Contents

University of California Santa Barbara

Tomoyuki ICHIBA

Associate Professor, Department of Statistics & Applied Probability and Center for Financial Mathematics and Actuarial Research

at University of California Santa Barbara

Current Research

# Large scale interactions in financial markets (NSF DMS-2008427)

The project is concerned with mathematical understanding of interactions among participants in large financial markets. In order to analyze both homogeneous interaction with the market average and heterogenous interactions with similar participants, graphical interactions are modeled and analyzed with the methods in the system of Stochastic (Partial) Differential Equations (SDEs and SPDEs) with probability distribution constraints. Four problems are considered: 1. Tree structure: extension from the simple, directed chain interaction, 2. Detection problem and beyond: SPDE approach, 3. Model of entangled stochastic volatility in financial markets, 4. Mean-field game and controlled directed chain equation.

In the project the interactions are considered through the pivotal quantities, such as average and ranking, as well as through the local structure of the system. Among various local structures, directed chain and associated system of SDEs with distributional constraints are made in the main focus of the analysis in the first problem. Here, a new class of infinite-dimensional SPDEs emerges in developing the theory of filtering to solve the second problem. As one of applications from the analyses of the first problem, the model of stochastic volatility with directed chain structure is investigated and then the deriving methods are used in the theory of the Mean-field games.

Directed chain (left) and Mean-field (Right) interactions Related papers:

``Linear-Quadratic Stochastic Differential Games on Directed Chain Networks" Journal of Mathematics and Statistical Science (2021) 7 25--67 arXiv: 2003.08840 (with Yichen Feng and Jean-Pierre Fouque).

``Linear-Quadratic Stochastic Differential Games on Random Directed Networks" Journal of Mathematics and Statistical Science (2021) 7 79--108 arXiv:2011.04279 (with Yichen Feng and Jean-Pierre Fouque).

# Information and Stochastic Differential Equations in Financial Markets (NSF DMS-1615229)

The proposed project is concerned with mathematical aspects of information flow in financial markets. Based on different aspects in the financial markets three systems of Stochastic Differential Equations (SDEs) are considered. The stochastic analysis on these systems of SDEs is performed with two problems:

# Financial Markets with Discontinuities (NSF DMS-1313373)

This project studies some topics in the system of Stochastic Differential Equations with possibly discontinuous and degenerate diffusion coefficients with applications to stochastic portfolio management and optimization problems. Here the coefficients of the equations are functions of the states and determine the stochastic behavior of the system. The coefficients, the initial states and the stochastic noise of the system are main inputs of the model. The output of the system is the stochastic process which can be construed as the dynamics of the financial system. The long-term goals of the proposed activity are two-folds: (i) to defend against financial crises and (ii) to provide efficient investment strategies in the financial markets.

Preprints

(1) ``Large deviations for interacting Bessel-like processes and applications to systemic risk" (with M. Shkolnikov) arXiv1303.3061

(2) ``Option pricing with delayed information'' arXiv: 1707.01600 (with Seyyed Mostafa Mousavi).

(3) ``Large Banking Systems with Default and Recovery: A Mean Field Game Model" arXiv: 2001.10206 (with Romuald Elie and Mathieu Lauriere).

(4) ``Degenerate Competing Three-Particle Systems" arXiv: 2006.04970 (with Ioannis Karatzas) to appear in Bernoulli.

(5) ``Relative Arbitrage Opportunities in N Investors and Mean-Field Regimes" arXiv:2006.15158 (with Tianjiao Yang)

(6) ``Convolutional Signature for Sequential Data" arXiv:2009.06719 (with Ming Min).

(7) ``Semimartingale properties of a generalized fractional Brownian motion and its mixtures with applications in finance" arXiv:2012.00975 (with Guodong Pang and Murad Taqqu)

(8) ``Investments with Heterogenous Risk Aversion" (with Y. Feng and J.-P. Fouque) https://ssrn.com/abstract=3907445

Recently Published Papers

[19] ``Linear-Quadratic Stochastic Differential Games on Random Directed Networks" Journal of Mathematics and Statistical Science (2021) 7 79--108 arXiv:2011.04279 (with Yichen Feng and Jean-Pierre Fouque).

[18] ``Linear-Quadratic Stochastic Differential Games on Directed Chain Networks" Journal of Mathematics and Statistical Science (2021) 7 25--67 arXiv: 2003.08840 (with Yichen Feng and Jean-Pierre Fouque).

[17] ``Path properties of a generalized fractional Brownian motion" Journal of Theoretical Probability (2021) arXiv: 2009.07788 (with Guodong Pang and Murad Taqqu)

[16] ``Directed chain stochastic differential equation'' Stochastci Processes and their Applications 130 (2020) 2519-2551 arXiv: 1805.01962 (with Nils Detering and Jean-Pierre Fouque)

[15] ``Dynamic Contagion in a Banking System with Births and Defaults'' Annals of Finance (2019) 15 489-538 , arXiv: 1807.09897 (with Michael Ludkovski and Andrey Sarantsev).

[14] ``Stationary distribution and convergence for Walsh diffusions'' Bernoulli Journal Volume 25 (2019) 2439-2478 (with Andrey Sarantsev) arXiv: 1706.07127

[13] ``Stochastic integral equations for Walsh semimartingales" Annales de l'Institut Henri Poincare (2018) Vol. 54, No. 2, 726-756 (with Ioannis Karatzas, Vilmos Prokaj and Minghan Yan) arXiv: 1505.02504

[12] ``Yet another condition for absence of collisions for competing Brownian particles'' Electronic Communications in Probability Volume 22 (2017), paper no. 8, 7 pp. (with Andrey Sarantsev) arXiv:1608.07220

[11] ``Skew-unfolding the Skorokhod reflection of a continuous semimartingale" Stochastic Analysis and Applications 2014 In Honor of Terry Lyons. Springer Proceedings in Mathematics & Statistics (2014) 349-376 (with Ioannis Karatzas) arXiv: 1404.4662.

[10] ``Planar diffusions with rank-based characteristics and perturbed Tanaka equations" Probability Theory and Related Fields 156 (2013) 343-374; a full version with additional topics of Transition probabilities, Time reversibility and Maximality is also available in (with E. R. Fernholz, V. Prokaj and I. Karatzas) arXiv:1108.3992 .

[9] ``Strong solutions of stochastic equations with rank-based coefficients" Probability Theory and Related Fields 156 (2013) 229-248 (with I. Karatzas and M. Shkolnikov) arXiv1109.3823 .

[8] ``Diffusions with rank-based characteristics and values in the nonnegative quadrant" Bernoulli 19 (2013) 2455-2493 (with Ioannis Karatzas and Vilmos Prokaj) arXiv:1202.0036.

[7] ``Convergence rates for rank-based models with applications to portfolio theory" Probability Theory and Related Fields 156 (2013) 415-448 (with S. Pal and M. Shkolnikov) arXiv:1108.0384.

[6] ``A second-order stock market model" Annals of Finance 9 (2013), 439-454 (with R. Fernholz and I. Karatzas) arXiv:1302.3870.

[5] ``Two Brownian particles with rank-based characteristics and skew-elastic collisions'' Stochastic Processes and their Applications 123 (2013) 2999-3026 (with E. Robert Fernholz and Ioannis Karatzas) arXiv:1206.4350 .

[4] ``Stability in a model of inter-bank lending" SIAM Journal of Financial Mathematics 4 (2013) 784-803 (with J-P. Fouque).

[3] ``Hybrid Atlas models" Annals of Applied Probability 21 no 2. (2011), 609-644. (with R. Fernholz, I. Karatzas, A. Banner and V. Papathanakos) arXiv:0909.0065.

[2] ``Collisions of Brownian particles" Annals of Applied Probability 20 no 3. (2010), 951-977 (with I. Karatzas) arXiv:0810.2149.

[1] ``Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation" Journal of Applied Probability 48 no 3. (2011) 699-712 (with Constantinos Kardaraz) arXiv:1008.1326.

Dissertation

``Topics in multi-dimensional diffusions: attainability, reflection, ergodicity and rankings'' Dissertation Columbia University. 2009. [924kb].

Conference Proceedings

[C4] ``Stochastic analysis for collision of Brownian particles'' March 2017 The Mathematical Society of Japan.

[C3] ``On mean-field approximation of particle systems with annihilation and spikes'' at Probability Symposium Dec. 2016, published in RIMS RIMS Kokyuroku Kyoto University, Kyoto, 2017.

[C2] `` Folding and Skew-Unfolding of One-dimensional Continuous Semimartingales" RIMS Kokyuroku No 1952: Symposium on Probability Theory, Kyoto 2014

[C1] ``On planar rank-based diffusions with skew-elastic collisions" RIMS Kokyuroku No 1903: Symposium on Probability Theory, Kyoto 2013

Posters

``Detecting Mean-field" Seminar on Stochastic Processes at Brown University (May 2018)

``Stochastic Integral Equations for Walsh Semimartingales" IMA Special Workshop: Reflected Brownian motions, Stochastic Networks and their applications at University of Minnesota (June 2015)

``Strong/Weak Solutions of 2D Diffusions with Rank-based Characterisits" Probabiliy, Control and Finance at Columbia University (May 2012)

Miscellaneous Papers of Statistics in Sports, Biology and Actuarial Science

(S4) ``Estimating the effect of the red card in soccer: when to commit an offense in exchange for preventing a goal opportunity'' Journal of Quantitative Analysis in Sports 5, no. 1 / 2009. (with Jan Vecer and Frantisek Kopriva).

(S3)``On probabilistic excitement of sports games.'' Journal of Quantitative Analysis in Sports 3, no. 6 / 2007. (with Jan Vecer and Mladen Laudanovic).

(S2) ``Assessing substitution variation across sites in grass chloroplast DNA.'' Journal of Molecular Evolution 64, no. 6 pp. 605-613 / June, 2007. (with Tian Zheng and Brian Morton) .

(S1) ``On multi-period statistical risk management methods and equity-linked life insurance.'' Journal of the Japan Statistical Society. Japanese issue 35, no. 2 pp. 103-123. / 2006. abstract. (with Naoto Kunitomo).

Last modified: September 2021.

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