English)

**Andy Hone**(University of Kent)An infinite sequence of Heron triangles with two rational medians (English)

[ Abstract ]

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, despite an assertion by Schubert that even two rational medians are impossible, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z x Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, despite an assertion by Schubert that even two rational medians are impossible, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E(Q) with Mordell-Weil group Z x Z/2Z, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel-Roberts-Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

### 2023/01/12

#### Information Mathematics Seminar

16:50-18:35 Room #123 (Graduate School of Math. Sci. Bldg.)

Theory of fault-tolerant quantum computing II (Japanese)

**Yasunari Suzuki**(NTT)Theory of fault-tolerant quantum computing II (Japanese)

[ Abstract ]

To demonstrate reliable quantum computing, we need to integrate

quantum error correction techniques and achieve fault-tolerant quantum

computing. In this seminar, I will explain the basics of fault-tolerant quantum

computing and recent progress toward its experimental realization.

To demonstrate reliable quantum computing, we need to integrate

quantum error correction techniques and achieve fault-tolerant quantum

computing. In this seminar, I will explain the basics of fault-tolerant quantum

computing and recent progress toward its experimental realization.

#### Lectures

16:00-17:00 Online

The Calder'on problem for nonlocal parabolic operators (English)

https://u-tokyo-ac-jp.zoom.us/j/82806510515?pwd=NEk1RDlMVEFOTEg4WE1MekRySlJpdz09

**Prof. Yi-Hsuan Lin**(National Yang Ming Chiao Tung University, Taiwan)The Calder'on problem for nonlocal parabolic operators (English)

[ Abstract ]

We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calder'on problems, respectively.

This is a joint work with Ching-Lung Lin and Gunther Uhlmann.

[ Reference URL ]We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce nonlocal parabolic inverse problems to the corresponding local inverse problems with the lateral boundary Cauchy data. In addition, we derive a new equation and offer a novel proof of the unique continuation property for this new equation. We also build both uniqueness and non-uniqueness results for both nonlocal isotropic and anisotropic parabolic Calder'on problems, respectively.

This is a joint work with Ching-Lung Lin and Gunther Uhlmann.

https://u-tokyo-ac-jp.zoom.us/j/82806510515?pwd=NEk1RDlMVEFOTEg4WE1MekRySlJpdz09

### 2023/01/11

#### Discrete mathematical modelling seminar

13:15-16:45 Room #056 (Graduate School of Math. Sci. Bldg.)

Determinantal expressions for Ohyama polynomials (English)

Discrete dynamics, continued fractions and hyperelliptic curves (English)

**Joe Harrow**(University of Kent) 13:15-14:45Determinantal expressions for Ohyama polynomials (English)

[ Abstract ]

The Ohyama polynomials provide algebraic solutions of the D7 case of the Painleve III equation at a particular sequence of parameter values. It is known that many special function solutions of Painleve equations are expressed in terms of tau functions that can be written in the form of determinants, but until now such a representation for the Ohyama polynomials was not known. Here we present two different determinantal formulae for these polynomials: the first, in terms of Wronskian determinants related to a Darboux transformation for a Lax pair of KdV type; and the second, in terms of Hankel determinants, which is related to the Toda lattice. If time permits, then connections with orthogonal polynomials, and with the recent Riemann-Hilbert approach of Buckingham & Miller, will briefly be mentioned.

The Ohyama polynomials provide algebraic solutions of the D7 case of the Painleve III equation at a particular sequence of parameter values. It is known that many special function solutions of Painleve equations are expressed in terms of tau functions that can be written in the form of determinants, but until now such a representation for the Ohyama polynomials was not known. Here we present two different determinantal formulae for these polynomials: the first, in terms of Wronskian determinants related to a Darboux transformation for a Lax pair of KdV type; and the second, in terms of Hankel determinants, which is related to the Toda lattice. If time permits, then connections with orthogonal polynomials, and with the recent Riemann-Hilbert approach of Buckingham & Miller, will briefly be mentioned.

**Andy Hone**(University of Kent) 15:15-16:45Discrete dynamics, continued fractions and hyperelliptic curves (English)

[ Abstract ]

After reviewing some standard facts about continued fractions for quadratic irrationals, we switch from the real numbers to the field of Laurent series, and describe some classical and more recent results on continued fraction expansions for the square root of an even degree polynomial, and other functions defined on the associated hyperelliptic curve. In the latter case, we extend results of van der Poorten on continued fractions of Jacobi type (J-fractions), and explain the connection with a family of discrete integrable systems (including Quispel-Roberts-Thompson maps and Somos sequences), orthogonal polynomials, and the Toda lattice. If time permits, we will make some remarks on current work with John Roberts and Pol Vanhaecke, concerning expansions involving the square root of an odd degree polynomial, Stieltjes continued fractions, and the Volterra lattice.

After reviewing some standard facts about continued fractions for quadratic irrationals, we switch from the real numbers to the field of Laurent series, and describe some classical and more recent results on continued fraction expansions for the square root of an even degree polynomial, and other functions defined on the associated hyperelliptic curve. In the latter case, we extend results of van der Poorten on continued fractions of Jacobi type (J-fractions), and explain the connection with a family of discrete integrable systems (including Quispel-Roberts-Thompson maps and Somos sequences), orthogonal polynomials, and the Toda lattice. If time permits, we will make some remarks on current work with John Roberts and Pol Vanhaecke, concerning expansions involving the square root of an odd degree polynomial, Stieltjes continued fractions, and the Volterra lattice.

### 2023/01/10

#### Algebraic Geometry Seminar

10:30-12:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)

Toric Fano varieties arising from posets and their combinatorial mutation equivalence (日本語)

**Akihiro Higashitani**(Osaka/Dept. of Inf. )Toric Fano varieties arising from posets and their combinatorial mutation equivalence (日本語)

[ Abstract ]

In 1986, Stanley introduced two polytopes arising from posets, called order polytopes and chain polytopes. Since then, those polytopes have been studied from viewpoints of combinatorics. Projective toric varieties arising from order polytopes are called Hibi toric varieties in these days. On the other hand, combinatorial mutations were introduced by Akhtar-Coates-Galkin-Kasprzyk in 2012 in the context of the classification problem of Fano varieties using mirror symmetry.

In this talk, after surveying two poset polytopes and combinatorial mutations, we discuss the combinatorial mutation equivalence of two poset polytopes. Those equivalence implies qG-deformation equivalence of projective toric varieties arising from two poset polytopes.

Moreover, it turns out that order polytopes, chain polytopes and their intermediate polytopes correspond to some toric Fano varieties.

In 1986, Stanley introduced two polytopes arising from posets, called order polytopes and chain polytopes. Since then, those polytopes have been studied from viewpoints of combinatorics. Projective toric varieties arising from order polytopes are called Hibi toric varieties in these days. On the other hand, combinatorial mutations were introduced by Akhtar-Coates-Galkin-Kasprzyk in 2012 in the context of the classification problem of Fano varieties using mirror symmetry.

In this talk, after surveying two poset polytopes and combinatorial mutations, we discuss the combinatorial mutation equivalence of two poset polytopes. Those equivalence implies qG-deformation equivalence of projective toric varieties arising from two poset polytopes.

Moreover, it turns out that order polytopes, chain polytopes and their intermediate polytopes correspond to some toric Fano varieties.

#### Lectures

16:00-17:00 Online

Logarithmic convexity of semigroups and inverse problems for parabolic equations (English)

[ Reference URL ]

https://u-tokyo-ac-jp.zoom.us/j/83149935801?pwd=OE5aanNBVGxvajNycXgyb2RKcW1kZz09

**Professor Salah-Eddine CHORFI**(Cadi Ayyad University, Faculty of Sciences, Morocco)Logarithmic convexity of semigroups and inverse problems for parabolic equations (English)

[ Reference URL ]

https://u-tokyo-ac-jp.zoom.us/j/83149935801?pwd=OE5aanNBVGxvajNycXgyb2RKcW1kZz09

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Some calculations of an earthquake map in the cross ratio coordinates and the earthquake theorem of cluster algebras of finite type (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Takeru Asaka**(The Univesity of Tokyo)Some calculations of an earthquake map in the cross ratio coordinates and the earthquake theorem of cluster algebras of finite type (JAPANESE)

[ Abstract ]

Thurston defined an earthquake, which cuts the Poincaré half-plane model and shifts it. Though it is a discontinuous bijective map, it can be extended to a homeomorphism of a circumference. Also, if an earthquake is equivalent relative to a Fuchsian group, the homeomorphism is equivalent, too. Moreover, Thurston proved the earthquake theorem saying that there uniquely exists an earthquake for any orient-preserving homeomorphism of a circumference, and Bonsante-Krasnov-Schlenker extended it to the case of marked surfaces. We calculate some earthquake maps by the cross ratio coordinates. The cross ratio coordinates are deeply related by the cluster algebra (Fock-Goncharov). We prove the earthquake theorem of cluster algebras of finite type. It is a joint work with Tsukasa Ishibashi and Shunsuke Kano.

[ Reference URL ]Thurston defined an earthquake, which cuts the Poincaré half-plane model and shifts it. Though it is a discontinuous bijective map, it can be extended to a homeomorphism of a circumference. Also, if an earthquake is equivalent relative to a Fuchsian group, the homeomorphism is equivalent, too. Moreover, Thurston proved the earthquake theorem saying that there uniquely exists an earthquake for any orient-preserving homeomorphism of a circumference, and Bonsante-Krasnov-Schlenker extended it to the case of marked surfaces. We calculate some earthquake maps by the cross ratio coordinates. The cross ratio coordinates are deeply related by the cluster algebra (Fock-Goncharov). We prove the earthquake theorem of cluster algebras of finite type. It is a joint work with Tsukasa Ishibashi and Shunsuke Kano.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

#### Seminar on Probability and Statistics

10:50-11:30 Room # (Graduate School of Math. Sci. Bldg.)

Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions

(現地参加) https://forms.gle/qwssLccVgsAWcfps7 (Zoom参加) (1/8迄) https://docs.google.com/forms/d/e/1FAIpQLSe7OYeMDfaZ7pTLO42k43Tn5dWKpsyw

**井口優雅**(University College London)Parameter Estimation with Increased Precision for Elliptic and Hypo-elliptic Diffusions

[ Abstract ]

Parametric inference for multi-dimensional diffusion processes has been studied over the past decades. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) over finite time periods. Especially in the setting of some class of hypo-elliptic models, recent research (Ditlevsen and Samson 2019, Gloter and Yoshida 2021) has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates. In our work, first, we develop two weak second order ‘sampling schemes' (to cover both the hypo-elliptic and elliptic classes) and generate accompanying ‘transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods, in both high and low frequency observations regime. We also present numerical results from carrying out classical or Bayesian inference. This is a joint work with Alexandros Beskos and Matthew Graham, and the preprint is available at https://arxiv.org/abs/2211.16384.

[ Reference URL ]Parametric inference for multi-dimensional diffusion processes has been studied over the past decades. Established approaches for likelihood-based estimation invoke a numerical time-discretisation scheme for the approximation of the (typically intractable) transition dynamics of the Stochastic Differential Equation (SDE) over finite time periods. Especially in the setting of some class of hypo-elliptic models, recent research (Ditlevsen and Samson 2019, Gloter and Yoshida 2021) has highlighted the critical effect of the choice of numerical scheme on the behaviour of derived parameter estimates. In our work, first, we develop two weak second order ‘sampling schemes' (to cover both the hypo-elliptic and elliptic classes) and generate accompanying ‘transition density schemes' of the SDE (i.e., approximations of the SDE transition density). Then, we produce a collection of analytic results, providing a complete theoretical framework that solidifies the proposed schemes and showcases advantages from their incorporation within SDE calibration methods, in both high and low frequency observations regime. We also present numerical results from carrying out classical or Bayesian inference. This is a joint work with Alexandros Beskos and Matthew Graham, and the preprint is available at https://arxiv.org/abs/2211.16384.

(現地参加) https://forms.gle/qwssLccVgsAWcfps7 (Zoom参加) (1/8迄) https://docs.google.com/forms/d/e/1FAIpQLSe7OYeMDfaZ7pTLO42k43Tn5dWKpsyw

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