Stochastic Process Ontology

flow "StochasticProcessOntology" v4.0

# ============================================================
# STOCHASTIC PROCESS ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/stochastic-mathematics/stochastic-process-ontology.html"
  title: "Stochastic Process Ontology"
  domain: "Probability and Statistics"
  category: "Stochastic Mathematics"
  entity: STOCHASTIC_PROCESS
  entity_type: RANDOM_PROCESS

DEFINE_STOCHASTIC_PROCESS:
  STOCHASTIC_PROCESS :=
    INDEXED_FAMILY_OF_RANDOM_VARIABLES(
      PROBABILITY_SPACE,
      INDEX_SET,
      STATE_SPACE
    )

CORE_STRUCTURE:
  PROBABILITY_SPACE:
    (Omega, F, P)

  INDEX_SET:
    T

  STATE_SPACE:
    S

  PROCESS:
    X := {X_t : t IN T}

  RANDOM_VARIABLE:
    X_t:
      Omega -> S

ENTITIES:
  SAMPLE_SPACE:
    type: OUTCOME_SPACE

  SIGMA_ALGEBRA:
    type: EVENT_STRUCTURE

  PROBABILITY_MEASURE:
    type: MEASURE

  INDEX_SET:
    type: PROCESS_PARAMETER_DOMAIN

  STATE_SPACE:
    type: PROCESS_VALUE_SPACE

  RANDOM_VARIABLE:
    type: MEASURABLE_FUNCTION

  SAMPLE_PATH:
    type: PROCESS_REALISATION

  FINITE_DIMENSIONAL_DISTRIBUTION:
    type: JOINT_DISTRIBUTION

  FILTRATION:
    type: INFORMATION_STRUCTURE

RELATIONSHIPS:
  STOCHASTIC_PROCESS
    -> DEFINED_ON
    -> PROBABILITY_SPACE

  STOCHASTIC_PROCESS
    -> INDEXED_BY
    -> INDEX_SET

  STOCHASTIC_PROCESS
    -> TAKES_VALUES_IN
    -> STATE_SPACE

  STOCHASTIC_PROCESS
    -> CONSISTS_OF
    -> RANDOM_VARIABLES

  FIXED_INDEX
    -> PRODUCES
    -> RANDOM_VARIABLE

  FIXED_OUTCOME
    -> PRODUCES
    -> SAMPLE_PATH

  PROCESS
    -> CHARACTERISED_BY
    -> FINITE_DIMENSIONAL_DISTRIBUTIONS

INDEX_CLASSIFICATION:
  DISCRETE_TIME:
    T := {0,1,2,...}

  CONTINUOUS_TIME:
    T subset_of REAL_NUMBERS

  GENERAL_INDEX:
    T:
      MAY_BE:
        SPATIAL
        TEMPORAL
        MULTIDIMENSIONAL
        ABSTRACT

STATE_SPACE_CLASSIFICATION:
  DISCRETE_STATE:
    S:
      FINITE_OR_COUNTABLE

  CONTINUOUS_STATE:
    S:
      UNCOUNTABLE
      OFTEN:
        SUBSET_OF_REAL_OR_VECTOR_SPACE

SAMPLE_PATH:
  FOR:
    omega IN Omega

  DEFINE:
    X(omega):
      t -> X_t(omega)

  interpretation:
    ONE_REALISATION_OF_PROCESS_EVOLUTION

FINITE_DIMENSIONAL_DISTRIBUTIONS:
  FOR:
    t_1,...,t_n IN T

  DEFINE:
    DISTRIBUTION_OF:
      (X_t1, X_t2, ..., X_tn)

  role:
    ENCODE:
      JOINT_PROBABILISTIC_BEHAVIOUR
      AT:
        FINITE_COLLECTIONS_OF_INDICES

MEAN_FUNCTION:
  m_X(t) :=
    E[X_t]

  require:
    EXPECTATION_EXISTS

SECOND_MOMENT:
  R_X(s,t) :=
    E[X_s X_t]
  FOR:
    REAL_VALUED_PROCESS
    WHEN:
      SECOND_MOMENTS_EXIST

COVARIANCE_FUNCTION:
  C_X(s,t) :=
    Cov(X_s, X_t)

  equivalent:
    C_X(s,t) :=
      E[
        (X_s - m_X(s))
        (X_t - m_X(t))
      ]

AUTOCOVARIANCE:
  same_process:
    Cov(X_s, X_t)

AUTOCORRELATION:
  rho_X(s,t) :=
    C_X(s,t)
    /
    sqrt(
      C_X(s,s)
      * C_X(t,t)
    )

  require:
    POSITIVE_FINITE_VARIANCES

STRICT_STATIONARITY:
  DEFINE:
    FOR_ANY:
      n
      t_1,...,t_n
      shift_h

    JOINT_DISTRIBUTION:
      (X_t1,...,X_tn)
      =
      (X_(t1+h),...,X_(tn+h))
    IN_DISTRIBUTION
    WHEN:
      SHIFTED_INDICES_VALID

WEAK_STATIONARITY:
  requirements:
    E[X_t] := CONSTANT

    Cov(X_t, X_(t+h)):
      DEPENDS_ONLY_ON:
        h

    FINITE_SECOND_MOMENTS = TRUE

  distinction:
    WEAK_STATIONARITY != STRICT_STATIONARITY_BY_DEFAULT

INDEPENDENT_PROCESS_VALUES:
  IF:
    X_t1,...,X_tn:
      MUTUALLY_INDEPENDENT
    FOR:
      DISTINCT_INDICES

  THEN:
    PROCESS_HAS:
      INDEPENDENT_VALUES

  caution:
    MOST_STOCHASTIC_PROCESSES:
      HAVE_DEPENDENCE_ACROSS_INDICES

INDEPENDENT_INCREMENTS:
  DEFINE:
    INCREMENTS_OVER_DISJOINT_INTERVALS:
      INDEPENDENT

  examples:
    - POISSON_PROCESS
    - BROWNIAN_MOTION

STATIONARY_INCREMENTS:
  DEFINE:
    DISTRIBUTION_OF:
      X_(t+h) - X_t
    DEPENDS_ON:
      h
    NOT:
      t

MARKOV_PROPERTY:
  DEFINE:
    FUTURE:
      CONDITIONALLY_INDEPENDENT_OF_PAST
    GIVEN:
      PRESENT_STATE

  symbolic:
    P(X_t IN A | HISTORY_UP_TO_s)
      =
    P(X_t IN A | X_s)
    FOR:
      s < t
    UNDER:
      MARKOV_ASSUMPTIONS

MARKOV_PROCESS:
  stochastic_process:
    SATISFIES:
      MARKOV_PROPERTY

MARKOV_CHAIN:
  common_context:
    DISCRETE_TIME_MARKOV_PROCESS

  transition:
    P_ij :=
      P(X_(n+1)=j | X_n=i)

POISSON_PROCESS:
  type:
    COUNTING_PROCESS

  parameter:
    lambda > 0

  properties:
    N_0 := 0
    INDEPENDENT_INCREMENTS = TRUE
    STATIONARY_INCREMENTS = TRUE
    N_t - N_s ~ POISSON(lambda*(t-s))

COUNTING_PROCESS:
  properties:
    N_t:
      NONNEGATIVE_INTEGER_VALUED
      NONDECREASING

    N_0:
      OFTEN_ZERO

BROWNIAN_MOTION:
  alias:
    WIENER_PROCESS

  properties:
    W_0 := 0
    INDEPENDENT_INCREMENTS = TRUE
    STATIONARY_GAUSSIAN_INCREMENTS = TRUE
    W_t-W_s ~ NORMAL(0,t-s)
    CONTINUOUS_SAMPLE_PATHS = TRUE

  covariance:
    Cov(W_s,W_t) :=
      min(s,t)

RANDOM_WALK:
  discrete_process:
    S_n :=
      SUM[k=1..n] Z_k

  increments:
    Z_k:
      RANDOM_VARIABLES

  common_case:
    IID_INCREMENTS

GAUSSIAN_PROCESS:
  DEFINE:
    EVERY_FINITE_COLLECTION:
      (X_t1,...,X_tn)
    HAS:
      MULTIVARIATE_NORMAL_DISTRIBUTION

  characterised_by:
    MEAN_FUNCTION
    COVARIANCE_FUNCTION

  under:
    VALID_GAUSSIAN_PROCESS_STRUCTURE

FILTRATION:
  notation:
    {F_t}

  requirement:
    IF:
      s <= t
    THEN:
      F_s subset_of F_t

  interpretation:
    INFORMATION_AVAILABLE_UP_TO_INDEX_t

ADAPTED_PROCESS:
  X_t:
    F_t_MEASURABLE
    FOR:
      EACH_t

PREDICTABLE_PROCESS:
  concept:
    PROCESS_VALUE:
      DETERMINED_BY_INFORMATION
      AVAILABLE_IMMEDIATELY_BEFORE_CURRENT_TIME
    UNDER:
      FORMAL_PREDICTABILITY_DEFINITION

MARTINGALE:
  adapted:
    TRUE

  integrability:
    E[abs(M_t)] < infinity

  property:
    E[M_t | F_s] :=
      M_s
    FOR:
      s <= t

SUBMARTINGALE:
  property:
    E[X_t | F_s] >= X_s

SUPERMARTINGALE:
  property:
    E[X_t | F_s] <= X_s

STOPPING_TIME:
  random_time:
    tau

  requirement:
    {tau <= t}:
      F_t_MEASURABLE

  role:
    STOPPING_DECISION_USES_NO_FUTURE_INFORMATION

FIRST_PASSAGE_TIME:
  tau_A :=
    inf{
      t >= 0:
      X_t IN A
    }

  relation:
    OFTEN:
      STOPPING_TIME
    UNDER:
      APPROPRIATE_PROCESS_AND_FILTRATION_CONDITIONS

ERGODICITY:
  role:
    CONNECT:
      LONG_RUN_TIME_AVERAGES
      TO:
      DISTRIBUTIONAL_OR_ENSEMBLE_PROPERTIES

  caution:
    EXACT_DEFINITION_DEPENDS_ON_PROCESS_CONTEXT

TRANSITION_PROBABILITY:
  MARKOV_CONTEXT:
    P(s,x;t,A) :=
      P(X_t IN A | X_s=x)

  role:
    DESCRIBE:
      PROBABILISTIC_STATE_EVOLUTION

SEMIGROUP:
  time_homogeneous_markov_process:
    P_t

  composition:
    P_(s+t) :=
      P_s P_t

  connection:
    CHAPMAN_KOLMOGOROV

CHAPMAN_KOLMOGOROV:
  relation:
    P(s,x;t,A)
      =
    INTEGRAL
      P(s,x;u,dy)
      P(u,y;t,A)

  for:
    s < u < t

STOCHASTIC_DIFFERENTIAL_EQUATION:
  process_model:
    dX_t :=
      a(X_t,t) dt
      + b(X_t,t) dW_t

  role:
    DEFINE_CONTINUOUS_TIME_STOCHASTIC_PROCESS
    UNDER:
      EXISTENCE_AND_UNIQUENESS_CONDITIONS

SIMULATION:
  INPUT:
    PROCESS_MODEL
    INITIAL_STATE
    PARAMETERS
    TIME_GRID_OR_EVENT_RULE

  FLOW:
    INITIALISE_STATE
      -> GENERATE_RANDOM_INPUT
      -> UPDATE_PROCESS_STATE
      -> RECORD_STATE
      -> REPEAT

  OUTPUT:
    SAMPLE_PATH

PROCESS_ESTIMATION:
  targets:
    - MEAN_FUNCTION
    - COVARIANCE_FUNCTION
    - TRANSITION_PARAMETERS
    - DRIFT
    - DIFFUSION
    - INTENSITY
    - AUTOCORRELATION

  depends_on:
    OBSERVATION_SCHEME
    MODEL_ASSUMPTIONS

MODEL_CONNECTIONS:
  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> STOCHASTIC_MATHEMATICS

  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> PROBABILITY_THEORY

  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> MARKOV_CHAINS

  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> MARTINGALES

  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> BROWNIAN_MOTION

  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> TIME_SERIES

  STOCHASTIC_PROCESS
    -> CONNECTS_TO
    -> STOCHASTIC_CALCULUS

APPLICATIONS:
  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> FINANCIAL_MODELLING

  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> QUEUEING_THEORY

  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> RELIABILITY

  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> SIGNAL_PROCESSING

  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> BIOLOGICAL_MODELLING

  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> PHYSICAL_SYSTEMS

  STOCHASTIC_PROCESS
    -> SUPPORTS
    -> MACHINE_LEARNING

SEMANTIC_EQUIVALENCE:
  "stochastic process"
    ~= "random process"

  "stochastic process"
    ~= "indexed family of random variables"

  "sample path"
    ~= "process realisation"

  "Wiener process"
    ~= "Brownian motion"

  EXCLUDE_EQUIVALENCE:
    STOCHASTIC_PROCESS != RANDOM_VARIABLE
    SAMPLE_PATH != PROCESS_DISTRIBUTION
    MARKOV_PROPERTY != INDEPENDENT_PROCESS_VALUES
    STATIONARY_PROCESS != INDEPENDENT_PROCESS
    STATIONARY_INCREMENTS != STATIONARY_PROCESS
    RANDOM_WALK != BROWNIAN_MOTION

VALIDATION_RULES:
  RULE_1:
    PROBABILITY_SPACE:
      MUST_BE:
        VALID

  RULE_2:
    INDEX_SET:
      MUST_BE:
        EXPLICIT

  RULE_3:
    STATE_SPACE:
      MUST_BE:
        DEFINED

  RULE_4:
    RANDOM_VARIABLES:
      MUST_BE:
        MEASURABLE

  RULE_5:
    DEPENDENCE_STRUCTURE:
      MUST_BE:
        EXPLICIT_OR_INFERABLE_FROM_MODEL

  RULE_6:
    STATIONARITY:
      MUST_NOT_BE:
        ASSUMED_WITHOUT_JUSTIFICATION

  RULE_7:
    MARKOV_PROPERTY:
      MUST_NOT_BE:
        CONFUSED_WITH_INDEPENDENCE

  RULE_8:
    SIMULATION:
      MUST_PRESERVE:
        DISTRIBUTIONAL_AND_DEPENDENCE_STRUCTURE

CONFIDENCE:
  ontology_identity: 0.99
  stochastic_process_definition: 0.99
  index_state_structure: 0.99
  stationarity_structure: 0.99
  markov_structure: 0.99
  filtration_structure: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    NEW_PROCESS_OBSERVATION_ARRIVES

  THEN:
    APPEND:
      PROCESS_HISTORY

    UPDATE_IF_REQUIRED:
      STATE_ESTIMATE
      PARAMETER_ESTIMATES
      COVARIANCE_ESTIMATES

  IF:
    NEW_INDEX_VALUE_ADDED

  THEN:
    EXTEND:
      INDEX_SET
      SAMPLE_PATH
      FILTRATION

  IF:
    MODEL_STRUCTURE_CHANGES

  THEN:
    REVALIDATE:
      STATE_SPACE
      DEPENDENCE_STRUCTURE
      STATIONARITY
      MARKOV_PROPERTY

  PRESERVE:
    PROCESS_IDENTITY
    INDEX_ORDER
    STATE_SPACE
    FILTRATION
    MODEL_PARAMETERS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_PROBABILITY_SPACE
    - IDENTIFY_INDEX_SET
    - IDENTIFY_STATE_SPACE
    - IDENTIFY_RANDOM_VARIABLE_FAMILY
    - IDENTIFY_SAMPLE_PATH_STRUCTURE
    - IDENTIFY_DEPENDENCE_STRUCTURE
    - IDENTIFY_FILTRATION_IF_PRESENT
    - TEST_STATIONARITY_IF_REQUIRED
    - TEST_MARKOV_PROPERTY_IF_REQUIRED
    - COMPUTE_PROCESS_MOMENTS
    - SIMULATE_OR_PREDICT_IF_REQUIRED
    - RETURN_PROCESS_INTERPRETATION

  constraints:
    - PRESERVE_INDEX_STRUCTURE
    - PRESERVE_STATE_SPACE
    - PRESERVE_DEPENDENCE_STRUCTURE
    - DO_NOT_EQUATE_MARKOV_PROPERTY_WITH_INDEPENDENCE
    - DO_NOT_ASSUME_STATIONARITY
    - DISTINGUISH_SAMPLE_PATH_FROM_PROCESS_DISTRIBUTION
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "stochastic-process"
  namespace: "probability-statistics/stochastic-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - INDEX_SET
    - STATE_SPACE
    - RANDOM_VARIABLE_FAMILY

END_FLOW