Capped-usage SaaS products -- LLM subscriptions such as Claude Code and ChatGPT, cloud platforms such as Vercel and Cloudflare Workers, corporate benefit platforms, identity-verification services with liability transfer -- share a structural signature with insurance products: a fixed premium decoupled from realized consumption, stochastic per-user demand with heavy-tailed severity, a non-fungible cap that resets on a fixed schedule, and a portfolio-level exposure that requires reserve adequacy under tail risk. We argue that this is not an analogy. It is the same operational problem actuarial science has been tooled for decades to address, restated with new dependent variables (tokens, bandwidth bytes, function-invocations, gym check-ins) in place of medical claims. This paper proposes a modeling framework for capped-usage SaaS pricing built from frequency-severity decomposition, premium calculation principles, and Monte Carlo reserve adequacy. We map the framework to publicly observable subscription tiers in two domains (LLM services and cloud platforms), ground it in canonical health-insurance economics (Arrow 1963; Pauly 1968; Manning et al. 1987; Brot-Goldberg et al. 2017), and demonstrate divergence from traditional unit economics through a worked example. The contribution is operational rather than theoretical: not a new theorem, but vocabulary and tools currently absent from cs.LG/stat.ML practice.
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Artificial Intelligence in Healthcare and Education
We study the Banach dual of the one-parameter stochastic integral δ_L(u) = ∫₀^T u_t dL_t for a symmetric γ-stable Lévy process with γ ∈ (1,2). The natural integrand exponent is p ∈ (1,γ): the small-jump integrability ∫|z|^p ν_γ(dz) < ∞ holds iff p < γ, so this is not an arbitrary L^p but the unique scale dictated by the singularity of the Lévy measure at the origin. On this scale, the operator-covariant derivative D_L := δ_L^* : L^q(Ω) → H_L^* is the Banach dual of the one-parameter integral. Since p < 2, the Riesz identification H_L^* ≅ H_L is unavailable, and the Banach setting is forced. The principal result is structural: D_L is strictly more restricted than the standard Malliavin add-a-point operator D_{t,z}F = F(ω + δ_(t,z)) − F(ω) on Poisson space, which is the dual of the full two-parameter compensated Poisson integral ∫∫ h(s,z) Ñ(ds,dz). By Lévy-Itô, the one-parameter integrand of δ_L has the special form h(s,z) = u(s) · z — linear in z — whereas full martingale representation on Lévy space uses general h(s,z). The representability obstruction quantifies the resulting gap precisely: centered functionals depending nonlinearly on jump sizes — canonically, the centered large-jump count #{|ΔL_s| > 1} − E[#{|ΔL_s| > 1}] — lie in ker(D_L) yet are detected by the standard add-a-point operator. The obstruction is a property of the one-parameter integral, not a feature of jump processes themselves. The factorization (Theorem A) holds on the closed proper subspace im(δ_L) ⊊ L^p_0(Ω) and characterizes precisely which functionals admit one-parameter representation. Theorem B (product rule with Leibniz defect) is a standalone duality identity: its proof uses only the definition of D_L, the Lévy-Itô formula, and Hölder's inequality, and it does not invoke (H3) or the factorization machinery. Theorem C — the strongest technical result — identifies ker(D_L) and the annihilator of im(δ_L) via L^q-L^p truncation in the jump variable, showing the annihilator is infinite-dimensional even within the first chaos. The framework has been formally verified in the Lean 4 proof assistant (2,439 lines, zero sorry, zero axioms) using Mathlib. To our knowledge, this is the first formalization of the operator-covariant derivative framework with its representability obstruction in any proof assistant. The formalization includes proved Poisson mean and variance identities, a constructed compound Poisson path, a compensated-integral interface with derived Banach-side consequences, a concrete first-chaos orthogonality model, and the full abstract theorem pipeline — all machine-checked from clearly isolated stochastic-analysis assumptions.
This deposit contains the Lean 4 formal verification companion (BanachLevyComplete.lean, 2,439 lines) for the paper "Operator Factorization Beyond Hilbert Spaces: Representability Obstructions, Leibniz Defects, and Chaos Characterizations for Stable Lévy Processes" by Ramiro Fontes. The file has zero sorry declarations and zero axiom declarations. It integrates three layers: Part 1 — Poisson infrastructure: The symmetric γ-stable Lévy measure density with proved symmetry and nonnegativity. The Poisson mean identity E[Poisson(λ)] = λ and variance identity Var(Poisson(λ)) = λ, proved as theorems via a recurrence lemma and HasSum assembly. A canonical Poisson random variable constructed on (ℕ, poissonMeasure(λT)) with its distribution proved by Measure.map_id. Stable measure moment computations and the Blumenthal–Getoor dichotomy. Quadratic defect sharpness for the variance swap payoff. Part 2 — Lévy–Itô framework: The Itô formula for compound Poisson processes proved as a finite telescoping sum via Finset.sum_range_sub. A compound Poisson path defined as a concrete function, proved to start at zero and to have the correct terminal value. The compensated Poisson integral constructed as an L² limit of compound Poisson finite sums, with linearity inherited from finite-sum linearity and centering derived via tendsto_nhds_unique. Truncation convergence, centering, the predictable module structure, and chaos orthogonality derived from the compensated-integral interface. The first Poisson chaos realized concretely on (ℕ, poissonMeasure) with orthogonality proved via tsum_mul_left. The L² Cauchy estimate for the ε → 0 approximation proved, with the M → ∞ direction documented as requiring Lp (not L²) convergence. Part 3 — Banach energy space framework: The operator-covariant derivative D constructed via mk_dual (not axiomatized). The fluctuation factorization (Theorem A), representability obstruction, product rule with jump defect (Theorem B), and chaos characterization (Theorem C) verified. The centered obstruction witness derived from primitive stable-noise data: evenness from absolute-jump structure, positive variance from λ > 0 and T > 0 via mul_pos, nonzero from positive variance, and the obstruction from representability_obstruction. The Hilbert bridge showing the Banach framework specializes when the jump defect vanishes. The remaining primitive inputs are concentrated in two places: the Banach-side Lp-convergence layer for the compensated integral as M → ∞, and a full bottom-up Poisson-random-measure realization. These are isolated as explicit structure fields rather than hidden proof gaps. Together with the companion OperatorDerivative.lean (5,184 lines, zero sorry, one axiom) for the Hilbert paper, this constitutes 7,623 lines of formally verified stochastic calculus. To our knowledge, the Poisson mean and variance identities, the first Poisson chaos orthogonality, and the compound Poisson Itô formula via finite telescoping are among the first such formalized results in Lean 4.
For a single event with finitely many mutually exclusive outcomes, the full Kelly problem is to maximize expected log wealth over nonnegative stakes together with an optional cash position. The optimal formula is classical, but the support-selection step is often presented via Lagrange multipliers. This note gives a shorter state-price derivation. A cash fraction $c$ acts as an implicit position in every outcome: in terminal-wealth terms, it is equivalent to a baseline stake $cq_i$ on outcome $i$, where $q_i$ is the state price. On any active support, explicit bets therefore only top up favorable outcomes from this baseline $cq_i$ to the optimal total stake $p_i$. This yields the formula $x_i = (p_i - c q_i)_+$, the threshold rule $p_i/q_i > c$, and, after sorting outcomes by $p_i/q_i$, a one-pass greedy algorithm for support selection. The result is standard in substance, but the implicit-position viewpoint gives a compact proof and a convenient way to remember the solution.
Abstract This paper considers option valuation under finite mixture models in a discrete-time economy. Specifically, the Esscher transform is employed to select a pricing kernel. Novel finite mixture models with negative-shifted Gamma and negative-shifted inverse Gaussian distributions are developed. A hybrid finite mixture model that allows different parametric forms for component distributions is introduced to incorporate model uncertainty. An empirical characteristic function estimation method is employed to estimate the finite mixture models. Closed-form pricing formulas for a European call option are obtained for some finite mixture models. Empirical examples using data on the Bitcoin-USD prices are provided to illustrate an application of the proposed models to value Bitcoin options.
Decentralized Federated Learning (DFL) enables collaborative model training without a central server, but it remains vulnerable to privacy leakage because shared model updates can expose sensitive information through inversion, reconstruction, and membership inference attacks. Differential Privacy (DP) provides formal safeguards, yet existing DP-enabled DFL methods operate as black-boxes that cannot track cumulative noise added across clients and rounds, forcing each participant to inject worst-case perturbations that severely degrade accuracy. We propose PrivateDFL, a new explainable and privacy-preserving framework that addresses this gap by combining a HyperDimensional Computing (HD) model with a transparent DP noise accountant tailored to decentralized learning. HD offers structured, noise-tolerant high-dimensional representations, while the accountant explicitly tracks cumulative perturbations so each client adds only the minimal incremental noise required to satisfy its (epsilon, delta) budget. This yields significantly tighter and more interpretable privacy-utility tradeoffs than prior DP-DFL approaches. Experiments on MNIST (image), ISOLET (speech), and UCI-HAR (wearable sensor) show that PrivateDFL consistently surpasses centralized DP-SGD and Renyi-DP Transformer and deep learning baselines under both IID and non-IID partitions, improving accuracy by up to 24.4% on MNIST, over 80% on ISOLET, and 14.7% on UCI-HAR, while reducing inference latency by up to 76 times and energy consumption by up to 36 times. These results position PrivateDFL as an efficient and trustworthy solution for privacy-sensitive pattern recognition applications such as healthcare, finance, human-activity monitoring, and industrial sensing. Future work will extend the accountant to adversarial participation, heterogeneous privacy budgets, and dynamic topologies.
Luiz Koodi Hotta, Carlos Trucíos, Pedro L. Valls Pereira, Mauricio Zevallos
Recent studies have suggested that more complex models than GARCH are better suited for forecasting cryptocurrency risk measures, such as Value-at-Risk and Expected Shortfall. Among these studies, some highlight the advantages of MSGARCH models over traditional GARCH models. While improvements over single-regime GARCH models have been observed by using MSGARCH, the literature has only focused on the MSGARCH specification proposed by Haas, Mittnik and Paolella (Journal of Financial Econometrics, 2004) overlooking several other well-established MSGARCH specification alternatives. In this paper, we illustrate that exploring alternative MSGARCH specifications can lead to improvements in risk measure performance, emphasizing the potential benefits of using several specifications.
This paper applies the Lévy-GJR-GARCH model to explore the empirical dynamics of Bitcoin, Ethereum, and Ripple. It highlights volatility clustering, pronounced skewness, and high kurtosis in cryptocurrency markets. The study finds that models integrating innovation distributions more accurately capture and explain the volatility processes and tail risks in these assets. Advanced models, especially those accounting for extreme tail-end and asymmetric jump effects, are better suited for adapting to market changes and providing precise risk indicators, effectively identifying potential losses.
As one of the most popular blockchain platforms supporting smart contracts, Ethereum has caught the interest of both investors and criminals. Differently from traditional financial scenarios, executing Know Your Customer verification on Ethereum is rather difficult due to the pseudonymous nature of the blockchain. Fortunately, as the transaction records stored in the Ethereum blockchain are publicly accessible, we can understand the behavior of accounts or detect illicit activities via transaction mining. Existing risk control techniques have primarily been developed from the perspectives of de-anonymizing address clustering and illicit account classification. However, these techniques cannot be used to ascertain the potential risks for all accounts and are limited by specific heuristic strategies or insufficient label information. These constraints motivate us to seek an effective rating method for quantifying the spread of risk in a transaction network. To the best of our knowledge, we are the first to address the problem of account risk rating on Ethereum by proposing a novel model called RiskProp, which includes a de-anonymous score to measure transaction anonymity and a network propagation mechanism to formulate the relationships between accounts and transactions. We demonstrate the effectiveness of RiskProp in overcoming the limitations of existing models by conducting experiments on real-world datasets from Ethereum. Through case studies on the detected high-risk accounts, we demonstrate that the risk assessment by RiskProp can be used to provide warnings for investors and protect them from possible financial losses, and the superior performance of risk score-based account classification experiments further verifies the effectiveness of our rating method.
Hansjörg Albrecher, Dina Finger, Pierre-Olivier Goffard
The resource-consuming mining of blocks on a blockchain equipped with a proof of work consensus protocol bears the risk of ruin, namely when the operational costs for the mining exceed the received rewards. In this paper we investigate to what extent it is of interest to join a mining pool that reduces the variance of the return of a miner for a specified cost for participation. Using methodology from ruin theory and risk sharing in insurance, we quantitatively study the effects of pooling in this context and derive several explicit formulas for quantities of interest. The results are illustrated in numerical examples for parameters of practical relevance.
Mining blocks on a blockchain equipped with a proof of work consensus protocol is well known to be resource consuming. A miner bears the operational cost, mainly electricity consumption and IT gear, of mining and is compensated by a capital gain when a block is discovered. This paper aims at quantifying the profitability of mining when the possible event of ruin is also considered. This is done by formulating a tractable stochastic model and using tools from applied probability and analysis, including the explicit solution of a certain type of advanced functional differential equation. The expected profit at a future time point is determined for the situation when the miner follows the protocol as well as when the miner withholds blocks. The obtained explicit expressions allow us to analyze the sensitivity with respect to the different model components and to identify conditions under which selfish mining is a strategic advantage.
Due to conclusion could not rely on only one test, in this study, we apply various approaches to verify the actuary of VaR model to find out whether VaR model, especially historical VaR and delta normal VaR model, can provide the accurate risk measurement results for cryptocurrencies risk, especially CRIX, BTC, ETH and XRP. We use Kupiec’s POF test, Independence Test - Christoffersen (1998) and Joint Test that widely use for backtesting VaR model. Performance test results for risk measurement by historical VaR provide a fairly accurate over delta normal VaR when we use Kupiec’s POF-test for the accuracy of VaR model. Christoffersen (1998) independence test, the exceptions (failures) of historical VaR and delta normal VaR model show independence exceptions in accordance with an only high confidence level of critical values (0.99). Otherwise, the low confidence level of critical values (0.90 and 0.95) appears dependence exceptions. For the Joint test, we combine POF-test and independence test because each model has different advantages and disadvantages. The results show that historical VaR model is suitable for measuring cryptocurrency risk over delta normal VaR only high confidence level of critical values.
Stake systems which issue stakes as well as coins are proposed. Two subadditive stake systems are studied: one is the radical stake system, the other is the logarithmic stake system. Securities of both systems are analysed.
Jeffrey Chu, Stephen Chan, Saralees Nadarajah, Joerg Osterrieder
With the exception of Bitcoin, there appears to be little or no literature on GARCH modelling of cryptocurrencies. This paper provides the first GARCH modelling of the seven most popular cryptocurrencies. Twelve GARCH models are fitted to each cryptocurrency, and their fits are assessed in terms of five criteria. Conclusions are drawn on the best fitting models, forecasts and acceptability of value at risk estimates.
We correct the double spend race analysis given in Nakamoto’s foundational Bitcoin article and find the exact closed-form formula for the probability of success of a double spend attack using the regularized incomplete beta function. We give the first proof of its exponential decay on the number of confirmations, often cited in the literature, and find an asymptotic formula. Larger number of confirmations are required compared to those given by Nakamoto. We also compute this probability conditional to the knowledge of the time of the confirmations. This provides a finer risk analysis than the classical one.