Papers

Stock price forecasting is a long-standing challenge in computational finance, driven by the inherent randomness of markets and complex temporal patterns. While recent deep-learning models have raised forecasting accuracy by jointly modeling inter-stock and temporal price dynamics, they conflate inter-stock relationships with intra-stock temporal dependencies and focus solely on the univariate objective of price movement. To address these limitations, we propose LiMT, a Hierarchical Multi-Task Learning framework that integrates liquidity-aware signals for stock price forecasting. LiMT employs a Market Regime Encoder (MRE) module that first extracts contemporaneous cross-stock dependencies, then models each stock's temporal dynamics, yielding a unified latent state. Building on this latent state, we introduce a Liquidity-Driven Learning (LDL) module, a mixture-of-experts architecture that features cross-task gating mechanisms to jointly predict price movement, volatility, and trading volume. We further design an Adaptive Portfolio Optimization (APO) mechanism that converts multi-task forecasts into executable portfolio weights under transaction-cost and liquidity constraints. Extensive experiments on the CSI300 and CSI500 benchmarks show that LiMT performs best among strong neural and tree-based baselines across the reported metrics. In realistic CSI300 backtests, APO improves annualized return from 3.99% to 10.01% and Sharpe ratio from 1.22 to 1.86 over equal weighting, showing that the multi-task forecasts translate into deployable portfolio gains.

In this paper, we study the diversification properties of convex combinations of iid infinitely divisible random variables. For Lévy processes with bounded variation sample paths, we characterize, in terms of subadditivity and concavity of the transformed Lévy tails, Lévy processes that exhibit the non-diversification phenomenon or the reverse diversification order with respect to the majorization order uniformly over all time horizons. For general symmetric Lévy processes without a Gaussian component, we show that the symmetric 1-stable Lévy process is the only nontrivial process exhibiting either phenomenon. We further investigate convex combinations of components of multivariate infinitely divisible distributions, allowing for dependent and heterogeneous components, and characterize the Lévy measures of multidimensional Lévy processes exhibiting the two adverse diversification phenomena uniformly over all time horizons. Explicit characterizations are obtained for the multidimensional symmetric Lévy processes, multidimensional $\alpha$-stable processes and multidimensional compound Poisson processes. Finally, we show that the non-diversification phenomenon extends beyond Lévy processes to running maxima and integrals of increasing convex functionals of Lévy processes, while both adverse diversification phenomena are preserved for Lévy-driven stochastic integrals with nonnegative deterministic kernels. Applications to ruin theory, storage processes and stochastic volatility are also discussed.

Lepto-regression is defined as the machine learning process of constructing a Regression Tree of a target feature on itself. It is a novel, model-free method potentially revealing information on important sample structure properties. But it is yet not clear what the informational content of lepto-variance is and how it is related to other well-known statistics of a sample. One significant finding is that 58% of the historical US stock return variability is 1-bit lepto-variance that can not be explained by any financial factor. The central question investigated in this paper is to use small normal N(0, 1) drawn samples to explore how the 1-bit sample lepto-variance and lepto-ratio relate to sample variance, skewness and excess kurtosis. Using a large sample simulation, the lepto ratio of a normal is found to converge to 36.3%. For smaller normally distributed simulated N(0, 1) samples, while lepto-variance itself is highly correlated to sample variance, lepto-variance as a fraction of total variance is highly correlated to excess kurtosis. Both lepto-variance and lepto-ratio are orthogonal to sample skew. Another finding is that while lepto-ratio is strongly correlated to lepto-variance it remains orthogonal to sample variance.

We study affine stochastic Volterra equations on the cone of symmetric positive semidefinite matrices. For scalar kernels acting entrywise on the matrix dynamics, we establish weak existence by exploiting stochastic invariance results for Volterra equations on convex domains and derive a conditional Fourier–Laplace transform formula characterized by matrix-valued Riccati–Volterra equations. As an application, we extend the Gibson–Schwartz commodity model by replacing its variance-covariance structure with a Volterra–Wishart process. The resulting model allows for memory in the variances and for stochastic instantaneous correlation, while retaining affine tractability. Its joint Fourier–Laplace transform admits an exponential-affine representation governed by a matrix Riccati–Volterra equation. While the existence theory considered here excludes kernels that are singular at the origin, shifted fractional kernels remain admissible and provide a tractable specification with power-law memory.

Standard valuation methods, including discounted cash flow, the income approach standard IDW S 1 of the Institute of Public Auditors in Germany, and market multiples, compress milestone probabilities, continuation options, and risk shifts into opaque aggregate parameters; none provides a structured protocol for decomposing AI integration into auditable option-level assumptions. We propose an industry-agnostic taxonomy separating AI Integrators from AI Providers. AI Integrators are further classified by their Integration Depth Level, ranging from no integration to AI at the core of the product or process. A milestone-gated real-options overlay decomposes milestone state value into five components, and an Analytic Hierarchy Process-based Success Readiness Index derives per-option probabilities from structured pairwise comparisons for scenario analysis. Applied to an AI-native energy software-as-a-service firm, the framework yields a coherent valuation band traceable to identifiable option-level assumptions. Risk concentrates in later-stage continuation options, matching the structural prediction for AI Providers. The protocol applies across the firm lifecycle, including mergers and acquisitions due diligence. The case is a single-firm demonstration of protocol coherence, not empirical validation; multi-case testing against realised post-exit valuations is left to future research.

The sooner we receive information, and the more accurate it is, the better planning decisions we can make. Every day, prediction markets let anyone bet on tomorrow's high temperature in cities around the world, creating a market-implied forecast built on dispersed information. We use the past five years of market data from the Kalshi exchange for seven American cities to extract, hour by hour, the market-implied forecast. We use this forecast as a measuring instrument to see how much information about the temperature the market makes public before the public forecasting system does. We race it against the leading American and European weather forecasts. In six of the seven cities we study, the market beats the most accurate single public forecast, the National Blend of Models (NBM). Aggregating every city-day, at the end of the market's first hour of trading it beats the best single public product by about 10 percent in root-mean-square error, and holds its lead through the day, overnight, and into the target day. Looking at how the forecasts move over time, we find the National Blend travels four times further toward the market between its postings than the market travels toward the NBM. The market does not react to new weather forecast updates; instead, the forecast slowly publishes information that the market had already shared publicly.

Financial time series evolve across multiple temporal resolutions, challenging forecasting systems to incorporate newly available information without repeatedly recomputing unchanged representations. We introduce HARN, a Hierarchical Associative Resonance Network for event-driven multi-timeframe forecasting. HARN maintains persistent representations across temporal levels and updates each level only when its corresponding completed bar becomes available. The architecture combines causal multi-scale temporal encoding, gated associative memory, cross-level resonance, and hierarchical evidence aggregation, with forecasting performed in basis-point space and reconstructed to the original price scale. We evaluate HARN on four assets spanning equity, foreign exchange, and commodity markets using multiple random seeds and component ablations. HARN achieves competitive reconstructed-price forecasting errors against single-timeframe PatchTST and TimeXer baselines, while ablations reveal the effects of removing individual components across assets and timeframes. A code-level audit further examines consistency between the implementation and the defined event-driven causal protocol. The results position HARN as a persistent multi-timeframe forecasting framework rather than evidence of universal predictive superiority.

We study optimal stopping under dynamic risk measures with simultaneous ambiguity in the probability model and the discount rate. We introduce a paired ambiguity framework combining Girsanov model uncertainty with cash subadditive risk evaluation and characterize the stopping value by an upper reflected backward stochastic differential equation (BSDE). We establish structural properties of the resulting stopping operator and study quadratic drivers associated with entropic risk measures, obtaining explicit stopping rules in several benchmark cases. We then develop a deep learning scheme for the reflected quadratic BSDE. The convergence analysis uses discrete reflection and truncation to reduce the quadratic problem to a globally Lipschitz system and combines reflected BSDE discretization estimates with neural network approximation errors. Numerical experiments for American options illustrate the effects of discount rate and entropic ambiguity on stopping values and exercise decisions.

Organizations invest heavily in internal mechanisms such as incentive systems, governance structures, performance measurement, analytics, and repeated reorganizations. Yet realized performance gains are often weak or short-lived, particularly in large and mature organizations. Standard explanations emphasize allocative inefficiency arising from incentive misalignment, information problems, or bounded rationality. While relevant, these explanations struggle to account for environments in which effort is high, optimization capabilities are sophisticated, and aggregate performance remains far below apparent potential. This paper develops a theory of structural inefficiency inside organizations. The central mechanism is the presence of internal externalities: unpriced cross-effects of organizational mechanisms acting simultaneously on multiple performance objectives. When such effects are mixed in sign, organizational effort is partially canceled internally, creating a structural ceiling on realized performance. The paper introduces a simple representation of organizational primitives and key performance indicators and derives a realized-performance law in which internal conflict enters as a multiplicative discount factor. Under weak and realistic conditions, structural inefficiency dominates allocative inefficiency, explaining why optimization often fails before organizational coherence is restored.

Financial language models can transform unstructured firm-specific news into structured decision signals, but financial AI research lacks an integrated deployment framework for evaluating whether those signals remain useful in financial decision systems. Computer science research has developed strong methods for time-series forecasting, text classification, multimodal stock prediction, graph-based market modeling, and machine-learning operations, yet these streams do not provide a domain-specific protocol that jointly tests financial language-model outputs under event-time observability, probability calibration, execution timing, transaction costs, liquidity constraints, capacity limits, operational diagnostics, and statistical inference. We introduce MFAST, a Market-Friction-Aware Sentiment-to-Trading framework that converts timestamped financial text into auditable, reproducible, and market-feasible trading decisions. The application is news-based trading, where firm-specific text must be linked to securities before portfolio decisions can be evaluated. The framework links Refinitiv News Analytics to Center for Research in Security Prices (CRSP) equity data, restricts the primary out-of-sample evaluation to post-release news outside disclosed foundation-model data-freshness periods, and adds a public replication arm using open financial text and public price data. Results show that decoder-only language models outperform encoder baselines and dictionary sentiment in classification, calibration, return prediction, and net portfolio performance, while operational diagnostics reveal trade-offs among accuracy, latency, memory, throughput, and inference cost. The paper shows that credible evaluation of financial language models requires an end-to-end engineering approach combining language understanding, temporal discipline, market-friction-aware deployment, and reproducible validation.

Intraday electricity markets enable participants to adjust energy positions close to delivery, with price forecasts necessary to support trading and the scheduling of flexible electricity resources as well as increasingly responsive consumers. Forecasting model specifications have progressed from using macro-features, such as renewable generation and load, to the micro-features of continuous orderbooks. A recent advanced deep learning model, OrderFusion, explicitly models micro-level buy-sell orderbook interactions. However, despite its superior comparative forecasting performance, it considers only one delivery product at a time, ignoring neighboring-product information. Moreover, when forecasting aggregated price indices such as the liquid German ID3, ID2, and ID1 products, it omits the information contained in price trajectories. In contrast, pretrained time-series foundation models have shown success in financial, renewable-energy, and day-ahead electricity price forecasting. However, their performance on intraday orderbook data remains an open research question of considerable practical importance. In this paper, we propose OrderFusion+, an open-source deep learning model that combines historical orders from the target and neighboring delivery products to forecast probabilistic buy-sell price trajectories. We benchmark OrderFusion+ against forecasting baselines and pretrained foundation models, and investigate dynamic market conditions through the designed dynamic masking mechanism, revealing insights into market efficiency. The implementation and forecasts can be found at: https://runyao-yu.com/OrderFusion/

We characterize the worst- and best-case values of a mean-variance functional over a 2-Wasserstein ball. Using quantile representations and the geometry of attainable means and standard deviations, we reduce both infinite-dimensional problems to scalar equations and construct the extremal laws as location-scale transformations of the reference distribution. Their values depend on the reference law only through its first two moments. We then derive dual representations and formulate proportional risk sharing under heterogeneous beliefs as a finite-dimensional optimization problem. Under homogeneous beliefs, we show that the classical proportional allocation remains optimal for every ambiguity radius and $\alpha$-maxmin weight. Finally, we study a coupled distortion-variance functional and characterize its worst-case quantile through a convex-envelope construction, allowing the extremal law to change in shape as well as location and scale.

Recursive differenced forecasting, the standard remedy for non-stationarity, predicts one-step changes and integrates them by cumulative summation. We show that this reconstruction is a discrete integrator with a pole on the unit circle, so the biased increment errors of a learned nonlinear model are summed without bound and the rollout diverges: at 336 steps its normalised MAE reaches 1.6-3.8 for every neural architecture tested, against 0.80 for a stable linear recursion. We then introduce leaky-integrator reconstruction, a training-free fix that moves the pole inside the unit circle with H(z) = 1/(1 - gamma z^-1), gamma < 1, bounding the accumulation of the model's own increment errors. Applied post hoc with a single fixed gamma=0.9 (no retraining, a two-line change to any deployed one-step or foundation-model forecaster), it beats the traditional recursive integrator at every horizon, with the mean gain over seven diverging architectures and twenty datasets growing from ~3% at H=24 to 23% at H=96, 37% at H=192 and 51% at H=336 (43-75% across those architectures; 78% with an oracle pole), bringing all of them to 0.87-0.97. Based on these extensive empirical experiments, adding a leaky integrator thus improves recursive differenced time-series forecasting.

Mitigating drawdown, the decline in wealth from its running peak, presents a canonical problem in path-dependent risk control. In this paper, we develop a finite-horizon control framework that enforces a prescribed maximum percentage drawdown limit in multi-asset stochastic systems. Our first result is an exact robust-invariance theorem characterizing every control action that preserves a prescribed drawdown limit against all supported returns. We show that every robustly safe control admits a drawdown-modulated form: the product of the current drawdown cushion and a feasible normalized direction. This yields a complete parameterization of robustly drawdown-safe policies. Additionally, under stagewise-independent returns, we show that optimizing over all robustly safe causal policies reduces to a one-dimensional Bellman recursion and yields an optimal robustly safe state-feedback policy. Finally, we characterize the linear time-invariant (LTI) gains satisfying a prescribed drawdown limit and prove that optimal drawdown modulation achieves no lower expected return under the same limit. Strict expected-return improvement holds for horizons of at least two stages whenever the LTI policy has positive expected one-stage net return.

Day-ahead electricity price forecasts support trading and storage decisions, but for battery arbitrage predicting intraday price spreads is more relevant than predicting individual hourly prices. Here we show that a temporal hierarchy forecasting (THieF) framework that jointly reconciles forecasts of hourly electricity prices and all intraday price spreads consistently improves performance across two major European electricity markets and three different forecasting architectures. Using five years of out-of-sample data from Germany and Spain, we obtain accuracy improvements of up to 19.7% and profit gains of up to 10.4% relative to unreconciled hourly price forecasts. The gains persist even for a highly accurate pretrained TabPFN foundation model. Our results demonstrate that exploiting coherent relationships between economically relevant forecasting targets can improve both predictive accuracy and decision value, and that better statistical forecasts do not necessarily imply better economic decisions.

Modeling the dynamics of option implied volatility surface (IVS) is crucial for pricing, hedging, and risk-managing option portfolios. We develop a universal conditional diffusion model that learns to jointly generate next-day IVS increments and the underlying stock's returns. The model is trained on pooled data from 50 stocks and evaluated on a test set comprising 50 in-sample stocks and 50 out-of-sample stocks excluded from training. The training objective is primarily the minimization of MSE, with the variants that jointly impose surface smoothness, and that penalize the presence of static-arbitrage, which are all economically meaningful constraints. Across in-sample and out-of-sample stocks, our diffusion model outperforms the VolGAN benchmark in reducing arbitrage violations, improving stock risk prediction, and aligning explained variance ratios by the first three principal components. The fact that the model extrapolates well to stocks that are excluded from training indicates that the learned dynamics can be shared across stocks. These findings support a universal conditional diffusion as a credible framework for multi-stock implied volatility surface scenario generation.

We study the design of priority pricing systems with heterogeneous agents in environments in which improving quality for some agents reduces the average quality that can be provided. Contrary to the equity-efficiency tradeoff emphasized in public debates, we show that under economically natural conditions priority pricing can Pareto-improve on an equal-allocation benchmark. Three priority tiers suffice for such an improvement, combining higher quality for a fee, lower quality with compensation, and an intermediate tier at the benchmark quality; two tiers are never enough. Our results provide a framework for overcoming equity-efficiency tensions in applications such as lane pricing, waiting-line design, public provision, and insurance.

Pairs trading exploits mean reversion in the relationship between related assets. We adapt this idea to political betting markets by modelling the combined implied probability of the two major-party nominees with a latent Ornstein-Uhlenbeck process whose mean-reversion level varies over time and whose observations contain additive noise. Model parameters are estimated from regularly sampled odds data using a state-space likelihood, with consecutive repeated values represented by a single retained observation and the elapsed number of sampling intervals preserved in the continuous-time transition. Parametric-bootstrap upper prediction bounds identify signal times at which the combined implied probability is likely to decline, and a no-intercept Bradley-Terry-type model selects the candidate-specific odds quote. The candidate-selection model is trained on 2020 U.S. presidential-election data and evaluated out of sample on 2024 data. The 2024 analysis produced 130 signals, empirical one-step coverage of 95.1%, a mean synthetic odds-price return of 1.86%, and an unannualized per-trade Sharpe-type ratio of 1.12. These returns are frictionless descriptive quantities rather than executable betting-exchange profits. The results support the integrated framework as a proof of concept for two-candidate electoral markets.

We examine log-optimal portfolio allocation when the long-run price of an asset follows a power-law trajectory, $P(t)=At^\alpha$, and its instantaneous return variance decays as $\sigma^2(t)=\sigma_0^2 t^{-2\gamma}$. Under a zero risk-free-rate benchmark, the continuous-time Kelly fraction scales as $K^\ast(t)=(\alpha/\sigma_0^2)t^{2\gamma-1}$. Exact temporal invariance therefore occurs when $\gamma=1/2$, whereas deviations from this value produce systematic age dependence in the allocation. We propose a scaling hypothesis connecting growth in network participation, effective market liquidity, and declining volatility. Under a specified set of scaling assumptions, this model predicts the benchmark exponent $\gamma=1/2$. Using historical daily Bitcoin prices, we estimate the power-law price exponent and examine the sensitivity of the volatility exponent to the length of the rolling window. For windows of four to nine years, the estimated volatility exponents have an arithmetic mean of 0.53 and a cross-window standard deviation of approximately 0.03. Because these estimates are obtained from overlapping observations and the same underlying price history, this spread is interpreted as a measure of model sensitivity rather than a formal confidence interval. Finally, we show that a time-dependent multiplicative contribution to return variance generally breaks exact Kelly invariance. We illustrate this result using a scenario in which Bitcoin transaction-fee variability affects the effective variance process. The results identify the conditions under which log-optimal allocation can remain stable under non-stationary power-law asset dynamics and clarify the assumptions required when applying this result to Bitcoin.

People often face environments where multiple models compete to explain the same observations. This paper examines how people update beliefs in such settings and how preferences over payoff-relevant states shape model selection and belief updating. This paper first develops a framework where preference-driven bias distorts the perceived model, affecting Bayesian and best-fit updating differently. In a laboratory experiment, most participants are classified as Bayesian updaters, who average across models, while a substantial minority are classified as best-fit updaters, who select the model that best fits the observed signal. Within-participant comparisons between the symmetric payoff and asymmetric payoff conditions indicate that asymmetric payoffs shift reported beliefs toward the preferred state, particularly among participants classified as best-fit updaters. Relative to symmetric payoffs, asymmetric payoffs increase the reported belief of the preferred state by about 8 percentage points among best-fit updaters, while the estimated effect among Bayesian updaters is close to zero. These findings help us better understand model-based learning and have implications for domains such as political polarization and financial investment, where competing narratives and strong preferences often coexist.

AI systems are becoming participants in the evaluation and use of scientific research. They encounter citation counts, download statistics and lists of popular articles developed around human readers, but the collective consequences of these signals for artificial readers remain uncertain. This paper adapts the Music Lab design to a market for academic attention. In the first experiment, 1,000 AI agents choose papers from the titles and abstracts of all 114 regular research articles published in the American Economic Review in 2025. The experiment has five independent-choice communities and five social-influence communities, each with 100 sequential agents. Only agents in the social-influence condition observe earlier selections within their community. Agents may select any number of papers. Social-information communities select 17.2 percent fewer papers per agent, concentrate their choices more heavily, and collectively cover 73 papers, compared with 90 independently. Between-community variation is greater under social information. In a second experiment with 200 agents across twenty social communities, randomly assigning papers five initial selections raises their subsequent selection rate by 45.55 percentage points (95% CI: 41.20 to 49.90). Choices have modest correspondence with external citations and little correspondence with download counts. The results show how a simple information rule shapes the volume, breadth and distribution of scientific attention in an artificial population.

Catastrophe (cat) bonds overcome some lack of reinsurance by sourcing capacity from the wider capital markets. We present a new type of cat bond addressing the known trade-off between moral hazard and basis risk. As our main contributions we propose a trigger mechanism which is entirely transparent and simpler to evaluate compared to indemnity modeling techniques, as well as a methodology to price this cat bond. This is relevant for insurers and public authorities in a world where natural disasters are occurring with increasing frequency and severity due to climate change, but also for players willing to enter the cat bond market for whom the lack of transparency of this asset class has been a significant obstacle. Our trigger is derived from a cost random field which separates the physical hazard, a vulnerability function and the exposure. This allows the trigger to take a flexible form between parametric and modeled loss, in case exposure is taken into account. We present a case study based on historical windstorm events impacting Germany. Using wind speed data from historical storms, we fit a max-stable random field on a resolution which is standard in the reinsurance industry. The availability of industry loss and exposure data allows us to calibrate the vulnerability component to historical observations. Besides measuring the basis risk associated with our trigger, we perform a full model assessment and discuss numerical results.

We study random signed Takagi–Landsberg bridges with independent Rademacher Faber–Schauder coefficients. At index $H=1/2$, we prove that every fixed deterministic refining sequence of partitions with vanishing mesh yields quadratic variation $t$, almost surely and uniformly in $t\in[0,1]$. For partitions that need not be refining, the mesh condition $o(1/\log n)$ is sufficient, and its order is sharp. Our proofs use an operator representation of the quadratic sums and moment bounds for Rademacher chaos. We also show that asymmetric signs can destroy this invariance. At every index $H\ne1/2$, we construct a deterministic refining sequence along which the critical $p$-power sums, with $p=1/H$, have two distinct finite accumulation points almost surely. At $H=1/4$, uniform thirds grids provide an explicit alternative to the dyadic limit. Thus $H=1/2$ is the unique index in the symmetric family at which critical variation is invariant along every fixed deterministic refining sequence. Nevertheless the variation index equals $1/H$ almost surely across a large sequence of partitions.

We study a representative-agent Epstein-Zin economy with geometric dividends and a hidden finite-state Markov drift. We allow the price-dividend ratio to contain an additional positive, absolutely continuous valuation factor and, within the class $\mathfrak C$ defined below and under the regularity, admissibility, and positivity conditions of our main theorem, equilibrium forces this factor to be constant, yielding belief-Markovian prices. In the two-state case, under the stated positivity condition and strictly positive transition intensities, we prove existence, uniqueness, endpoint smoothness, interior analyticity, and uniform bounds for the positive solution of the pricing equation. For $0<\theta\leq1$, this solution supports an equilibrium under any continuous short rate satisfying the model's one-sided portfolio condition. Finally, in the two-state subregion $\eta>0$, we derive belief-dependent stock volatility, a European option-pricing PDE, and a leading short-maturity conditional risk-neutral log-return skewness expansion.

We estimate housing externalities in a high-density city, exploiting the staggered rollout of Singapore's nationwide Main Upgrading Programme for public housing. Controlling for nonrandom neighborhood exposure, we find that upgrading raises treated buildings' prices by 11.5% upon completion and neighboring buildings' resale prices by about 2% within 500 meters, decaying to zero beyond. A model with distance-decaying externalities shows that in dense settings spillovers justify the distortions of in-kind provision; this advantage diminishes and reverses at lower densities. Administrative data on over 2 million residents show that upgrading disproportionately retains older incumbents, suggesting age-specific amenities as an underexplored externality channel.

People increasingly compete against AI agents rather than other human opponents. We distinguish two channels: an opponent effect and an information effect. These are different elements with different consequences: the opponent effect is specific to a given computational system, the information effect a property of the information environment that an organisation or policymaker can control. We separate them in a preregistered experiment (N = 1,395) using a dynamic all-pay auction, a repeated contest in which escalation of commitment arises from the incentives. What participants are told about the opponent (human, an AI trained to imitate people, or an AI trained to compete well) is varied and crossed with who they actually face, in a deception-free design. What people are told influences escalation: the median price rises by 6.7 points when a human might be the opponent and falls by 8.8 when an optimising machine might be, a spread of about 15% of the prize value of the competition, produced by information alone. Competing against the AI agents lowers prices, yet reduces the chance that both sides finish with positive earnings, showing distinct effects of the opponent channel. The information effect is not explained by articulated strategy, or individual differences, and is consistent with a competitive response engaged when a human is a live possibility. This shows that describing an AI competitor is not behaviourally neutral.

The standard Euler discretization schemes for numerical option pricing under stochastic volatility models are known to exhibit high biases and potential unreliability. The alternative use of the exact (unbiased) simulation approach invariably involves numerical evaluation of integrated variance (and / or volatility) conditional on terminal variance (volatility) value. To resolve the technical challenge, most simulation schemes either employ the tedious Fourier inversion of conditional characteristic function or numerical approximation by moment matched distribution. We propose a general framework of constructing efficient and reliable simulation schemes for stochastic volatility models via the Strang operator splitting approximation. The simulation procedure completely circumvents the necessity of evaluation of conditional integrated variance (and / or) volatility. Our simulation schemes compete favorably well with most existing exact simulation schemes and the biased Euler schemes in terms of accuracy, efficiency, reliability and ease of implementation. Extensive numerical tests were conducted to illustrate the versatility and success of our operator splitting approach for most common stochastic volatility models, such as the Heston-type models, lifted Heston model, Hull-White model, and Barndorff-Nielsen and Shephard model. We also establish the proof of second-order convergence of the operator splitting schemes.

We develop an efficient Monte Carlo simulation scheme for pricing options under the Ornstein-Uhlenbeck driven stochastic volatility model via the operator splitting approach. With an ingenious splitting of the governing stochastic differential equations, our operator splitting scheme admits analytic solutions in all sub-steps, so its implementation is simplified to require simulation of a few normal variates. This resolves the two typical numerical challenges in other simulation schemes, namely, sampling of conditional integrated variance and pathwise inverse integral transform of characteristic functions. There are three pioneering simulation schemes that attempt to overcome the above two numerical challenges. These include the Hilbert interpolation scheme of Zeng et al. (2023), Karhunen-Lo`eve expansion scheme of Choi (2025) and moment matching scheme based on the Inverse Gaussian distribution of Brignone and Sgarra (2026). We performed numerical tests to compare accuracy-speed performance of pricing options using our operator splitting scheme with these three pioneering schemes. We found that our scheme competes favorably well in terms of accuracy-speed tradeoff among all these schemes, in particular for pricing path dependent options with a large number of monitoring instants. The performance of our scheme can be well enhanced by martingale-preserving control variates and variance reduction via conditioning.

Nested clustered optimization allocates within each cluster from the cluster's own covariance block and then across the resulting cluster portfolios. Block inversion says the unconstrained minimum-variance portfolio has the same two-tier shape, with each block replaced by its Schur complement against every other asset. Conditioning instead on one knot from each other cluster truncates the conditioning set in the manner of a Vecchia approximation, and we give the rank-one model of cross-cluster dependence under which it is exact. Damping the complement by $\gamma\in[0,1]$ then gives a bridge with nested clustered optimization at $\gamma=0$ and the global optimum at $\gamma=1$, with no linear solve larger than a cluster or the number of clusters. Under estimation error the optimal $\gamma$ can be strictly interior and full coupling can remain optimal, and we give the local theorem at the minimum-variance end with exact examples of both, including a symmetric family in which the optimum is a closed form.

A substantial body of empirical evidence suggests that stochastic mortality models ignoring long range dependence tend to underestimate life expectancy, which may lead to profound implications for pension schemes and funding arrangements. This paper addresses the modelling of stochastic mortality via a mixture of a fractional Lévy process and standard Brownian motion. Our stochastic mortality model exhibits nice analytical tractability in actuarial valuations and flexibility in the choice of underlying Lévy specifications. The long range dependence feature embedded in our stochastic mortality model is well reflected in our empirical studies on the mortality shocks during World War II and COVID-19. For efficient numerical pricing of longevity derivatives, we construct an effective singular value decomposition approximation scheme to overcome the computational challenges arising from the non-Markovian nature of the fractional Lévy process. Truncation errors in singular value decomposition approximation can be reduced by an effective residual correction scheme.