Econophysics & Complex Systems

We study expected utility maximization problem with constant relative risk aversion utility function in a complete market under the reinforcement learning framework. To induce exploration, we introduce the Tsallis entropy regularizer, which generalizes the commonly used Shannon entropy. Unlike the c

A property of data which is common across a wide range of instruments, markets and time periods is known as stylized empirical fact in the financial statistics literature. This paper first presents a wide range of stylized facts studied in literature which include some univariate distributional prop

Factor models characterize the joint behavior of large sets of financial assets through a smaller number of underlying drivers. We develop a network-based framework in which factors emerge naturally from the structure of interactions among assets rather than being imposed statistically. The market i

The deviation of the efficient market hypothesis (EMH) for the practical economic system allows us gain the arbitrary or risk premium in finance markets. We propose the triplet $(R,H,σ)$ theory to give the local and global optimal portfolio, which eneralize from the $(R,σ)$ model. We present the for

We use a $φ^{“4”}$ quantum field theory with inhomogeneous couplings and explicit symmetry-breaking to model an ensemble of financial time series from the S$&$P 500 index. The continuum nature of the $φ^4$ theory avoids the inaccuracies that occur in Ising-based models which require a discretization

Large language models (LLMs) produce fluent but unsupported answers - hallucinations - limiting safe deployment in high-stakes domains. We propose ECLIPSE, a framework that treats hallucination as a mismatch between a model’s semantic entropy and the capacity of available evidence. We combine entrop

Scaling laws offer a powerful lens to understand complex transactional behaviors in decentralized systems. This study reveals distinctive statistical signatures in the transactional dynamics of ERC20 tokens on the Ethereum blockchain by examining over 44 million token transfers between July 2017 and

We investigate exploratory randomization for an extended linear-exponential-quadratic-Gaussian (LEQG) control problem in discrete time. This extended control problem is related to the structure of risk-sensitive investment management applications. We introduce exploration through a randomization of

The Fractional Stochastic Regularity Model (FSRM) is an extension of Black-Scholes model describing the multifractal nature of prices. It is based on a multifractional process with a random Hurst exponent $H_t$, driven by a fractional Ornstein-Uhlenbeck (fOU) process. When the regularity parameter $

The objective of this work is the investigation of complexity, asymmetry, stochasticity and non-linearity of the financial and economic systems by using the tools of statistical mechanics and information theory. More precisely, this thesis concerns statistical-based modeling and empirical analyses w

We use the statistical properties of Shannon entropy estimator and Kullback-Leibler divergence to study the predictability of ultra-high frequency financial data. We develop a statistical test for the predictability of a sequence based on empirical frequencies. We show that the degree of randomness

In 2001, Rama Cont introduced a now-widely used set of ‘stylized facts’ to synthesize empirical studies of financial price changes (returns), resulting in 11 statistical properties common to a large set of assets and markets. These properties are viewed as constraints a model should be able to repro

The problem of order execution is cast as a relative entropy-regularized robust optimal control problem in this article. The order execution agent’s goal is to maximize an objective functional associated with his profit-and-loss of trading and simultaneously minimize the execution risk and the marke

When tackling binary optimization problems using quantum algorithms, the conventional Ising representation and Quantum Approximate Optimization Algorithm (QAOA) encounter difficulties in efficiently handling errors for large-scale problems involving multiple constraints. To address these challenges,

We analyze the scaled voter model, which is a generalization of the noisy voter model with time-dependent herding behavior. We consider the case when the intensity of herding behavior grows as a power-law function of time. In this case, the scaled voter model reduces to the usual noisy voter model,

We consider a class of stochastic processes with rough stochastic volatility, examples of which include the rough Bergomi and rough Stein-Stein model, that have gained considerable importance in quantitative finance. A basic question for such (non-Markovian) models concerns efficient numerical schem

In retrospect, the experimental findings on competitive market behavior called for a revival of the old, classical, view of competition as a collective higgling and bargaining process (as opposed to price-taking behaviors) founded on reservation prices (in place of the utility function). In this pap

This is a review about financial dependencies which merges efforts in econophysics and financial economics during the last few years. We focus on the most relevant contributions to the analysis of asset markets’ dependencies, especially correlational studies, which in our opinion are beneficial for

We study a benchmarked risk-sensitive portfolio problem in a factor-based setting to bring together three strands of the literature: benchmarked risk-sensitive investment management, the Kuroda-Nagai change-of-measure method, and the free energy-entropy duality of Dai Pra et al. (1996). We show that

We develop spectral portfolio theory by establishing a direct identification: neural network weight matrices trained on stochastic processes are portfolio allocation matrices, and their spectral structure encodes factor decompositions and wealth concentration patterns. The three forces governing sto

Cross-validation is one of the most widely used methods for model selection and evaluation; its efficiency for large covariance matrix estimation appears robust in practice, but little is known about the theoretical behavior of its error. In this paper, we derive the expected Frobenius error of the

Cross-validation is a statistical tool that can be used to improve large covariance matrix estimation. Although its efficiency is observed in practical applications and a convergence result towards the error of the non linear shrinkage is available in the high-dimensional regime, formal proofs that

Short-term patterns in financial time series form the cornerstone of many algorithmic trading strategies, yet extracting these patterns reliably from noisy market data remains a formidable challenge. In this paper, we propose an entropy-assisted framework for identifying high-quality, non-overlappin

In this paper, we introduce a robust market making framework based on Wasserstein distance, utilizing a stochastic policy approach enhanced by entropy regularization. We demonstrate that, under mild assumptions, the robust market making problem can be reformulated as a convex optimization question.

We analyze the stability of financial investment networks, where financial institutions hold overlapping portfolios of assets. We consider the effect of portfolio diversification and heterogeneous investments using a random matrix dynamical model driven by portfolio rebalancing. While heterogeneity

We study the relationship between model complexity and out-of-sample performance in the context of mean-variance portfolio optimization. Representing model complexity by the number of assets, we find that the performance of low-dimensional models initially improves with complexity but then declines

We study the fluctuations, particularly the inequality of fluctuations, in cryptocurrency prices over the last ten years. We calculate the inequality in the price fluctuations through different measures, such as the Gini and Kolkata indices, and also the $Q$ factor (given by the ratio between the hi

The skew-stickiness-ratio (SSR), examined in detail by Bergomi in his book, is critically important to options traders, especially market makers. We present a model-free expression for the SSR in terms of the characteristic function. In the diffusion setting, it is well-known that the short-term lim

The standard approach for constructing a Mean-Variance portfolio involves estimating parameters for the model using collected samples. However, since the distribution of future data may not resemble that of the training set, the out-of-sample performance of the estimated portfolio is worse than one

As is widely known, the stock market is a complex system in which a multitude of factors influence the performance of individual stocks and the market as a whole. One method for comprehending – and potentially predicting – stock market behavior is through network analysis, which can offer insights i

The assessment of risk based on historical data faces many challenges, in particular due to the limited amount of available data, lack of stationarity, and heavy tails. While estimation on a short-term horizon for less extreme percentiles tends to be reasonably accurate, extending it to longer time

Time irreversibility in neuronal dynamics has recently been demonstrated to correlate with various indicators of cognitive effort in living systems. Using Landauer’s principle, which posits that time-irreversible information processing consumes energy, we establish a thermodynamically consistent mea

In recent years, high-frequency trading has emerged as a crucial strategy in stock trading. This study aims to develop an advanced high-frequency trading algorithm and compare the performance of three different mathematical models: the combination of the cross-entropy loss function and the quasi-New

This paper uses the concepts of entropy to study the regularity/irregularity of the returns from the Indian Foreign exchange (forex) markets. The Approximate Entropy and Sample Entropy statistics which measure the level of repeatability in the data are used to quantify the randomness in the forex re

The Lillo-Mike-Farmer (LMF) model is an established econophysics model describing the order-splitting behaviour of institutional investors in financial markets. In the original article (LMF, Physical Review E 71, 066122 (2005)), LMF assumed the homogeneity of the traders’ order-splitting strategy an

We propose a new theoretical framework that exploits convolution kernels to transform a Volterra-type path-dependent (non-Markovian) stochastic process into a standard (Markovian) diffusion process. Remarkably, it is also possible to go back, i.e., the transformation is reversible. We discuss existe

We study the eigenvalue of the Wishart matrix, which is created from a time series with temporal correlation. When there is no correlation, the eigenvalue distribution of the Wishart matrix is known as the Marchenko-Pastur distribution (MPD) in the double scaling limit. When there is temporal correl

The concept of entropy is not uniquely relevant to the statistical mechanics but among others it can play pivotal role in the analysis of a time series, particularly the stock market data. In this area sudden events are especially interesting as they describe abrupt data changes which may have long-

This paper extends our previous work to continuous-time optimal stopping, focusing on American options in an exploratory setting. Our first contribution is an entropy-regularized penalization scheme, inspired by classical penalization techniques for reflected BSDEs. It yields a smooth approximation

The geometric Brownian motion (GBM) is widely employed for modeling stochastic processes, yet its solutions are characterized by the log-normal distribution. This comprises predictive capabilities of GBM mainly in terms of forecasting applications. Here, entropy corrections to GBM are proposed to go

Volatility clustering is one of the most robust stylized facts of financial markets, yet it is typically detected using moment-based diagnostics or parametric models such as GARCH. This paper shows that clustered volatility also leaves a clear imprint on the time-reversal symmetry of horizontal visi

The critical dynamics of conformal field theories on random surfaces is investigated beyond the previously studied dynamics of the overall area and the genus. It is found that the evolution of the order parameter in physical time performs a generalization of the multifractal random walk. Accordingly

We present a possible approach to measuring inequality in a system of coupled Fokker-Planck-type equations that describe the evolution of distribution densities for two populations interacting pairwise due to social and/or economic factors. The macroscopic dynamics of their mean values follow a Lotk

We formulate and solve an optimal trading problem with alpha signals, where transactions induce a nonlinear transient price impact described by a general propagator model, including power-law decay. Using a variational approach, we demonstrate that the optimal trading strategy satisfies a nonlinear

In statistical analysis, many classic results require the assumption that models have finite mean or variance, including the most standard versions of the laws of large numbers and the central limit theorems. Such an assumption may not be completely innocent, and it may not be appropriate for datase

We use random walks to simulate the fluid limit of two coupled diffusive limit order books to model correlation emergence. The model implements the arrival, cancellation and diffusion of orders coupled by a pairs trader profiting from the mean-reversion between the two order books in the fluid limit

This paper identifies the cryptocurrency market crashes and analyses its dynamics using the complex network. We identify three distinct crashes during 2017-20, and the analysis is carried out by dividing the time series into pre-crash, crash, and post-crash periods. Partial correlation based complex

We analyze correlation structures in financial markets by coarse graining the Pearson correlation matrices according to market sectors to obtain Guhr matrices using Guhr’s correlation method according to Ref. [“P. Rinn {"\it et. al.”}, Europhysics Letters 110, 68003 (2015)"]. We compare the results

Not a matter of serious contention, Pearson’s correlation coefficient is still the most important statistical association measure. Restricted to just two variables, this measure sometimes doesn’t live up to users’ needs and expectations. Specifically, a multivariable version of the correlation coeff

We present a standard form of master equations (ME) for general one-dimensional non-Markovian (history-dependent) jump processes, complemented by an asymptotic solution derived from an expanded system-size approach. The ME is obtained by developing a general Markovian embedding using a suitable set

The aim of this paper is to analyse the Bitcoin in order to shed some light on its nature and behaviour. We select 9 cryptocurrencies that account for almost 75% of total market capitalisation and compare their evolution with that of a wide variety of traditional assets: commodities with spot and fu

In optimal covariance cleaning theory, minimizing the Frobenius norm between the true population covariance matrix and a rotational invariant estimator is a key step. This estimator can be obtained asymptotically for large covariance matrices, without knowledge of the true covariance matrix. In this

Recently, we developed a theory of a geometrically growing system. Here we show that the theory can explain some phenomena of power-law distribution including classical demographic and economic and novel pandemic instances, without introduction of delicate economic models but only on the statistical

We study the problem of optimal portfolio selection under stochastic volatility within a continuous time reinforcement learning framework with portfolio constraints. Exploration is modeled through entropy-regularized relaxed controls, where the investor selects probability distributions over admissi

We investigate the impossibility of universally winning trading strategies – those generating strict profit across all market trajectories – through three distinct mathematical paradigms. Fundamentally, under standard admissibility constraints, the existence of such a strategy is a strict subset o

This paper bridges reinforcement learning (RL) and risk-sensitive stochastic control by introducing a tractable exploration mechanism for policy search in risk-sensitive portfolio management, with known and unknown model parameters, that yields an endogenous relative-entropy regularization. We const

Recent advances in continuous-time optimal stopping have been driven by entropy-regularized formulations of randomized stopping problems, with most existing approaches relying on partial differential equation methods. In this paper, we propose a fully probabilistic framework based on the Doob-Meyer-

Horizon risk (see arXiv:2301.04971) is studied in the context of cash non-additive fully-dynamic risk measures induced by BSDEs. Furthermore, we introduce a risk measure based on generalized Tsallis entropy which can dynamically evaluate the riskiness of losses considering both horizon risk and inte

In this paper, we present analytical proof demonstrating that the Sandwiched Volterra Volatility (SVV) model is able to reproduce the power-law behavior of the at-the-money implied volatility skew, provided the correct choice of the Volterra kernel. To obtain this result, we assess the second-order

We present analytical results for the Gini coefficient of economic inequality under the dynamics of a modified Yard-Sale Model of kinetic asset exchange. A variant of the Yard-Sale Model is introduced by modifying the underlying binary transaction of the classical system. It is shown that the Gini c

We discuss our simulation tool, fintech-kMC, which is designed to generate synthetic data for machine learning model development and testing. fintech-kMC is an agent-based model driven by a kinetic Monte Carlo (a.k.a. continuous time Monte Carlo) engine which simulates the behaviour of customers usi

We introduce a mean-reverting stochastic differential equation with a three-component stochastic term and show that it generates a hierarchy of steady-state (stationary) distributions. At the top level, the hierarchy is described by a modified-Beta distribution, while one- and two-parameter reductio

Let (P) be a fixed physical law and let (Q) be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps (Q) to (T_LQ), where (d(T_LQ)=L,dQ) and (L) is the terminal likelihood ratio. The forward relative-entr

Robust portfolio rules that reconstruct confidence sets after learning need not preserve the evaluator obtained by prior-by-prior Bayesian transport. In the Gaussian model, this discrepancy is summarized by natural-coordinate displacement: inherited transport preserves it whereas fresh reconstructio

Transient impact models compose a nonlinearity with a memory kernel, and the order of composition determines the criterion for absence of price manipulation. We classify both orders. If an arbitrary instantaneous law $f$ acts on the trading rate before any nonzero integrable Volterra kernel, nonnega

The analysis of financial markets using models inspired by statistical physics offers a fruitful approach to understand collective and extreme phenomena [“3, 14, 15”] In this paper, we present a study based on a 2D Ising network model where each spin represents an agent that interacts only with its

This paper develops a comprehensive theoretical framework that imports concepts from stochastic thermodynamics to model price impact and characterize the feasibility of round-trip arbitrage in financial markets. A trading cycle is treated as a non-equilibrium thermodynamic process, where price impac

We develop a theoretical framework that aims to link micro-level option hedging and stock-specific factor exposure with macro-level market turbulence and explain endogenous volatility amplification during gamma-squeeze events. By explicitly modeling market-maker delta-neutral hedging and incorporati

We introduce Entropy-Guided Multiplicative Updates (EGMU), a convex optimization framework for constructing multi-factor target-exposure portfolios by minimizing Kullback-Leibler divergence from a benchmark under linear factor constraints. We establish feasibility and uniqueness of strictly positive

Volatility, as a primary indicator of financial risk, forms the foundation of classical frameworks such as Markowitz’s Portfolio Theory and the Efficient Market Hypothesis (EMH). However, its conventional use rests on assumptions-most notably, the Markovian nature of price dynamics-that often fail t

Models for spin systems, known from statistical physics, are applied analogously in econometrics in the form of agent-based models. The models discussed in the econophysics literature all use the state variable $T$, which, in physics, represents the temperature of a system. However, there is little

In [“Han & Schied, 2023, \textit{“arXiv 2307.02582”}”], an easily computable scale-invariant estimator $\widehat{"\mathscr{R"}}^s_n$ was constructed to estimate the Hurst parameter of the drifted fractional Brownian motion $X$ from its antiderivative. This paper extends this convergence result by pr

The stock market closely monitors macroeconomic policy announcements, such as annual budget events, due to their substantial influence on various economic participants. These events tend to impact the stock markets initially before affecting the real sector. Our study aims to analyze the effects of

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N ,/, M,$, $m ,/, M$ as well as $n ,/, N$ converge to fixed ratios. Our method makes

Traditional Markowitz portfolio optimization constrains daily portfolio variance to a target value, optimising returns, Sharpe or variance within this constraint. However, this approach overlooks the relationship between variance at different time scales, typically described by $σ(Δt) \propto (Δt)^{

Conformal field theories with central charge $c\le1$ on random surfaces have been extensively studied in the past. Here, this discussion is extended from their equilibrium distribution to their critical dynamics. This is motivated by the conjecture that these models describe the time evolution of ce

This paper compares and contrasts stationarity between the conventional stock market and cryptocurrency. The dataset used for the analysis is the intraday price indices of the S&P500 from 1996 to 2023 and the intraday Bitcoin indices from 2019 to 2023, both in USD. We adopt the definition of `wide s

Debt recycling is an aggressive equity extraction strategy that potentially permits faster repayment of a mortgage. While equity progressively builds up as the mortgage is repaid monthly, mortgage holders may obtain another loan they could use to invest on a risky asset. The wealth produced by a suc

Investors and regulators can greatly benefit from a realistic market simulator that enables them to anticipate the consequences of their decisions in real markets. However, traditional rule-based market simulators often fall short in accurately capturing the dynamic behavior of market participants,

We present a formal framework for the aggregation of financial markets mediated by arbitrage. Our main tool is to characterize markets via utility functions and to employ a one-to-one correspondence to limit order book states. Inspired by the theory of thermodynamics, we argue that the arbitrage-med

The open availability of the entire history of the Bitcoin transactions opens up the possibility to study this system at an unprecedented level of detail. This contribution is devoted to the analysis of the mesoscale structural properties of the Bitcoin User Network (BUN), across its entire history

In this paper we study the complexity in the information traffic that occurs in the peruvian financial market, using the Shannon entropy. Different series of prices of shares traded on the Lima stock exchange are used to reconstruct the unknown dynamics. We present numerical simulations on the recon

The structure of many financial networks is protected by privacy and has to be inferred from aggregate observables. Here we consider one of the most successful network reconstruction methods, producing random graphs with desired link density and where the observed constraints (related to the market

The optimal portfolio size for a venture capital (VC) fund is a topic often debated, but there is no consensus on the best strategy. This is because it is a function of many factors. It is not easy to find a general formula that can be applied to all situations, and it largely depends on the goal of

The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the $q$-distribution family, which is characterized by

We associate backward and forward Kolmogorov equations to a class of fully nonlinear Stochastic Volterra Equations (SVEs) with convolution kernels $K$ that are singular at the origin. Working on a carefully chosen Hilbert space $\mathcal{H}_1$, we rigorously establish a link between solutions of SVE

This paper investigates an optimal consumption-investment problem featuring recursive utility via Tsallis relative entropy. We establish a fundamental connection between this optimization problem and a quadratic backward stochastic differential equation (BSDE), demonstrating that the value function

We consider a model for systemic risk comprising of a system of diffusion processes, interacting through their empirical mean. Each process is subject to a confining double-well potential with some uncertainty in the coefficients, corresponding to fluctuations in height of the potential barrier sepe

We introduce a family of particle systems on sparse graphs where local interactions occur via hitting times, providing a dynamic and tractable model for default cascades in large sparsely-connected financial networks. Building on the framework of Lacker, Ramanan and Wu (2023), we extend convergence

We start with the idea that open quantum systems can be used to represent financial markets by modelling events from the external environment and their impact on the market price. We show how to characterize distinct orbits of the time evolution, and look at the development of the reduced density ma

In this paper, we develop the lower and upper bounds of worst-case distortion riskmetrics and weighted entropy for unimodal, and symmetric unimodal distributions when mean and variance information are available. We also consider the sharp upper bounds of distortion riskmetrics and weighted entropy f

We consider a tick-by-tick model of price formation, in which buy and sell orders are modeled as self-exciting point processes (Hawkes process), similar to the one in [Bacry, Delattre, Hoffmann, Muzy, Modelling microstructure noise with mutually exciting point processes, Quantitative Finance, 2013]

We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overla

Quantum machine learning has demonstrated significant potential in solving practical problems, particularly in statistics-focused areas such as data science and finance. However, challenges remain in preparing and learning statistical models on a quantum processor due to issues with trainability and

This paper provides a general method to translate a standard economic model with a large number of agents into a field-formalism model. This formalism preserves the system’s interactions and microeconomic features at the individual level but reveals the emergence of collective states.We apply this m

We have shown, in a series of articles, that a classical description of a large number of economic agents can be replaced by a statistical fields formalism. To better understand the accumulation and allocation of capital among different sectors, the present paper applies this statistical fields desc

We study a reinsurer who faces multiple sources of model uncertainty. The reinsurer offers contracts to $n$ insurers whose claims follow compound Poisson processes representing both idiosyncratic and systemic sources of loss. As the reinsurer is uncertain about the insurers’ claim severity distribut

Kyle model in continuous time where the insider may be subject to legal penalties is considered. In equilibrium the insider internalises this legal risk by trading less aggressively. The equilibrium is characterised via the solution of a backward stochastic differential equation (BSDE) whose termina

Stochastic resetting has attracted significant attention in recent years due to its wide-ranging applications across physics, biology, and search processes. In most existing studies, however, resetting events are governed by an external timer and remain decoupled from the system’s intrinsic dynamics

We study the nonlinear chaotic dynamics in a system of linear oscillators coupled by social network links with an additional stratification of oscillator energies, or frequencies, and supplementary nonlinear interactions. It is argued that this system can be viewed as a model of social stratificatio