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preprint2021arXiv

Group Quantization of Quadratic Hamiltonians in Finance

The Group Quantization formalism is a scheme for constructing a functional space that is an irreducible infinite dimensional representation of the Lie algebra belonging to a dynamical symmetry group. We apply this formalism to the construction of functional space and operators for quadratic potentials -- gaussian pricing kernels in finance. We describe the Black-Scholes theory, the Ho-Lee interest rate model and the Euclidean repulsive and attractive oscillators. The symmetry group used in this work has the structure of a principal bundle with base (dynamical) group a semi-direct extension of the Heisenberg-Weyl group by SL(2,R), and structure group (fiber) the positive real line.

preprint2026arXiv

Learning Time-Inhomogeneous Markov Dynamics in Financial Time Series via Neural Parameterization

Modeling the dynamics of non-stationary stochastic systems requires balancing the representational power of deep learning with the mathematical transparency of classical models. While classical Markov transition operators provide explicit, theoretically grounded rules for system evolution, their empirical estimation collapses due to severe data sparsity when applied to high-resolution, high-noise environments. We explore this statistical barrier using financial time series as a canonical, real-world testbed. To overcome the degeneracy of empirical counting, we introduce a framework that utilizes neural networks strictly as parameterization engines to generate explicit, time-varying Markov transition matrices. By constraining the neural network to output its predictions as a formal stochastic operator, we maintain complete structural interpretability. We demonstrate that these learned operators successfully capture complex regime shifts: the state-conditioned model achieves mean row heterogeneity $\barρ = 0.0073$ while the state-free ablation collapses to exactly zero, and operator row entropy correlates with realized variance at $r = -0.62$ ($p \approx 10^{-251}$), revealing that high-volatility regimes homogenize transition dynamics rather than diversify them. Furthermore, rather than enforcing the Chapman-Kolmogorov equations as a rigid structural requirement, we repurpose them as a localized diagnostic tool to pinpoint specific temporal windows where first-order memory assumptions break down. Ultimately, this framework demonstrates how neural networks can be constrained to make rigorous, classical operator analysis viable for complex real-world time series.

preprint2026arXiv

Enhancing a Risk Model by Adding Transient Statistical Factors

Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

preprint2026arXiv

Yield Curves Dynamics Using Variational Autoencoders Under No-arbitrage

This paper introduces a physics-informed generative framework that resolves the fundamental conflict between the statistical flexibility of deep learning and the rigorous theoretical constraints of fixed-income modeling. We demonstrate that standard generative models and unconstrained statistical extrapolations suffer from "manifold collapse" and severe arbitrage violations when forecasting term structures across diverse macroeconomic regimes. To overcome this, we propose a two-stage architecture. First, a Student-t Conditional Variational Autoencoder with Dynamic Level Injection (CVAEsT+LS) extracts a robust, heavy-tailed term structure manifold, effectively decoupling macroeconomic shape dynamics from absolute base rates. Second, the latent dynamic evolution is governed by a continuous-time Neural Stochastic Differential Equation (SDE) strictly penalized by a No-Arbitrage Partial Differential Equation (PDE). Empirical results across multiple sovereign currencies (USD, GBP, JPY) confirm that our synergistic approach drastically reduces out-of-sample forecasting errors -- achieving an exceptional 6.58 bps Mean Tenor RMSE -- and successfully overcomes the massive parallel drift and zero-lower-bound violations exhibited by the classical HJM model in extreme environments. Furthermore, through phase space vector field analysis, we demonstrate the model's superior capability in unsupervised macroeconomic regime detection and high-quality continuous-time scenario generation. Ultimately, this research provides a highly scalable, mathematically sound evolutionary engine for term structure modeling.

preprint2026arXiv

SURGE: Approximation-free Training Free Particle Filter for Diffusion Surrogate

Diffusion-based generative models increasingly rely on inference-time guidance, adding a drift term or reweighting mixture of experts, to improve sample quality on task-specific objectives. However, most existing techniques require repeated score or gradient evaluations, introducing bias, high computational overhead, or both. We introduce \texttt{URGE}, Unbiased Resampling via Girsanov Estimation, a derivative-free inference-time scaling algorithm that performs path-wise importance reweighting via a Girsanov change of measure. Instead of computing gradient-based particle weights in previous work, \texttt{URGE} attaches a simple multiplicative weight to each simulated trajectory and periodically resamples. No score, no Hessian, and no PDE evaluation is required. We establish an equivalence between path-wise and particle-wise SMC: the Girsanov path weight admits a backward conditional expectation that recovers the previous particle-level weights, guaranteeing that both schemes produce the same unbiased terminal law. Empirically, \texttt{URGE} outperforms existing inference-time guidance baselines on synthetic tests and diffusion-model benchmarks, achieving better generation quality, while being significantly simpler to implement and fully gradient-free.

preprint2026arXiv

SNAPO: Smooth Neural Adjoint Policy Optimization for Optimal Control via Differentiable Simulation

Many real-world problems require sequential decisions under uncertainty: when to inject or withdraw gas from storage, how to rebalance a pension portfolio each month, what temperature profile to run through a pharmaceutical reactor chain. Dynamic programming solves small instances exactly but scales exponentially in state dimensions. Black-box reinforcement learning handles high-dimensional states but trains slowly and produces no sensitivities. We introduce SNAPO (Smooth Neural Adjoint Policy Optimization), a framework that embeds a neural policy inside a known, differentiable simulator, replaces hard constraints with smooth approximations, and computes exact gradients of the objective with respect to all policy parameters and all inputs in a single adjoint pass. We demonstrate SNAPO on three domains: natural gas storage (training in under a minute, 365 forward curve sensitivities at no additional cost per sensitivity), pension fund asset-liability management (6.5x-200x sensitivity speedup over bump-and-revalue, scaling with the number of risk factors), and pharmaceutical manufacturing (cross-unit sensitivities through a 4-unit process chain, with 20 ICH Q8 regulatory sensitivities from 5 adjoint passes in 74.5 milliseconds). All sensitivities are produced by the same backward pass that trains the policy, at a cost proportional to one reverse pass regardless of how many sensitivities are computed.

preprint2024arXiv

An arbitrage driven price dynamics of Automated Market Makers in the presence of fees

We present a model for price dynamics in the Automated Market Makers (AMM) setting. Within this framework, we propose a reference market price following a geometric Brownian motion. The AMM price is constrained by upper and lower bounds, determined by constant multiplications of the reference price. Through the utilization of local times and excursion-theoretic approaches, we derive several analytical results, including its time-changed representation and limiting behavior.

preprint2022arXiv

Utility Indifference Pricing with High Risk Aversion and Small Linear Price Impact

We consider the Bachelier model with linear price impact. Exponential utility indifference prices are studied for vanilla European options and we compute their non-trivial scaling limit for a vanishing price impact which is inversely proportional to the risk aversion. Moreover, we find explicitly a family of portfolios which are asymptotically optimal.

preprint2022arXiv

An optional decomposition of $\mathscr{Y}^{g,ξ}-submartingales$ and applications to the hedging of American options in incomplete markets

In the recent paper \cite{DESZ}, the notion of $\mathscr{Y}^{g,ξ}$-submartingale processes has been introduced. Within a jump-diffusion model, we prove here that a process $X$ which satisfies the simultaneous $\mathscr{Y}^{\mathbb{Q},g,ξ}$ -submartingale property under a suitable family of equivalent probability measures $\mathbb{Q}$, admits a \textit{nonlinear optional decomposition}. This is an analogous result to the well known optional decomposition of simultaneous (classical and $\mathscr{E}^g$-)supermartingales. We then apply this decomposition to the super-hedging problem of an American option in a jump-diffusion model, from the buyer's point of view. We obtain an \textit{infinitesimal characterization} of the buyer's superhedging price, this result being completely new in the literature. Indeed, it is well known that the seller's superheding price of an American option admits an infinitesimal representation in terms of the \textit{minimal supersolution of a constrained reflected BSDE}. To the best of our knowledge, no analogous result has been established for the buyer of the American option in an incomplete market. Our results fill this gap, and show that the buyer's super-hedging price admits an infinitesimal charcaterization in terms of the \textit{maximal subsolution of a constrained reflected BSDE}.

preprint2024arXiv

Exploratory Control with Tsallis Entropy for Latent Factor Models

We study optimal control in models with latent factors where the agent controls the distribution over actions, rather than actions themselves, in both discrete and continuous time. To encourage exploration of the state space, we reward exploration with Tsallis Entropy and derive the optimal distribution over states - which we prove is $q$-Gaussian distributed with location characterized through the solution of an FBS$Δ$E and FBSDE in discrete and continuous time, respectively. We discuss the relation between the solutions of the optimal exploration problems and the standard dynamic optimal control solution. Finally, we develop the optimal policy in a model-agnostic setting along the lines of soft $Q$-learning. The approach may be applied in, e.g., developing more robust statistical arbitrage trading strategies.

preprint2023arXiv

Acceptable Bilateral Gamma Parameters

The purpose of this paper is to utilize statistical methodologies to infer from market prices of assets and their derivatives the magnitude of the set of a measure M that defines acceptance sets of risky future cash flows. Specifically, we estimate upper and lower boundaries of the compensation needed for a given bilateral gamma distributed future cash flow to be acceptable. We show that prospects theory provides a natural interpretation of the behaviors implied by such boundaries, which are not compatible with expected utility theory over terminal wealth. Boundaries for bilateral gamma risk neutral scale parameters for given speed parameters are also estimated and tested against market data and, in particular, comparisons are made with known empirical facts about the magnitude of the acceptance set of a common class of risk measures.

preprint2023arXiv

All AMMs are CFMMs. All DeFi markets have invariants. A DeFi market is arbitrage-free if and only if it has an increasing invariant

In a universal framework that expresses any market system in terms of state transition rules, we prove that every DeFi market system has an invariant function and is thus by definition a CFMM; indeed, all automated market makers (AMMs) are CFMMs. Invariants connect directly to arbitrage and to completeness, according to two fundamental equivalences. First, a DeFi market system is, we prove, arbitrage-free if and only if it has a strictly increasing invariant, where arbitrage-free means that no state can be transformed into a dominated state by any sequence of transactions. Second, the invariant is, we prove, unique if and only if the market system is complete, meaning that it allows transitions between all pairs of states in the state space, in at least one direction. Thus a necessary and sufficient condition for no-arbitrage (respectively, for completeness) is the existence of the increasing (respectively, the uniqueness of the) invariant, which, therefore, fulfills in nonlinear DeFi theory the foundational role parallel to the existence (respectively, uniqueness) of the pricing measure in the Fundamental Theorem of Asset Pricing for linear markets. Moreover, a market system is recoverable by its invariant if and only if it is complete; and in all cases, complete or incomplete, every market system has, and is recoverable by, a multi-invariant. A market system is arbitrage-free if and only if its multi-invariant is increasing. Our examples illustrate (non)existence of various specific types of arbitrage in the context of various specific types of market systems -- with or without fees, with or without liquidity operations, and with or without coordination among multiple pools -- but the fundamental theorems have full generality, applicable to any DeFi market system and to any notion of arbitrage expressible as a strict partial order.

preprint2023arXiv

Walraswap: a solution to uniform price batch auctions

Consider a finite set of trade orders and automated market makers (AMMs) at some state. We propose a solution to the problem of finding an equilibrium price vector to execute all the orders jointly with corresponding optimal AMMs swaps. The solution is based on Brouwer's fixed-point theorem. We discuss computational aspects relevant for realistic situations in public blockchain activity.

preprint2023arXiv

Epstein-Zin Utility Maximization on a Random Horizon

This paper solves the consumption-investment problem under Epstein-Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize the optimal consumption and investment strategies using backward stochastic differential equations with superlinear growth on unbounded random horizons. This characterization, compared with the classical fixed-horizon result, involves an additional stochastic process that serves to capture the randomness of the horizon. As demonstrated in two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies.

preprint2023arXiv

A Càdlàg Rough Path Foundation for Robust Finance

Using rough path theory, we provide a pathwise foundation for stochastic Itô integration, which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called Property (RIE) for càdlàg paths, which is shown to imply the existence of a càdlàg rough path and of quadratic variation in the sense of Föllmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type, and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Cover's universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.

preprint2022arXiv

Delta family approach for the stochastic control problems of utility maximization

In this paper, we propose a new approach for stochastic control problems arising from utility maximization. The main idea is to directly start from the dynamical programming equation and compute the conditional expectation using a novel representation of the conditional density function through the Dirac Delta function and the corresponding series representation. We obtain an explicit series representation of the value function, whose coefficients are expressed through integration of the value function at a later time point against a chosen basis function. Thus we are able to set up a recursive integration time-stepping scheme to compute the optimal value function given the known terminal condition, e.g. utility function. Due to tensor decomposition property of the Dirac Delta function in high dimensions, it is straightforward to extend our approach to solving high-dimensional stochastic control problems. The backward recursive nature of the method also allows for solving stochastic control and stopping problems, i.e. mixed control problems. We illustrate the method through solving some two-dimensional stochastic control (and stopping) problems, including the case under the classical and rough Heston stochastic volatility models, and stochastic local volatility models such as the stochastic alpha beta rho (SABR) model.

preprint2023arXiv

Fixed-point iterative algorithm for SVI model

The stochastic volatility inspired (SVI) model is widely used to fit the implied variance smile. Presently, most optimizer algorithms for the SVI model have a strong dependence on the input starting point. In this study, we develop an efficient iterative algorithm for the SVI model based on a fixed-point and least-square optimizer. Furthermore, we present the convergence results in certain situations for this novel iterative algorithm. Compared with the quasi-explicit SVI method, we demonstrate the advantages of the fixed-point iterative algorithm using simulation and market data.

preprint2022arXiv

Choquet regularization for reinforcement learning

We propose \emph{Choquet regularizers} to measure and manage the level of exploration for reinforcement learning (RL), and reformulate the continuous-time entropy-regularized RL problem of Wang et al. (2020, JMLR, 21(198)) in which we replace the differential entropy used for regularization with a Choquet regularizer. We derive the Hamilton--Jacobi--Bellman equation of the problem, and solve it explicitly in the linear--quadratic (LQ) case via maximizing statically a mean--variance constrained Choquet regularizer. Under the LQ setting, we derive explicit optimal distributions for several specific Choquet regularizers, and conversely identify the Choquet regularizers that generate a number of broadly used exploratory samplers such as $ε$-greedy, exponential, uniform and Gaussian.

preprint2023arXiv

Lévy bandits under Poissonian decision times

We consider a version of the continuous-time multi-armed bandit problem where decision opportunities arrive at Poisson arrival times, and study its Gittins index policy. When driven by spectrally one-sided Lévy processes, the Gittins index can be written explicitly in terms of the scale function, and is shown to converge to that in the classical Lévy bandit of Kaspi and Mandelbaum (1995).

preprint2023arXiv

Efficient Pricing and Hedging of High Dimensional American Options Using Recurrent Networks

We propose a deep Recurrent neural network (RNN) framework for computing prices and deltas of American options in high dimensions. Our proposed framework uses two deep RNNs, where one network learns the price and the other learns the delta of the option for each timestep. Our proposed framework yields prices and deltas for the entire spacetime, not only at a given point (e.g. t = 0). The computational cost of the proposed approach is linear in time, which improves on the quadratic time seen for feedforward networks that price American options. The computational memory cost of our method is constant in memory, which is an improvement over the linear memory costs seen in feedforward networks. Our numerical simulations demonstrate these contributions, and show that the proposed deep RNN framework is computationally more efficient than traditional feedforward neural network frameworks in time and memory.

preprint2023arXiv

European baskets in discrete-time continuous-binomial market models

We consider a discrete-time incomplete multi-asset market model with continuous price jumps. For a wide class of contingent claims, including European basket call options, we compute the bounds of the interval containing the no-arbitrage prices. We prove that the lower bound coincides, in fact, with Jensen's bound. The upper bound can be computed by restricting to a binomial model for which an explicit expression for the bound is known by an earlier work of the authors. We describe explicitly a maximal hedging strategy which is the best possible in the sense that its value is equal to the upper bound of the price interval of the claim. Our results show that for any $c$ in the interval of the non-arbitrage contingent claim price at time $0$, one can change the boundaries of the price jumps to obtain a model in which $c$ is the upper bound at time $0$ of this interval. The lower bound of this interval remains unaffected.

preprint2022arXiv

Analysis of stock index with a generalized BN-S model: an approach based on machine learning and fuzzy parameters

In this paper we implement a combination of data-science and fuzzy theory to improve the classical Barndorff-Nielsen and Shephard model, and implement this to analyze the S&P 500 index. We pre-process the index data based on fuzzy theory. After that, S&P 500 stock index data for the past ten years are analyzed, and a deterministic parameter is extracted using various machine and deep learning methods. The results show that the new model, where fuzzy parameters are incorporated, can incorporate the long-term dependence in the classical Barndorff-Nielsen and Shephard model. The modification is based on only a few changes compared to the classical model. At the same time, the resulting analysis effectively captures the stochastic dynamics of the stock index time series.

preprint2022arXiv

Robust replication of barrier-style claims on price and volatility

We show how to price and replicate a variety of barrier-style claims written on the $\log$ price $X$ and quadratic variation $\langle X \rangle$ of a risky asset. Our framework assumes no arbitrage, frictionless markets and zero interest rates. We model the risky asset as a strictly positive continuous semimartingale with an independent volatility process. The volatility process may exhibit jumps and may be non-Markovian. As hedging instruments, we use only the underlying risky asset, zero-coupon bonds, and European calls and puts with the same maturity as the barrier-style claim. We consider knock-in, knock-out and rebate claims in single and double barrier varieties.

preprint2024arXiv

Quantitative Fundamental Theorem of Asset Pricing

In this paper we provide a quantitative analysis to the concept of arbitrage, that allows to deal with model uncertainty without imposing the no-arbitrage condition. In markets that admit ``small arbitrage", we can still make sense of the problems of pricing and hedging. The pricing measures here will be such that asset price processes are close to being martingales, and the hedging strategies will need to cover some additional cost. We show a quantitative version of the Fundamental Theorem of Asset Pricing and of the Super-Replication Theorem. Finally, we study robustness of the amount of arbitrage and existence of respective pricing measures, showing stability of these concepts with respect to a strong adapted Wasserstein distance.