Generating realistic financial time series is challenging as training data is often limited to a single historical path. With such scarce data, overfitting is hard to avoid, especially under adversarial training where a trained discriminator can memorize the training samples. To mitigate this, recent approaches train generators to minimize the discrepancy between untrained feature representations of real and generated time series. In these works, the feature maps are based on path signatures, which can fail to capture relevant time series properties at tractable truncation depths. In this work, we instead train generators by matching random convolutional features of real and generated time series. Existing random convolutional feature maps, such as Rocket and Hydra, have been shown to provide informative representations of real-world time series, but cannot supervise generative models because they are non-differentiable. We introduce SOCK (SOft Competing Kernels), a fully differentiable random convolutional feature map, suited to train generative time series models. We show that generators trained by matching random SOCK features consistently outperform signature and diffusion baselines across a wide range of small-sample financial datasets. We further demonstrate SOCK's expressiveness on two-sample hypothesis testing and time series classification tasks, where SOCK matches or outperforms existing unsupervised feature maps.

LLMs can appear cautious in risk decision-making tasks, yet cautious-looking outputs do not necessarily indicate alignment with human decision-making mechanisms. We investigate this distinction using the St. Petersburg game as a controlled testbed, a classical paradox in which the expected payoff is infinite, yet humans typically report low, finite willingness to pay. We evaluate 28 LLMs with a structured prompt suite that includes the original game; controlled decision variants that perturb truncation, repeated play, numeric endowment, and occupational identity; a human-perspective prompt that asks models to reason as human decision makers; and paired comparisons between base models and their instruction-tuned counterparts. In the original game, most models generate finite bids, creating the appearance of human-like risk behavior. However, this outcome-level resemblance masks substantial mechanism-level differences. The controlled variants reveal that rather than maintaining human-like behavior seen in the original game, models often shift to conditionally and computationally rational behavior. Human-cue prompting and instruction tuning often lower bids and reduce some visible pathologies, but most mechanism-level response patterns remain largely unchanged. These findings show that behavioral alignment in risk decision-making can be surface-level: LLMs may produce human-like risk decisions without exhibiting human-consistent mechanisms. High-stakes evaluations of LLM decision-making should therefore move beyond outcome similarity and examine whether the alignment is supported by mechanism-level consistency.

No deployed automated market maker lets its liquidity providers vote on the trading function. We show this is structural, not an oversight. On the weighted-product family with $n \geq 3$ assets, no aggregation rule is at once fair and strategy-proof. Arrovian fairness forces a unique form, the weighted Aitchison centroid, the weighted geometric mean of the providers' preferred pools. But fairness forces mean-type aggregation and strategy-proofness forces median-type, and the only rule that is both is a single-provider dictator. The obstruction is sharp: it vanishes at $n = 2$, where a fair strategy-proof rule exists. Under the Frongillo--Papireddygari--Waggoner equivalence, the centroid is Genest's logarithmic opinion pool, and the impossibility transfers to externally Bayesian pooling.

Worker utility is not observed -- only its consequence is. Each gig transaction produces a single bit: accepted or rejected. We argue this structure points directly to the Preisach hysteresis model as the natural representation of latent worker preferences. The Preisach operator models aggregate output as an integral over a population of binary threshold elements -- precisely the structure that emerges when heterogeneous workers each carry a private acceptance wage. We estimate two latent utility surfaces: acceptance utility U_1(X) and rejection utility U_0(X), via a dual-output neural network (shared layers 256->128, margin loss enforcing U_1 >= U_0). Classification reduces to the Preisach gap U_1(X) - U_0(X), passed into an XGBoost classifier alongside clip-stabilised price-to-threshold encodings. On 36,891 gig transactions, this pipeline achieves Jaccard = 0.827 and ROC AUC = 0.799. The price-to-threshold encoding accounts for +11.0 pp AUC over raw utility features. The model confirms the directional asymmetry hysteresis predicts: price decreases depress completion rates more than equivalent increases raise them. Applied to the full dataset, the model's recommendations simultaneously reduce the total wage bill by 21.3% and increase expected fill rate by 9.7 pp. For 74.2% of transactions, P(accept) already exceeds 0.80; reducing the wage keeps it above threshold (mean post-cut P = 0.972), releasing cost savings (median 31%). For the remaining 25.4%, a median 7% wage increase recovers +43 pp acceptance. A model without an explicit indifference zone cannot execute both moves simultaneously.

The surplus of a life insurance policy depends on both systematic changes in mortality risk and financial changes. We propose to decompose the surplus by the axiomatically justified IASU decomposition, which is a continuous time version of the Shapley value. However, life tables are not updated continuously, but rather, only once per year. In this yearly update cycle of the life tables, we apply different interpolation methods to perform the IASU decomposition and analyze the effects of these methods on the surplus decomposition. Our results show that Lee-Carter and linear interpolation yield almost identical decompositions, whereas constant approximations results in substantially different decompositions. As a consequence, reporting standards and regulators should clarify how to interpolate mortality risks.

Learning Value-at-Risk (VaR) and Expected Shortfall (ES) is important for managing financial risks effectively. Existing approaches with limited parameters are vulnerable to model misspecification in the era of big data. To address this limitation, we propose a large tail risk model, the retrieval-enhanced self-grouping autoencoder (ReSGA), which is designed with millions of parameters to exploit the rich cross-sectional dependence and long-term temporal dynamics of assets using their characteristics. Applied to monthly US equity returns from 1926 to 2023 with 153 firm characteristics, ReSGA outperforms twelve econometric and machine learning competitors in terms of out-of-sample loss and statistical backtesting. In addition, its forecast advantages can translate into significant economic gains from long-short decile portfolios that are constructed by a new size-enhanced left-side momentum strategy. To clarify the role of complexity, we further conduct a systematic scaling analysis and demonstrate that improvements in joint VaR-ES forecasting are primarily driven by data complexity rather than model complexity. Finally, our analyses of group-importance and transfer-learning exhibit the interpretability and cross-market generalizability of ReSGA.

This study aims to determine whether the application of Deep Reinforcement Learning (DRL) as a specialized execution overlay can enhance pair trading in highly volatile cryptocurrency markets. Although classical implementations of the strategy have proven successful in traditional equities, they frequently exhibit rigidity and suffer from severe divergence risks when applied to high-variance environments. To address this need, this research introduces novel concepts. To construct a robust system, we developed a hierarchical "Filter-then-Rank" pair selection methodology and a proprietary "Fixed Risk, Adaptive Mean" execution model. The system employs a Proximal Policy Optimization (PPO) agent with a Long Short-Term Memory (LSTM) layer to govern execution decisions within strict deterministic risk management boundaries. Evaluated on 1-hour interval data from the Binance USD-M Futures market, the optimized RL policy achieved an out-of-sample performance that substantially outperformed the heuristic baseline. A stationary circular block bootstrap robustness check confirms that the agent's risk-adjusted outperformance is statistically significant at the 10 percent level. Although falling marginally short of the stricter 5 percent threshold, this result highlights the extreme idiosyncratic variance characteristic of digital assets. Ultimately, this thesis contributes to the quantitative finance literature by introducing a hybrid architecture that combines statistical arbitrage with DRL execution policies. Furthermore, it delivers a novel framework for safe reinforcement learning via deterministic shielding, proving that anchoring a neural policy to statistically robust boundaries successfully mitigates severe divergence risks.

A portfolio is \emph{anticipatory} when its optimizer acts on a richer model than the myopic, price-taking estimator used to calibrate it. Enrichment may be informational, via enlarged filtrations; dynamic, via horizon forecasts; or performative, via the deployment law induced by market impact. We give a decision-theoretic definition for all three cases and measure anticipation by the realized control gap between enriched controller and restricted estimator. The same quadratic geometry separates information, planning value, impact correction, and overfitting. For log utility under initial enlargement, value is the information-drift energy $\frac12\mathbb{E} \int_0^Tα_t^2\,dt$, equivalently mutual information or relative entropy. In mean-variance form, signal value is $\frac{1}{2γ}{\rm tr}(Σ^{-1}Ω)$. Dynamic forecast anticipation gives a finite-horizon quadratic premium in the forecast stack, while permanent impact changes the price-taking allocation $θ_{\rm na} =(Λ+γΣ)^{-1}μ$ into $θ_{\rm an} = (2Λ+γΣ)^{-1}μ$ and reveals a spectral phase transition for naive recalibration. The main result is a stacked finite-horizon LQG decomposition: information, forecast, and impact combine into an information trace plus one inverse-precision norm, whose expansion yields the impact term, forecast term, and signed forecast-impact interaction. Sharp angle bounds and an orthogonal nonnegative projection identity resolve the signed term. The stationary extension endogenizes information covariance as Kalman error reduction and carries impact anticipation to an infinite-horizon Lyapunov trace with transaction costs. Finally, the penalty $\frac{1}{2}{\rm tr}(H^{-1}Σ_\varepsilon)$ shows that correctly specified anticipation creates value, vacuous anticipation has zero value, and misspecified anticipation is harmful when estimated structure is optimized as true.

Collateral allocation for uncleared derivatives is a legally constrained and operationally discrete optimization problem. Institutions must satisfy margin requirements while respecting CSA eligibility rules, valuation percentages, rounding, transfer thresholds, concentration limits, custody conditions, inventory, and VM, IM, or IA side constraints. This manuscript develops CR-HO-QAOA, a certified higher-order quantum candidate-generation framework for margin- and CSA-aware collateral allocation. The framework is adapter-first: official SIMM, proxy SIMM, legacy IA, VM-only, RQV, or hybrid margin sources are normalized into a common MarginRequirement, so the optimizer does not calculate or replace official SIMM. Given the requirement, CSA terms, and inventory, the optimizer builds a bounded active neighborhood of pledge, recall, substitution, batch, and slack actions. These actions define a higher-order binary model whose hyperedges capture concentration pressure, custody batches, substitution tickets, chunky lots, liquidity effects, overshoot, and side-specific requirements. The quantum layer maps hyperedges into a Pauli-Z cost Hamiltonian and uses collateral-specific feasible-subspace mixers to preserve one-hot choices, movement budgets, side assignments, and substitution structure. Candidates are decoded, repaired if needed, evaluated under an eight-term production objective, and certified by a deterministic CP-SAT master solver before any recommendation is reported. Synthetic benchmarks show that higher-order, constraint-preserving candidate generation can improve certified sample quality relative to QUBO-style and generic-mixer baselines, while CP-SAT remains the feasibility and governance arbiter. These results are synthetic workflow-validation evidence only, not evidence of hardware quantum advantage or production bank savings.

Changes in volatility contain valuable information about the likelihood of positive versus negative returns. We propose a new approach to modeling financial returns that exploits this insight by decomposing returns into sign and magnitude (absolute value) components, with magnitude closely related to volatility. The joint distribution used to compute expected returns combines a model for the marginal distribution of magnitude with a model for the distribution of the sign, conditional on the contemporaneous magnitude. Unlike traditional linear predictive regressions, this decomposition framework captures nonlinear predictability in return dynamics. An out-of-sample forecasting evaluation using monthly U.S. stock market excess returns demonstrates substantial statistical and economic gains relative to linear regression and complete subset regression, while delivering performance that is competitive with copula-based return-decomposition methods and other nonlinear benchmarks.

This paper studies how workers' beliefs about pay shape the tradeoffs between pay and workplace amenities. We design a multi-stage incentivized survey experiment that combines hypothetical choice experiments with elicited beliefs about starting salaries in real jobs and randomly varies the provision of explicit pay information. Although stated preferences imply sizable willingness to pay for amenities consistent with prior literature, baseline beliefs about salaries in real jobs are systematically biased along two margins: respondents under-predict starting salaries by 18% and expect higher-amenity jobs to pay more, substantially over-predicting the amenity-pay gradient. Exposure to pay information raises mean pay beliefs for similar jobs by 4% and reduces belief dispersion by 15%, but does not alter the strong positive association between perceived pay and advertised amenities, leaving the amenity-pay tradeoffs in stated choices essentially unchanged. While workers have strong preferences for workplace amenities, the tradeoffs they perceive deviate sharply from those present under full information.

We provide a brief primer for the idea behind formalising hierarchical causality in the context of complex systems. Here actors are not simply agents. Actors instantiate causation classes. Agents implement local dynamics in given levels or organisation in a given system. Hierarchical causality then describes how actor-level roles constrain, select, and organise agent-level behaviour across levels. The system then necessarily requires three additional structures. First, causation classes to abstract a given form of causal influence that an actor instantiates. Second, aggregation operators to move across the levels. Third, discrete event-time maps are required because the system comprises events, and the relation between local event counts and any global clock must be specified. Our formulation here is purposefully simple and discrete.

We audit the operational decomposition of D-Wave's hybrid quantum-classical portfolio-optimization service on cardinality-constrained mean-variance-turnover instances spanning N=10 to 640, with the constraint-native LeapHybridCQM interface, the penalty-encoded LeapHybridBQM interface, and Gurobi MIQP and simulated-annealing classical anchors. We report all three SDK timing fields (t_run, t_charge, t_QPU) and define a candidate four-metric audit protocol for hybrid quantum-classical solvers. Three findings. First, the LeapHybridCQM service matches Gurobi's proven optimum on all 54 head-to-head instances at N <= 120, but the mean QPU access time is 0.034 seconds out of the 5-second nominal wall-clock budget -- 0.68% of the nominal budget, approximately 0.72% of measured run time -- and the remaining ~99% is the service's classical decomposition and feasibility-aware reassembly. Second, in a CPU-only matched-wall-clock counterfactual, TabuSampler on the penalty-encoded BQM reaches final exact-K objectives within mean absolute delta 0.001 of hybrid CQM on 24 tested instances; this does not ablate the LeapHybridCQM pipeline internals, but it shows that these objective levels are reproducible by a classical heuristic at the same wall-clock budget. Third, the cardinality penalty contributes a dense rank-one term that fully connects the encoded logical graph independent of the input covariance density, an effect we prove as a structural theorem; the resulting density-axis collapse explains the BQM degradation observed in the empirical comparison. Out-of-sample on Fama-French 49 industry portfolios, the QPU-selected portfolios deliver a mean Sharpe ratio of 1.94 versus 2.22 for the 1/N baseline. The practical implication is that reported D-Wave hybrid wins on this problem class are constraint-native classical pipelines, not quantum-sampling wins.

Financial structures transform stochastic cash-flow representations of underlying economic activities into ordered payments across multiple claims through systems of contractual allocation rules. While sophisticated computational models of such structures are widely used in practice, their allocation logic is typically embedded in proprietary implementations and deal-specific documentation, making systematic analysis, comparison, and verification across transactions difficult. This paper develops a computational framework that formalizes financial structures as structured allocation systems mapping stochastic inflows into hierarchically ordered payments through explicit and state-dependent allocation operators. The framework distinguishes between economic activities, financeable representations, admissible structures, feasible designs, and sustainable designs. It thereby specifies the inputs on which allocation mechanisms operate, the structural requirements that distinguish financial structures from generic contracts, and the feasibility restrictions under which a design may be undertaken.

The price of a financial derivative can be expressed as an iterated conditional expectation, where the inner term conditions on the future of an auxiliary process. We show that this inner conditional expectation solves an SPDE (a 'conditional Feynman-Kac formula'). The problem requires conditioning on a backward filtration generated by the noise of the auxiliary process and enlarged by its terminal value, leading us to search for a backward Brownian motion here. This adds a source of irregularity to the SPDE which we tackle with new techniques. Lastly, we establish a new class of mixed Monte-Carlo PDE numerical methods.

We develop a general model of discrete choice that incorporates peer effects in preferences and consideration sets. We characterize the equilibrium behavior and establish conditions under which all parts of the model can be recovered from a sequence of choices. We allow peers to affect preferences, consideration, or both. We show that these peer-effect mechanisms have different behavioral implications in the data. This allows us to recover the set and the type of connections between the agents in the network. We then use this information to recover each agent's preferences and consideration mechanisms. These nonparametric identification results allow for general forms of heterogeneity across agents and do not rely on the variation of either exogenous covariates or the set of available options (menus). We apply our results to model expansion decisions by tea chains and find evidence of limited consideration. We simulate counterfactual predictions and show how limited consideration slows market penetration and competition.

Extracting the joint risk-neutral distribution from option prices has remained an open problem since Ross (1976). We propose a projection-based estimator that approximates target payoffs over the entire state space using portfolios of observed options. The method estimates correlations and joint crash probabilities even when markets are incomplete, improves univariate estimates relative to Carr and Madan (2001(, and yields an explicit finite-sample bound interpretable as a measure of market incompleteness. Applying the method to two unexpected Swiss National Bank announcements about the EUR/CHF floor, we find that dependence, rather than marginal volatility, accounts for about two-thirds of the change in joint crash risk. A policy targeting one exchange rate reshaped the broader CHF currency market.

The concentration of digital payment transactions in just two UPI apps like PhonePe and Google Pay has raised concerns of duopoly in India s digital financial ecosystem. To address this, the National Payments Corporation of India (NPCI) has mandated that no single UPI app should exceed 30 percent of total transaction volume. Enforcing this cap, however, poses a significant computational challenge: how to redistribute user transactions across apps without causing widespread user inconvenience while maintaining capacity limits? In this paper, we formalize this problem as the Minimum Edge Activation Flow (MEAF) problem on a bipartite network of users and apps, where activating an edge corresponds to a new app installation. The objective is to ensure a feasible flow respecting app capacities while minimizing additional activations. We further prove that Minimum Edge Activation Flow is NP-Complete. To address the computational challenge, we propose scalable heuristics, named Decoupled Two-Stage Allocation Strategy (DTAS), that exploit flow structure and capacity reuse. Experiments on large semi-synthetic transaction network data show that DTAS finds solutions close to the optimal ILP within seconds, offering a fast and practical way to enforce transaction caps fairly and efficiently.

We establish an explicit approximation formula for European put option prices within a general stochastic volatility model with time-dependent parameters. Our methodology is based on expansions of the mixing representation of the put option price as an expectation of the Black-Scholes formula, in which the resulting terms are calculated explicitly by Malliavin calculus. We obtain an explicit representation of the error generated by the expansion procedure, and bound it in terms of moments of functionals of the underlying volatility process. Under the assumption of piecewise-constant parameters, our approximation formulas become closed-form, and compatible with a proposed fast calibration scheme. Finally, we perform a numerical sensitivity analysis to investigate the quality of our approximation formula in the so-called Stochastic Verhulst model, and show that the errors are well within the acceptable range for application purposes.

This paper develops a novel framework for modeling variance swap of multi-asset stochastic volatility models by employing determinant-based instantaneous generalized variance. In this setting the determinant of the covariance matrix captures the joint dispersion of the multivariate log-return dynamics. By specifying the distribution of the log returns of the underlying assets under the Heston and Barndorff-Nielsen & Shephard (BNS) stochastic volatility frameworks, we obtain an analytical pricing expression for multi-asset Heston formulation, while BNS formulation is treated through a tractable approximation. To evaluate the robustness of the proposed model, we conduct simulations using nine different assets generated via the quantmod package. For a three-asset portfolio, analytical expressions for the generalized variance swap are obtained under both the Heston and BNS models. Numerical experiments further demonstrate the effectiveness of the proposed model through parameter testing, calibration, and validation.

We propose multi-agent reinforcement learning as a new method for modeling fake news in social networks. This method allows us to model human behavior in social networks both in unaccustomed populations and in populations that have adapted to the presence of fake news. In particular the latter is challenging for existing methods. We find that a fake-news attack is more effective if it targets highly connected people and people with weaker private information. Attacks are more effective when the disinformation is spread across several agents than when the disinformation is concentrated with more intensity on fewer agents. Furthermore, fake news spread less well in balanced networks than in clustered networks. We test a part of our findings in a human-subject experiment. The experimental evidence provides support for the predictions from the model, suggesting that the model is suitable to analyze the spread of fake news in social networks.

This study explores the behavioral dynamics of illiquid stock prices in a listed stock market. Illiquidity, characterized by wide bid and ask spreads affects price formation by decoupling prices from standard risk and return relationships and increasing sensitivity to market sentiment. We model the prices at the Uganda Securities Exchange (USE) which is illiquid in that the prices remain constant much of the time thus complicating price modelling. We circumvent this challenge by combining the Markov model (MM) with two models; the exponential Ornstein Uhlenbeck model (XOU) and geometric Brownian motion (gBm). In the combined models, the MM was used to capture the constant prices in the stock prices while the XOU and gBm captured the stochastic price dynamics. We modelled stock prices using the combined models, as well as XOU and gBm alone. We found that USE stocks appeared to have low correlation with one another. Using theoretical analysis, simulation study and empirical analysis, we conclude that this apparent low correlation is due to illiquidity. In particular data simulated from combined MM-gBm, in which the gBm portion were highly correlated resulted in a low measured correlation when the Markov chain had a higher transition from zero state to zero state.

How does the climatic experience of past generations affect today's attitudes towards environmental issues? Using empirical evidence spanning multiple contemporary surveys and ethnic group level cultural records, we show that the intensity of ancestral climate anomalies has a persistent effect on the perceived stakes of environmental considerations in decision-making. The relationship is U-shaped: descendants of groups who faced more stable or more volatile climates attribute higher importance to environmental concerns, with a dip at intermediate levels. Consistent with a cultural transmission channel, environmental content in folklore and other cultural narratives displays the same U-shape. We propose a general model in which environmental attention is a costly choice made before climate conditions are realized, and perceptions of its stakes are shaped by realized gains and losses through an evolutionary process. Because attention is chosen ex ante, selection pressure is coarse: it only disciplines perceptions through average payoffs under the specific climate distribution a group experiences, generating heterogeneous bias across ethnic groups. When environmental attention serves two functions, using typical conditions effectively and protecting against extreme events, the model rationalizes the U-shaped dependence of perceived stakes on ancestral climate anomalies.

We present a variation of the well-known binomial model of asset prices. This variation is based on the interaction of a finite number of investors (grouped in two kinds, bulls and bears) with a single market maker, in an environment with bounds to short selling. We prove a formula for all the moments of the logarithmic returns and from that we derive the corresponding formula for the binomial model. As an application of the model, we show how to compute parameters in order to approximate given moments, enabling the modeling of skewness and excess kurtosis. We provide a concrete example of this procedure, using real data from a stock, and we present two plots revealing its convergence pattern. Finally, we generalize the framework to account for the possibility of more heterogeneous investors, and give the corresponding formula for the moments of the logarithmic returns, and the algorithm for fitting given moments.

The key characteristic of a true free market economy is that exchanges are entirely voluntary. When there is a monopoly in the creation of currency as we have in today's markets, you no longer have a true free market. Features of the current economic system such as central banking and taxation would be nonexistent in a free market. This paper examines how currency monopoly leads to the instabilities and imbalances that we see in today's economy. It also proposes that currencies should emerge from the voluntary exchange of goods and services, and studies economic interaction across all scales, by considering economic action in cases where the self-interests of individuals are coincident. By examining the voluntary exchange of goods and services at the scale of an entire society, it is shown that a new currency system, which resolves a lot of the problems caused by the current fiat currency system, emerges naturally from the free market. The new currency system is robust and efficient, and provides a way for public goods and services to be provided, and its providers compensated, without the need for direct taxation.

There is a great need to stock materials for production, but storing materials comes at a cost. Lack of organization in the inventory can result in a very high cost for the final product, in addition to generating other problems in the production chain. In this work we present mathematical and statistical methods applicable to stock management. The stock analysis using ABC curves serves to identify which are the priority items, the most expensive and with the highest turnover (demand), and thus determine, through stock control models, the purchase lot size and the periodicity that minimize the total costs of storing these materials. Using the Economic Order Quantity (EOQ) model and the (Q,R) model, the inventory costs of a company were minimized. The comparison of the results provided by the models was performed.

Connections appear to be helpful in many contexts, such as obtaining a job, a promotion, a grant, a loan, or publishing a paper. This may be due either to favoritism or to information conveyed by connections. Attempts at identifying both effects have relied on measures of true quality, generally built from data collected long after promotion. Building on earlier work on discrimination, we propose a new method to identify favors and information from data collected at the time of promotion. Under weak assumptions, we show that promotion decisions for connected candidates look more random to the econometrician due to the information channel. We derive new identification results and estimate the strength of the two effects. We adapt the control function approach to address the issue of the selection into connections. Applying our methodology to academic promotions in Spain and Italy, as well as political advancements in China, we find evidence that connections may both convey information and attract favors.

The goal of this paper is to review some analytic techniques that are potentially useful to shed light on the determinacy question that arises in New Keynesian models as result of a combination of several monetary policy rules; in these models, we provide conditions to guarantee existence and uniqueness of equilibrium by means of results that are obtained from theoretical analysis. In particular, these methods confirm the well known fact that Taylor--like rules in interest rate setting are not the only way to reach determinacy of the rational expectations equilibrium in the New Keynesian setting. The key technical tool we use for that purposes is the so--called Budan--Fourier Theorem, that we review along the paper. All the ideas and techniques presented have been already used, our contribution that might be original here are the organization and emphasis.

We consider a financial market in which traders potentially face restrictions in trading some of the available securities. Traders are heterogeneous with respect to their beliefs and risk profiles, and the market is assumed thin: traders strategically trade against their price impacts. We prove existence and uniqueness of a corresponding equilibrium, and provide an efficient algorithm to numerically obtain the equilibrium prices and allocations given market's inputs. We find that restrictions may increase the market's welfare if traders have different views regarding the covariance matrix of securities returns. The latter heterogeneity regarding covariance matrix disagreement is essential in modelling; for instance, when traders agree on the covariance matrix, restricting participation in some securities for some traders leaves equilibrium prices unaltered in the unrestricted securities, a certainly undesirable model effect.

A group of people wishes to use money to exchange goods efficiently over several time periods. However, there are disadvantages to using any of the goods as money, and in addition fiat money issued in the form of notes or coins will be valueless in the final time period, and hence in all earlier periods. Also, Walrasian market prices are determined only up to an arbitrary rescaling. Nevertheless we show that it is possible to devise a system which uses money to exchange goods and in which money has a determinate positive value. In this system, tokens are initially supplied to all traders by a central authority and recovered by a purchase tax. All trades must be made using tokens or promissory notes for tokens. This mechanism controls the flow rather than the stock of money: it introduces some trading frictions, some redistribution of wealth, and some distortion of prices, but these effects can all be made small.