About

Tomoyuki Ichiba is a professor in the Department of Statistics & Applied Probability and the Center for Financial Mathematics and Actuarial Research at the University of California Santa Barbara. His research focuses on probability theory and stochastic processes, including stochastic differential equations, collisions of Brownian particles, local time of semimartingales, interacting particle systems, stochastic partial differential equations, stochastic filtering and control, fractional Brownian motions, and rough path signatures. Additionally, his work encompasses applications in mathematical economics and finance, such as stochastic portfolio theory and systemic risk problems, as well as interdisciplinary areas like quantum computing, data science, statistics in finance, molecular biology, and sports. His research has been supported by several NSF grants. In teaching, he offers courses in probability, statistics, and actuarial science.

Research topics

• Mathematical economics
• Mathematical optimization
• Mathematical analysis
• Computer Science
• Physics
• Mathematics
• Applied mathematics
• Econometrics
• Economics
• Finance

Selected publications

• Feynman Formula for Discrete-Time Quantum Walks

  Journal of Statistical Physics · 2026-05-09

  articleOpen accessCorresponding

  Abstract We explicitly connect (discrete-time) quantum walks on $$\mathbb {Z} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> with a four-state Markov additive process via a Feynman-type formula (13). Using this representation, we derive a relation between the spectral decomposition of the Markov additive process and the limiting density of the homogeneous quantum walk. In addition, we consider a space-time rescaling of quantum walks, which leads to a system of quantum transport PDEs in continuous time and space with a phase interaction term. Our probabilistic representation for this type of PDE offers an efficient Monte Carlo computational technique.

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• A Unified Approach to Compound Poisson Process and its Time-Fractional Versions

  Journal of Theoretical Probability · 2026-02-23

  articleSenior author
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• Relative Arbitrage Opportunities With Interactions Among <i>N</i> Investors

  Mathematical Finance · 2026-05-09

  article1st authorCorresponding

  ABSTRACT The relative arbitrage portfolio outperforms a benchmark portfolio over a given time‐horizon with probability one. With market price of risk processes depending on the market portfolio and investors, this paper analyzes the multi‐agent optimization of relative arbitrage opportunities in the coupled system of market and wealth dynamics. We construct a well‐posed market dynamical system of McKean–Vlasov type under an empirical measure of investors, where each investor seeks for relative arbitrage with respect to a benchmark dependent on market and all the agents. We show the conditions to guaranty relative arbitrage opportunities among competitive investors through the Fichera drift. Under mild conditions, we derive the optimal strategies for investors and the unique Nash equilibrium that depends on the smallest nonnegative solution of a Cauchy problem.

  PublisherDOI

• Semimartingale properties of a generalised fractional Brownian motion and its mixtures with applications in asset pricing

  Finance and Stochastics · 2025-04-07 · 1 citations

  article1st authorCorresponding
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• Short Communication: Finding the Nonnegative Minimal Solutions of Cauchy PDEs in a Volatility-Stabilized Market

  SIAM Journal on Financial Mathematics · 2025-10-14 · 2 citations

  articleSenior author
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• Unbiased Rough Integrators and No Free Lunch in Rough-Path-Based Market Models

  ArXiv.org · 2025-09-18

  preprintOpen access1st authorCorresponding

  Built to generalise classical stochastic calculus, rough path theory provides a natural and pathwise framework to model continuous non-semimartingale assets. This paper investigates the capacity of this framework to support frictionless continuous No-Free-Lunch markets à la Kreps-Yan. We establish a "Rough Kreps-Yan" theorem, which links a No Controlled Free Lunch (NCFL) condition to the unbiasedness of the driver of the price process as a rough integrator. The central work of this paper is a classification of these unbiased rough integrators with respect to different classes of controlled paths as integrands, under some assumptions. As the admissible strategies are enlarged from Markovian-type portfolios to signature-type and adaptedly scaled signature-type portfolios, the admissible random rough paths collapse first to Gaussian-Hermite rough paths, and ultimately to the Itô rough path lift of a standard Brownian motion, up to a time change. Notably, simple strategies do not appear in the theory. This implies that within our framework, continuous frictionless markets based on rough path theory are inevitably constrained to the classical semimartingale paradigm, clarifying the limits of this approach. Our framework covers $α-$Hölder continuous rough paths for $α&gt;0$ arbitrarily small in the tensor algebra setting.

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• Rank-Based Stochastic Differential Inclusions and Diffusion Limits for a Load-Balancing Model

  Mathematics of Operations Research · 2025-11-27

  articleSenior author

  Banerjee, Budhiraja and Estevez (2025) studied a randomized load balancing model in a heavy traffic asymptotic regime where the load balancing stream is thin compared to the total arrival stream. It was shown that the limit is given by a system of rank-based Brownian particles on the half-line. In this paper we extend this result from the case of exponential service time to an invariance principle, where service times have finite second moment. The main tool is a new notion of rank-based stochastic differential inclusion, which may be of interest in its own right. Funding: R. Atar is partially supported by the Israel Science Foundation [Grant 1035/20]. T. Ichiba was supported in part by the National Science Foundation [Grant NSF-DMS-2008427].

  PublisherDOI

• Optimal Investment with Insider Information Using Skorokhod &amp; Russo-Vallois Integration

  Journal of Optimization Theory and Applications · 2025-08-13 · 1 citations

  articleOpen accessSenior author

  Abstract We study the maximization of the logarithmic utility for an insider with different anticipating techniques. Our aim is to compare the utilization of Russo-Vallois forward integral and Skorokhod integral in this context. Theoretical analysis and illustrative numerical examples showcase that the Skorokhod insider outperforms the forward insider. This remarkable observation stands in contrast to the scenario involving risk-neutral traders. Furthermore, an ordinary trader could surpass both insiders if a significant negative fluctuation in the driving stochastic process leads to a sufficiently negative final value. These findings underline the intricate interplay between anticipating stochastic calculus and nonlinear utilities, which may yield non-intuitive results from the financial viewpoint.

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• Heterogenous Macro-Finance Model: A Mean-field Game Approach

  ArXiv.org · 2025-02-15

  preprintOpen accessSenior author

  We investigate the full dynamics of capital allocation and wealth distribution of heterogeneous agents in a frictional economy during booms and busts using tools from mean-field games. Two groups in our models, namely the expert and the household, are interconnected within and between their classes through the law of capital processes and are bound by financial constraints. Such a mean-field interaction explains why experts accumulate a lot of capital in the good times and reverse their behavior quickly in the bad times even in the absence of aggregate macro-shocks. When common noises from the market are involved, financial friction amplifies the mean-field effect and leads to capital fire sales by experts. In addition, the implicit interlink between and within heterogeneous groups demonstrates the slow economic recovery and characterizes the deviating and fear-of-missing-out (FOMO) behaviors of households compared to their counterparts. Our model also gives a fairly explicit representation of the equilibrium solution without exploiting complicated numerical approaches.

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• Stochastic Volterra integral equations with ranks as scaling limits of parallel infinite-server queues under weighted shortest queue policy

  Journal of Applied Probability · 2025-12-02

  articleOpen access1st authorCorresponding

  Abstract We study a queueing system with a fixed number of parallel service stations of infinite servers, each having a dedicated arrival process, and one flexible arrival stream that is routed to one of the service stations according to a ‘weighted’ shortest queue policy. We consider the model with general arrival processes and general service time distributions. Assuming that the dedicated arrival rates are of order n and the flexible arrival rate is of order $\sqrt{n}$ , we show that the diffusion-scaled queueing processes converge to a stochastic Volterra integral equation with ‘ranks’ driven by a continuous Gaussian process. It reduces to the limiting diffusion with a discontinuous drift in the Markovian setting.

  PublisherDOI

Recent grants

• Financial markets with discontinuities

  NSF · $85k · 2013–2017

• Large-Scale Interactions in Financial Markets

  NSF · $148k · 2020–2024

• Information and Stochastic Differential Equations in Financial Markets

  NSF · $97k · 2016–2020

Frequent coauthors

• Ioannis Karatzas

  28 shared

• Vilmos Prokaj

  Eötvös Loránd University

  11 shared

• Jean‐Pierre Fouque

  10 shared

• Robert Fernholz

  9 shared

• Mykhaylo Shkolnikov

  7 shared

• Andrey Sarantsev

  7 shared

• E. Robert Fernholz

  6 shared

• Guodong Pang

  5 shared

Labs

• Center for Financial Mathematics and Actuarial ResearchPI