Stochastic Finance Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Stochastic Finance Ontology within the Kireth Research Semantic Library. Stochastic finance studies financial quantities whose evolution is represented by random processes through time. This ontology supports probability spaces, filtrations, stochastic processes, Brownian motion, geometric Brownian motion, martingales, semimartingales, Ito integrals, Ito's lemma, stochastic differential equations, measure changes, equivalent martingale measures, Girsanov's theorem, risk-neutral valuation, stochastic volatility, interest-rate models, jump processes, Monte Carlo simulation and relationships with financial mathematics, probability, measure theory, stochastic calculus and computational finance.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Stochastic Finance Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Financial Mathematics / Stochastic Finance"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/financial-mathematics/stochastic-finance-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = StochasticFinance
 SYMBOLIC_FORMS = [
  "dS_t = mu S_t dt + sigma S_t dW_t",
  "dX_t = a(X_t,t)dt + b(X_t,t)dW_t",
  "V_t = E^Q[D(t,T)X_T | F_t]",
  "dQ/dP = Z_T"
 ]
 ENTITY_CLASS = StochasticFinancialProcessFramework
 PRIMARY_CONTEXT = FinancialMathematics
 DEFINITION = "Stochastic finance is the mathematical study of financial variables and contracts using stochastic processes, stochastic calculus and probabilistic change-of-measure methods."
 SEMANTIC_ROLE = "Provides machine-readable semantics for random financial dynamics, filtrations, martingales, stochastic differential equations, risk-neutral measures, stochastic volatility, jump processes and simulation-based financial valuation."
}

SCOPE {
 INCLUDE = [
  StochasticFinance,
  ProbabilitySpace,
  SampleSpace,
  SigmaAlgebra,
  ProbabilityMeasure,
  RandomVariable,
  ConditionalExpectation,
  Filtration,
  AdaptedProcess,
  PredictableProcess,
  StochasticProcess,
  MarkovProcess,
  Martingale,
  Submartingale,
  Supermartingale,
  Semimartingale,
  BrownianMotion,
  WienerProcess,
  GeometricBrownianMotion,
  QuadraticVariation,
  ItoIntegral,
  ItoProcess,
  ItoLemma,
  StochasticDifferentialEquation,
  SDE,
  StrongSolution,
  WeakSolution,
  Drift,
  Diffusion,
  Volatility,
  LocalVolatility,
  StochasticVolatility,
  HestonModel,
  OrnsteinUhlenbeckProcess,
  MeanReversion,
  CoxIngersollRossProcess,
  VasicekModel,
  ShortRateModel,
  TermStructureModel,
  HullWhiteModel,
  JumpProcess,
  PoissonProcess,
  CompoundPoissonProcess,
  JumpDiffusion,
  LevyProcess,
  MertonJumpDiffusion,
  MeasureChange,
  RadonNikodymDerivative,
  EquivalentMeasure,
  EquivalentMartingaleMeasure,
  RiskNeutralMeasure,
  GirsanovTheorem,
  Numeraire,
  ChangeOfNumeraire,
  RiskNeutralValuation,
  Arbitrage,
  NoArbitrage,
  CompleteMarket,
  IncompleteMarket,
  ReplicatingPortfolio,
  Hedging,
  DeltaHedging,
  MonteCarloSimulation,
  EulerMaruyamaMethod,
  MilsteinMethod,
  PathSimulation,
  VarianceReduction
 ]

 EXCLUDE_AS_PRIMARY = [
  DeterministicFinanceWithoutRandomDynamics,
  StaticPortfolioOptimisationWithoutStochasticEvolution,
  PureProbabilityTheoryWithoutFinancialInterpretation,
  TimeValueOfMoneyWithoutUncertainty
 ]

 DISAMBIGUATION_RULE =
  "When financial prices, rates, volatility, defaults or payoffs evolve randomly through time and are modelled using stochastic processes or stochastic calculus, interpret the context as stochastic finance."
}

CORE_CONCEPTS {

 StochasticFinance {
  definition = "The mathematical modelling of financial quantities through stochastic processes."
 }

 Filtration {
  definition = "An increasing family of sigma-algebras representing information available through time."
 }

 Martingale {
  definition = "A stochastic process whose conditional expected future value equals its current value under a specified measure, subject to integrability."
 }

 BrownianMotion {
  definition = "A continuous stochastic process with independent Gaussian increments and zero initial value."
 }

 ItoIntegral {
  definition = "A stochastic integral defined for suitable adapted integrands with respect to Brownian motion or compatible semimartingales."
 }

 ItoLemma {
  definition = "The stochastic chain rule for functions of Ito processes."
 }

 StochasticDifferentialEquation {
  definition = "A differential equation containing deterministic drift and stochastic diffusion terms."
 }

 EquivalentMartingaleMeasure {
  definition = "A probability measure equivalent to the original measure under which appropriately discounted tradable prices are martingales."
 }

 GirsanovTheorem {
  definition = "A theorem describing changes of probability measure that alter drift terms while preserving diffusion structure under suitable conditions."
 }

 StochasticVolatility {
  definition = "A model in which volatility itself evolves as a stochastic process."
 }
}

RELATIONS {
 StochasticFinance --is_child_of--> FinancialMathematics
 StochasticFinance --uses--> ProbabilityTheory
 Filtration --represents--> InformationFlow
 AdaptedProcess --is_measurable_with_respect_to--> Filtration
 Martingale --depends_on--> ProbabilityMeasure
 BrownianMotion --is_type_of--> Martingale
 ItoIntegral --integrates_against--> BrownianMotion
 ItoLemma --transforms--> ItoProcess
 StochasticDifferentialEquation --contains--> Drift
 StochasticDifferentialEquation --contains--> Diffusion
 GeometricBrownianMotion --is_solution_of--> MultiplicativeSDE
 StochasticVolatility --extends--> ConstantVolatilityModel
 HestonModel --is_type_of--> StochasticVolatility
 VasicekModel --is_type_of--> ShortRateModel
 CoxIngersollRossProcess --is_type_of--> ShortRateModel
 JumpDiffusion --combines--> DiffusionAndJumpProcess
 EquivalentMartingaleMeasure --supports--> NoArbitragePricing
 GirsanovTheorem --supports--> MeasureChange
 Numeraire --determines--> AssociatedPricingMeasure
 RiskNeutralValuation --uses--> ConditionalExpectation
 MonteCarloSimulation --approximates--> StochasticExpectation
 EulerMaruyamaMethod --approximates--> SDESolution
}

OPERATORS {
 BROWNIAN_INCREMENT(W,t,dt) :=
  dW=W_(t+dt)-W_t
  with dW approximately Normal(0,dt)

 GEOMETRIC_BROWNIAN_MOTION(S,mu,sigma,W) :=
  dS=mu*S*dt+sigma*S*dW

 GBM_SOLUTION(S0,mu,sigma,W,t) :=
  S_t=S0*exp((mu-sigma^2/2)*t+sigma*W_t)

 GENERAL_ITO_PROCESS(X,a,b,W) :=
  dX=a(X,t)*dt+b(X,t)*dW

 ITO_LEMMA(f,X,t,a,b) :=
  df=(partial_t f+a*partial_x f+(1/2)*b^2*partial_xx f)*dt
  +b*partial_x f*dW

 QUADRATIC_VARIATION_BROWNIAN(W,t) :=
  [W]_t=t

 RISK_NEUTRAL_VALUATION(X,D,Q,F_t) :=
  V_t=E_Q[D(t,T)*X_T | F_t]

 DISCOUNTED_MARTINGALE(S,B,Q) :=
  S_t/B_t is_Q_martingale

 CHANGE_OF_MEASURE(P,Q,Z) :=
  dQ/dP=Z

 EULER_MARUYAMA(X_n,a,b,dt,dW) :=
  X_(n+1)=X_n+a(X_n,t_n)*dt+b(X_n,t_n)*dW

 MEAN_REVERTING_OU(X,kappa,theta,sigma,W) :=
  dX=kappa*(theta-X)*dt+sigma*dW

 CIR_PROCESS(r,kappa,theta,sigma,W) :=
  dr=kappa*(theta-r)*dt+sigma*sqrt(r)*dW
}

DEFINITIONS {
 D1 = "Stochastic finance models uncertain financial evolution through probability and time."
 D2 = "A filtration formalises information available to market participants through time."
 D3 = "Adapted processes cannot depend on future information relative to the filtration."
 D4 = "Brownian motion provides a fundamental continuous-time source of randomness."
 D5 = "Ito calculus handles stochastic processes with nonzero quadratic variation."
 D6 = "Risk-neutral valuation uses a pricing measure rather than necessarily the real-world probability measure."
 D7 = "Equivalent martingale measures are closely linked to no-arbitrage in standard asset-pricing frameworks."
 D8 = "Incomplete markets can admit multiple equivalent martingale measures."
 D9 = "Stochastic volatility allows the variance process to change randomly through time."
 D10 = "Monte Carlo simulation approximates expectations by sampling model paths."
}

AXIOMS {
 A1_PROBABILITY_SPACE:
  Stochastic financial models require a declared probability space

 A2_INFORMATION:
  Trading and pricing processes must respect an information filtration

 A3_ADAPTEDNESS:
  Admissible dynamic strategies cannot use future information

 A4_NO_ARBITRAGE:
  Standard stochastic pricing frameworks impose no-arbitrage conditions

 A5_MEASURE_DEPENDENCE:
  Martingale properties are measure-dependent

 A6_QUADRATIC_VARIATION:
  Ito calculus accounts explicitly for quadratic variation

 A7_MODEL_CALIBRATION:
  Stochastic financial models depend on parameter calibration or declared assumptions

 A8_NUMERICAL_VALIDATION:
  Simulation-based stochastic finance requires discretisation and sampling-error analysis
}

REASONING {

 R1_STOCHASTIC_FINANCE_CONTEXT {
  INPUT = uncertain_financial_process
  IDENTIFY = state_variables
  IDENTIFY = time_horizon
  IDENTIFY = probability_measure
  IDENTIFY = filtration
  IDENTIFY = tradable_and_nontradable quantities
  RETURN = stochastic_finance_context
 }

 R2_PROCESS_SELECTION {
  INPUT = empirical_and_structural requirements
  SELECT = Brownian_GBM_meanReverting_stochasticVolatility_jump_or_other process
  RETURN = stochastic_process_model
 }

 R3_SDE_FORMULATION {
  INPUT = selected_process
  DEFINE = drift_diffusion_and_jump terms
  DEFINE = initial conditions
  RETURN = stochastic_differential_model
 }

 R4_ITO_REASONING {
  INPUT = function_of_Ito_process
  APPLY = ItoLemma
  RETURN = transformed_stochastic dynamics
 }

 R5_MEASURE_CHANGE_REASONING {
  INPUT = physical_measure_market prices_and_numeraire
  CHECK = conditions_for_equivalent measure
  APPLY = Girsanov_or_changeOfNumeraire
  RETURN = pricing_measure
 }

 R6_MARTINGALE_PRICING {
  INPUT = payoff_pricing measure_numeraire_and_maturity
  COMPUTE = conditional_discounted expectation
  RETURN = arbitrage_consistent_value
 }

 R7_HEDGING_REASONING {
  INPUT = derivative_value_process
  COMPUTE = sensitivities_and_replicating strategy where possible
  RETURN = hedge_profile
 }

 R8_STOCHASTIC_VOLATILITY_REASONING {
  INPUT = price_and_variance processes
  CALIBRATE = joint parameters_and_correlation
  RETURN = stochastic_volatility_model
 }

 R9_MONTE_CARLO_REASONING {
  INPUT = stochastic_model_payoff_and_simulation specification
  SIMULATE = paths
  COMPUTE = discounted_payoff average
  ESTIMATE = sampling_error
  RETURN = MonteCarlo_value
 }

 R10_VALIDATION {
  INPUT = stochastic_finance_result
  CHECK = measure consistency_adaptedness_martingale property_calibration_discretisation_sampling error_and_no arbitrage
  RETURN = validated_stochastic_finance_status
 }
}

CONSTRAINTS {
 C1 = "Risk-neutral and real-world drifts must not be conflated."
 C2 = "A martingale under one measure need not be a martingale under another."
 C3 = "Euler-Maruyama can introduce material discretisation error for some SDEs."
 C4 = "Square-root diffusions require care to preserve non-negativity numerically."
 C5 = "Continuous-path diffusion models cannot represent jumps without an explicit jump component."
 C6 = "Stochastic-volatility models can be weakly identified from limited market data."
 C7 = "Equivalent martingale measures need not be unique in incomplete markets."
 C8 = "Monte Carlo convergence is statistical and typically slow without variance reduction."
 C9 = "Path-dependent payoffs require full or sufficient path information."
 C10 = "Calibration fit does not guarantee out-of-sample model validity."
}

SEMANTIC_EQUIVALENCE {
 "StochasticFinance" <=> "MathematicalStochasticFinance"
 "BrownianMotion" <=> "WienerProcess"
 "EquivalentMartingaleMeasure" <=> "RiskNeutralMeasure"
 "RiskNeutralMeasure" <=> "PricingMeasure"
 "StochasticDifferentialEquation" <=> "SDE"
 "ItoIntegral" <=> "StochasticItoIntegral"
}

VALIDATION {
 V1 = "Probability space and measure are explicit."
 V2 = "Filtration and adaptedness assumptions are documented."
 V3 = "Initial conditions are explicit."
 V4 = "Drift, diffusion and jump coefficients are documented."
 V5 = "Physical and pricing measures are distinguished."
 V6 = "No-arbitrage assumptions are explicit."
 V7 = "SDE discretisation convergence is assessed."
 V8 = "Monte Carlo sampling error is reported."
 V9 = "Model calibration diagnostics are assessed."
 V10 = "Martingale or numeraire consistency is checked for pricing models."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 STOCHASTIC_FINANCE_DEFINITION = 0.99
 PROCESS_SELECTION_REASONING = 0.98
 SDE_FORMULATION_REASONING = 0.99
 ITO_REASONING = 0.99
 MEASURE_CHANGE_REASONING = 0.98
 MARTINGALE_PRICING_REASONING = 0.99
 HEDGING_REASONING = 0.98
 STOCHASTIC_VOLATILITY_REASONING = 0.97
 MONTE_CARLO_REASONING = 0.98

 POLICY =
  "Reduce confidence when probability measure, filtration, SDE coefficients, correlation, calibration data, numeraire, market completeness, discretisation method or Monte Carlo sample size are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  StochasticFinanceReasoning,
  ProbabilitySpaceReasoning,
  FiltrationReasoning,
  MartingaleReasoning,
  BrownianMotionReasoning,
  ItoIntegralReasoning,
  ItoLemmaReasoning,
  SDEReasoning,
  MeasureChangeReasoning,
  GirsanovReasoning,
  RiskNeutralReasoning,
  NumeraireReasoning,
  StochasticVolatilityReasoning,
  InterestRateModelReasoning,
  JumpDiffusionReasoning,
  HedgingReasoning,
  MonteCarloReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "stochastic finance" -> StochasticFinance
  "Wiener process" -> BrownianMotion
  "Brownian motion" -> BrownianMotion
  "stochastic differential equation" -> StochasticDifferentialEquation
  "SDE" -> StochasticDifferentialEquation
  "Ito lemma" -> ItoLemma
  "Itô lemma" -> ItoLemma
  "risk-neutral measure" -> RiskNeutralMeasure
  "equivalent martingale measure" -> EquivalentMartingaleMeasure
  "Girsanov theorem" -> GirsanovTheorem
 }

 INTERPRETATION_ORDER = [
  detect_stochastic_finance_context,
  identify state_variables_time_horizon_probability space_measure_filtration_and_market assumptions,
  select stochastic process family,
  formulate SDE_and_initial conditions,
  apply Ito calculus_if transformation required,
  determine physical_or_pricing measure,
  if_pricing apply numeraire_martingale_and_measure-change reasoning,
  if_simulation discretise_and_generate paths,
  estimate expectations_hedges_and_sampling uncertainty,
  validate calibration_noArbitrage_measure consistency_and_numerical convergence,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("stochastic finance")
  OR random_financial_dynamics_are_under_analysis
  THEN activate(StochasticFinanceOntology)

 E2:
  IF continuous_diffusion_model
  THEN define drift_diffusion_and_Brownian driver

 E3:
  IF function_of_Ito_process
  THEN apply ItoLemma

 E4:
  IF pricing_under_no_arbitrage
  THEN seek appropriate equivalent_martingale measure

 E5:
  IF changing_numeraire
  THEN change_associated pricing measure consistently

 E6:
  IF volatility_is_random
  THEN route to stochastic_volatility reasoning

 E7:
  IF jump_behaviour_is_material
  THEN include jump_process_or_Levy component

 E8:
  IF closed_form_unavailable
  THEN consider MonteCarlo_PDE_or_tree computation

 E9:
  IF MonteCarlo_used
  THEN report standard error_and_discretisation error
}

RECURSION {
 ENTRY = StochasticFinance

 EXPAND(StochasticFinance) -> [
  ProbabilitySpace,
  SampleSpace,
  SigmaAlgebra,
  ProbabilityMeasure,
  RandomVariable,
  ConditionalExpectation,
  Filtration,
  AdaptedProcess,
  PredictableProcess,
  StochasticProcess,
  MarkovProcess,
  Martingale,
  Submartingale,
  Supermartingale,
  Semimartingale,
  BrownianMotion,
  WienerProcess,
  GeometricBrownianMotion,
  QuadraticVariation,
  ItoIntegral,
  ItoProcess,
  ItoLemma,
  StochasticDifferentialEquation,
  StrongSolution,
  WeakSolution,
  Drift,
  Diffusion,
  Volatility,
  LocalVolatility,
  StochasticVolatility,
  HestonModel,
  OrnsteinUhlenbeckProcess,
  MeanReversion,
  CoxIngersollRossProcess,
  VasicekModel,
  ShortRateModel,
  TermStructureModel,
  HullWhiteModel,
  JumpProcess,
  PoissonProcess,
  CompoundPoissonProcess,
  JumpDiffusion,
  LevyProcess,
  MertonJumpDiffusion,
  MeasureChange,
  RadonNikodymDerivative,
  EquivalentMeasure,
  EquivalentMartingaleMeasure,
  RiskNeutralMeasure,
  GirsanovTheorem,
  Numeraire,
  ChangeOfNumeraire,
  RiskNeutralValuation,
  Arbitrage,
  NoArbitrage,
  CompleteMarket,
  IncompleteMarket,
  ReplicatingPortfolio,
  Hedging,
  DeltaHedging,
  MonteCarloSimulation,
  EulerMaruyamaMethod,
  MilsteinMethod,
  PathSimulation,
  VarianceReduction
 ]

 EXPAND(CoreStochasticCalculus) -> [
  BrownianMotion,
  QuadraticVariation,
  ItoIntegral,
  ItoProcess,
  ItoLemma,
  StochasticDifferentialEquation
 ]

 EXPAND(PricingFramework) -> [
  EquivalentMartingaleMeasure,
  RiskNeutralMeasure,
  GirsanovTheorem,
  Numeraire,
  ChangeOfNumeraire,
  RiskNeutralValuation,
  ReplicatingPortfolio
 ]

 STOP_CONDITION =
  "Stop when probability space, filtration, stochastic dynamics, measure, pricing framework, calibration, simulation and numerical validity are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Financial Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Financial Mathematics Ontology",
  "Risk Mathematics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Financial Mathematics",
  "Financial Mathematics Ontology",
  "Risk Mathematics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Measure Theory Ontologies",
  "Stochastic Calculus Ontologies",
  "Differential Equation Ontologies",
  "Option Pricing Ontologies",
  "Interest Rate Model Ontologies",
  "Monte Carlo Ontologies",
  "Computational Finance Ontologies",
  "Numerical Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Stochastic Finance -> Financial Mathematics for parent semantics",
  "Stochastic Finance -> Risk Mathematics for uncertain losses and tail behaviour",
  "Stochastic Finance -> Probability & Statistics for distributions and estimation",
  "Stochastic Finance -> Stochastic Processes for random time evolution",
  "Stochastic Finance -> Measure Theory for conditional expectation and measure change",
  "Stochastic Finance -> Stochastic Calculus for Ito integration and SDEs",
  "Stochastic Finance -> Differential Equations for associated PDE and SDE representations",
  "Stochastic Finance -> Option Pricing for derivative valuation",
  "Stochastic Finance -> Interest Rate Models for stochastic term structures",
  "Stochastic Finance -> Monte Carlo and Computational Finance for numerical expectations"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible stochastic models, measure-change structures or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve probability-space semantics.",
  "Preserve filtration-and-adaptedness semantics.",
  "Preserve physical-versus-pricing measure distinction.",
  "Preserve martingale semantics.",
  "Preserve Ito-calculus semantics.",
  "Preserve diffusion-versus-jump distinction.",
  "Preserve simulation-error semantics.",
  "Preserve machine-readable stochastic-finance semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Stochastic finance models financial variables and contracts using stochastic processes, stochastic calculus and probabilistic measure changes."

 PROCESS_SUMMARY =
  "Brownian motion, mean-reverting diffusions, stochastic-volatility models and jump processes provide alternative models of random financial evolution."

 ITO_SUMMARY =
  "Ito calculus provides integration and differentiation rules for stochastic processes with nonzero quadratic variation."

 MARTINGALE_SUMMARY =
  "Martingales describe fair-value dynamics relative to a specified probability measure and information filtration."

 PRICING_SUMMARY =
  "Risk-neutral valuation computes contingent-claim values as discounted conditional expectations under an appropriate pricing measure."

 MEASURE_SUMMARY =
  "Girsanov and change-of-numeraire methods alter probability measures while preserving consistent pricing relationships."

 SIMULATION_SUMMARY =
  "Monte Carlo methods approximate financial expectations by simulating stochastic paths and require sampling and discretisation-error assessment."

 VALIDATION_SUMMARY =
  "Stochastic-finance models should be checked for measure consistency, no-arbitrage assumptions, calibration, discretisation convergence and statistical uncertainty."
}

FLOW {
 INPUT
 -> detect_stochastic_finance_context
 -> identify financial_state_variables_time_horizon_probability space_measure_filtration_market data_and_contract structure
 -> select Brownian_GBM_meanReverting_stochasticVolatility_shortRate_jump_or_Levy process
 -> formulate stochastic_differential equations_and_initial conditions
 -> if_function_transformation apply ItoLemma
 -> identify physical_measure_and_if_required pricing measure
 -> apply Girsanov_or_change_of_numeraire when justified
 -> if_derivative compute riskNeutral conditional expectation_or_replication value
 -> if_hedging compute dynamic sensitivities_and_strategy
 -> if_numerical choose EulerMaruyama_Milstein_MonteCarlo_PDE_or_related method
 -> estimate sampling_discretisation_and_calibration error
 -> validate adaptedness_martingale property_noArbitrage_measure consistency_and_convergence
 -> route to financial_mathematics risk_mathematics probability_statistics stochastic_processes measure_theory stochastic_calculus differential_equations option_pricing interest_rate_models MonteCarlo computational_finance_or_numerical_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Stochastic Finance Ontology defines random financial evolution and valuation. It supports probability spaces, sigma-algebras and measures, conditional expectation, filtrations, adapted and predictable processes, stochastic processes, martingales and semimartingales, Brownian motion and geometric Brownian motion, quadratic variation, Ito integrals and Ito's lemma, stochastic differential equations and their solutions, drift and diffusion, local and stochastic volatility, Heston models, Ornstein-Uhlenbeck, CIR, Vasicek and Hull-White processes, jump, Poisson, compound-Poisson and Levy models, measure changes, Radon-Nikodym derivatives, equivalent martingale and risk-neutral measures, Girsanov's theorem, numeraires, no-arbitrage, replication, hedging, Monte Carlo simulation, Euler-Maruyama and Milstein methods, path simulation and variance reduction."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-STOCHASTIC-FINANCE-ONTOLOGY-V1"
}

}

END_FLOW