How do we extract an explicit form for a data asset's price (Pi)? We reformulate the stochastic control problem as a saddle point problem with multivariate constraints—unlocking a new level of analytical clarity in data pricing. #QuantFinance #StochasticControl #DataValuation
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The core idea: The investor's optimization reduces to a constrained saddle point problem involving portfolio (xπ), consumption (xc), and market dynamics (xµ, xΣ). This shifts complexity into tractable mathematical form. #QuantitativeFinance #FinanceResearch #MathematicalModeling
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Functions F1 and F2 define the objective: F1, F2 capture trade-offs between return, risk, consumption, and ambiguity aversion. The data asset price Pi emerges where information-driven ambiguity reduction balances utility. #UtilityTheory #RiskManagement #DataEconomy #AssetPricing
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Existence and Uniqueness of Pi Using Lemma 3.1 and Theorems from prior literature, we prove: A saddle point always exists The value function Ji is uniquely defined This gives Pi a rigorous mathematical foundation. #MathematicalFinance #StochasticOptimization
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Saddle Point Equilibrium (3.2): For each xq, the equilibrium satisfies: sup (xπ, xc) inf (xµ, xΣ) = inf (xµ, xΣ) sup (xπ, xc) An elegant duality—where ambiguity meets investor behavior. #GameTheory #SaddlePoint #QuantitativeModels #DecisionScience
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Value function transformation: The stochastic control solution Ji(t, x; B) reduces to: A power utility form (i = 1) A log utility form (i = 2) These are linked to G1, G2—functions of the saddle point. #FinanceTheory #Mathematics #DataScienceInFinance
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Dynamic solution evolution: g1(t; B), g21(t), and g22(t; B) evolve over [t, T] via integral equations involving G1 and G2. They encode how uncertainty, time preference, and ambiguity shape Pi. #DynamicModels #MathematicalEconomics #TimeSeries
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Remark - Bounded Rationality Realism Our pricing model respects: Non-uniqueness of strategies Uniqueness of the outcome (Gi) That is, many ways to act, but one coherent value of information. #BehavioralFinance #DecisionTheory #InformationValue
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Consumption complicates things. Under joint investment and consumption, Pi’s form becomes more nuanced. Adding consumption blurs boundaries of risk-return optimization. #PortfolioTheory #ConsumptionModel #QuantitativeThinking
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Conclusion: We now have a mathematically solid, explicit pricing model for data assets via: Stochastic control Saddle point reduction Robust utility theory Setting the stage for future equilibrium pricing and DeFi integration. #DeFiResearch #FinancialInnovation #DataAssets #Quant
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