Brownian Motion Ontology

flow "BrownianMotionOntology" v4.0

# ============================================================
# BROWNIAN MOTION 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/brownian-motion-ontology.html"
  title: "Brownian Motion Ontology"
  domain: "Probability and Statistics"
  category: "Stochastic Mathematics"
  entity: BROWNIAN_MOTION
  aliases:
    - WIENER_PROCESS
    - STANDARD_BROWNIAN_MOTION
  entity_type: CONTINUOUS_TIME_STOCHASTIC_PROCESS

DEFINE_BROWNIAN_MOTION:
  BROWNIAN_MOTION :=
    STOCHASTIC_PROCESS(
      CONTINUOUS_TIME,
      CONTINUOUS_STATE,
      GAUSSIAN_INCREMENTS,
      INDEPENDENT_INCREMENTS,
      CONTINUOUS_SAMPLE_PATHS
    )

CORE_STRUCTURE:
  PROCESS:
    W := {W_t : t >= 0}

  INITIAL_STATE:
    W_0 := 0
    ALMOST_SURELY

  INCREMENT:
    W_t - W_s

  FOR:
    0 <= s < t

  DISTRIBUTION:
    W_t - W_s
      ~
    NORMAL(
      mean=0,
      variance=t-s
    )

ENTITIES:
  BROWNIAN_MOTION:
    type: STOCHASTIC_PROCESS

  TIME:
    type: CONTINUOUS_INDEX

  STATE:
    type: REAL_VALUE

  INCREMENT:
    type: RANDOM_VARIABLE

  SAMPLE_PATH:
    type: CONTINUOUS_RANDOM_FUNCTION

  GAUSSIAN_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  FILTRATION:
    type: INFORMATION_STRUCTURE

  QUADRATIC_VARIATION:
    type: PATHWISE_LIMIT_QUANTITY

  HITTING_TIME:
    type: RANDOM_TIME

RELATIONSHIPS:
  BROWNIAN_MOTION
    -> IS_A
    -> STOCHASTIC_PROCESS

  BROWNIAN_MOTION
    -> HAS
    -> CONTINUOUS_SAMPLE_PATHS

  BROWNIAN_MOTION
    -> HAS
    -> INDEPENDENT_INCREMENTS

  BROWNIAN_MOTION
    -> HAS
    -> STATIONARY_INCREMENTS

  BROWNIAN_MOTION
    -> HAS
    -> GAUSSIAN_INCREMENTS

  BROWNIAN_MOTION
    -> IS
    -> MARKOV_PROCESS

  BROWNIAN_MOTION
    -> IS
    -> MARTINGALE
    WITH_RESPECT_TO:
      NATURAL_OR_COMPATIBLE_FILTRATION

DEFINING_PROPERTIES:
  PROPERTY_1:
    W_0 = 0
    ALMOST_SURELY

  PROPERTY_2:
    FOR_DISJOINT_TIME_INTERVALS:
      INCREMENTS:
        INDEPENDENT

  PROPERTY_3:
    FOR:
      0 <= s < t

    W_t - W_s
      ~
    N(0,t-s)

  PROPERTY_4:
    t -> W_t:
      CONTINUOUS
    ALMOST_SURELY

MARGINAL_DISTRIBUTION:
  FOR:
    t >= 0

  W_t:
    ~ N(0,t)

  expectation:
    E[W_t] := 0

  variance:
    Var(W_t) := t

COVARIANCE:
  FOR:
    s,t >= 0

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

  derivation_context:
    IF:
      s <= t

    THEN:
      W_t :=
        W_s
        + (W_t-W_s)

      Cov(W_s,W_t)
        =
      Var(W_s)
        =
      s

CORRELATION:
  FOR:
    s,t > 0

  Corr(W_s,W_t) :=
    min(s,t)
    /
    sqrt(s*t)

STATIONARY_INCREMENTS:
  FOR:
    h >= 0

  W_(t+h) - W_t
    ~
  N(0,h)

  implication:
    INCREMENT_DISTRIBUTION:
      DEPENDS_ON:
        INTERVAL_LENGTH
      NOT:
        ABSOLUTE_TIME

INDEPENDENT_INCREMENTS:
  FOR:
    0 <= t_0 < t_1 < ... < t_n

  increments:
    W_t1-W_t0
    W_t2-W_t1
    ...
    W_tn-W_t(n-1)

  are:
    MUTUALLY_INDEPENDENT

GAUSSIAN_PROCESS:
  BROWNIAN_MOTION:
    IS_A:
      GAUSSIAN_PROCESS

  finite_dimensional_distributions:
    MULTIVARIATE_NORMAL

  mean_function:
    m(t) := 0

  covariance_function:
    k(s,t) := min(s,t)

MARKOV_PROPERTY:
  FUTURE:
    CONDITIONALLY_INDEPENDENT_OF:
      PAST
    GIVEN:
      PRESENT_STATE

  transition_distribution:
    W_t | W_s=x
      ~
    N(
      x,
      t-s
    )
    FOR:
      t > s

TRANSITION_DENSITY:
  p(s,x;t,y) :=
    1 / sqrt(2*pi*(t-s))
    *
    exp(
      -(y-x)^2
      /
      (2*(t-s))
    )

  FOR:
    t > s

MARTINGALE_PROPERTY:
  WITH_RESPECT_TO:
    APPROPRIATE_FILTRATION {F_t}

  FOR:
    s <= t

  E[W_t | F_s] :=
    W_s

  integrability:
    E[abs(W_t)] < infinity

NATURAL_FILTRATION:
  F_t^W :=
    sigma(
      W_s :
      0 <= s <= t
    )

  role:
    INFORMATION_GENERATED_BY_PROCESS_UP_TO_TIME_t

SCALING_PROPERTY:
  FOR:
    c > 0

  process:
    {W_(c*t) : t >= 0}

  equal_in_distribution_to:
    {sqrt(c) * W_t : t >= 0}

  interpretation:
    BROWNIAN_MOTION_IS_SELF_SIMILAR
    WITH:
      HURST_EXPONENT = 1/2

SYMMETRY:
  process:
    {-W_t}

  equal_in_distribution_to:
    {W_t}

TIME_HOMOGENEITY:
  transition_distribution:
    DEPENDS_ON:
      t-s
    NOT:
      ABSOLUTE_TIME

QUADRATIC_VARIATION:
  FOR_PARTITION:
    0 = t_0 < t_1 < ... < t_n = T

  sum:
    Q_n :=
      SUM[i=1..n]
        (W_ti - W_t(i-1))^2

  AS:
    MAX_PARTITION_WIDTH -> 0

  Q_n:
    -> T
    IN:
      PROBABILITY
    AND:
      UNDER_STANDARD_PARTITION_SEQUENCES
      ALMOST_SURELY

  notation:
    [W]_T := T

TOTAL_VARIATION:
  Brownian_sample_path:
    HAS:
      INFINITE_TOTAL_VARIATION
    ON:
      EVERY_NONTRIVIAL_INTERVAL
    ALMOST_SURELY

DIFFERENTIABILITY:
  Brownian_sample_path:
    CONTINUOUS = TRUE

  BUT:
    NOWHERE_DIFFERENTIABLE
    ALMOST_SURELY

  implication:
    ORDINARY_CLASSICAL_DERIVATIVE:
      NOT_AVAILABLE_PATHWISE

ITO_CALCULUS_CONNECTION:
  Brownian_motion:
    DRIVES:
      ITO_STOCHASTIC_INTEGRALS
      STOCHASTIC_DIFFERENTIAL_EQUATIONS

  notation:
    INTEGRAL H_t dW_t

  role:
    FUNDAMENTAL_NOISE_SOURCE
    IN:
      STOCHASTIC_CALCULUS

ITO_ISOMETRY:
  E[
    (INTEGRAL_0^T H_t dW_t)^2
  ]
  =
  E[
    INTEGRAL_0^T H_t^2 dt
  ]

  under:
    APPROPRIATE_SQUARE_INTEGRABILITY_AND_ADAPTEDNESS

ITO_FORMULA:
  for:
    dX_t :=
      a_t dt
      + b_t dW_t

  function:
    f(t,X_t)

  differential:
    df :=
      (
        partial_t_f
        + a_t*partial_x_f
        + 0.5*b_t^2*partial_xx_f
      ) dt
      + b_t*partial_x_f dW_t

FIRST_PASSAGE_TIME:
  FOR_LEVEL:
    a

  tau_a :=
    inf{
      t >= 0 :
      W_t = a
    }

  type:
    STOPPING_TIME
    UNDER:
      NATURAL_FILTRATION

REFLECTION_PRINCIPLE:
  FOR:
    a > 0
    t > 0

  relation:
    P(
      max_{0<=s<=t} W_s >= a
    )
    =
    2 * P(W_t >= a)

  role:
    DERIVE:
      HITTING_PROBABILITIES
      MAXIMUM_DISTRIBUTIONS

MAXIMUM_PROCESS:
  M_t :=
    max{
      W_s :
      0 <= s <= t
    }

  distribution:
    RELATED_TO:
      ABSOLUTE_NORMAL_DISTRIBUTION
    THROUGH:
      REFLECTION_PRINCIPLE

BROWNIAN_MOTION_WITH_DRIFT:
  process:
    X_t :=
      mu*t
      + sigma*W_t

  expectation:
    E[X_t] :=
      mu*t

  variance:
    Var(X_t) :=
      sigma^2*t

  note:
    STANDARD_BROWNIAN_MOTION:
      mu = 0
      sigma = 1

GEOMETRIC_BROWNIAN_MOTION:
  SDE:
    dS_t :=
      mu*S_t dt
      + sigma*S_t dW_t

  solution:
    S_t :=
      S_0
      * exp(
          (mu - 0.5*sigma^2)*t
          + sigma*W_t
        )

  applications:
    FINANCIAL_MODELLING

DONSKER_CONNECTION:
  random_walk_scaling:
    APPROPRIATELY_NORMALISED_RANDOM_WALK
      -> CONVERGES_IN_DISTRIBUTION_TO
      -> BROWNIAN_MOTION

  theorem:
    FUNCTIONAL_CENTRAL_LIMIT_THEOREM

DIFFUSION_CONNECTION:
  Brownian_motion:
    BASIC_DIFFUSION_PROCESS

  generator:
    L :=
      0.5 * second_derivative_operator

  forward_density_equation:
    partial_t p :=
      0.5 * partial_xx p

  relation:
    HEAT_EQUATION

SIMULATION:
  TIME_GRID:
    0=t_0 < t_1 < ... < t_n=T

  INITIALISE:
    W_0 := 0

  FOR_EACH_STEP_i:
    Delta_t :=
      t_i - t_(i-1)

    Z_i :=
      STANDARD_NORMAL_RANDOM_VARIABLE

    Delta_W_i :=
      sqrt(Delta_t) * Z_i

    W_ti :=
      W_t(i-1)
      + Delta_W_i

  OUTPUT:
    DISCRETE_APPROXIMATION_TO_SAMPLE_PATH

MULTIDIMENSIONAL_BROWNIAN_MOTION:
  process:
    W_t :=
      [W_t^(1), ..., W_t^(d)]^T

  standard_case:
    components:
      INDEPENDENT_STANDARD_BROWNIAN_MOTIONS

  covariance:
    Cov(W_t) :=
      t * I_d

CORRELATED_BROWNIAN_MOTION:
  construct:
    X_t :=
      L W_t

  covariance:
    Cov(X_t) :=
      t * L L^T

  use:
    MODEL_CORRELATED_DIFFUSION_COMPONENTS

APPLICATIONS:
  BROWNIAN_MOTION
    -> SUPPORTS
    -> STOCHASTIC_CALCULUS

  BROWNIAN_MOTION
    -> SUPPORTS
    -> FINANCIAL_MATHEMATICS

  BROWNIAN_MOTION
    -> SUPPORTS
    -> DIFFUSION_MODELLING

  BROWNIAN_MOTION
    -> SUPPORTS
    -> PHYSICS

  BROWNIAN_MOTION
    -> SUPPORTS
    -> STOCHASTIC_DIFFERENTIAL_EQUATIONS

  BROWNIAN_MOTION
    -> SUPPORTS
    -> FIRST_PASSAGE_MODELS

MODEL_CONNECTIONS:
  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> STOCHASTIC_PROCESS

  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> GAUSSIAN_PROCESS

  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> MARKOV_PROCESS

  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> MARTINGALE

  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> ITO_CALCULUS

  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> RANDOM_WALK

  BROWNIAN_MOTION
    -> CONNECTS_TO
    -> DIFFUSION_PROCESS

SEMANTIC_EQUIVALENCE:
  "Brownian motion"
    ~= "Wiener process"

  "standard Brownian motion"
    ~= "zero-drift unit-variance Brownian motion"

  "Brownian increment"
    ~= "Gaussian process increment with variance equal to elapsed time"

  EXCLUDE_EQUIVALENCE:
    BROWNIAN_MOTION != RANDOM_WALK
    BROWNIAN_MOTION != GEOMETRIC_BROWNIAN_MOTION
    BROWNIAN_MOTION != WHITE_NOISE
    CONTINUOUS_SAMPLE_PATH != DIFFERENTIABLE_SAMPLE_PATH
    INDEPENDENT_INCREMENTS != INDEPENDENT_PROCESS_VALUES
    STATIONARY_INCREMENTS != STATIONARY_PROCESS

VALIDATION_RULES:
  RULE_1:
    INITIAL_STATE:
      MUST_SATISFY:
        W_0 = 0
      FOR:
        STANDARD_BROWNIAN_MOTION

  RULE_2:
    INCREMENTS:
      MUST_BE:
        INDEPENDENT
      ON:
        DISJOINT_INTERVALS

  RULE_3:
    INCREMENT_VARIANCE:
      MUST_EQUAL:
        ELAPSED_TIME
      FOR:
        STANDARD_BROWNIAN_MOTION

  RULE_4:
    SAMPLE_PATHS:
      MUST_BE:
        CONTINUOUS
      ALMOST_SURELY

  RULE_5:
    PROCESS_VALUES:
      MUST_NOT_BE:
        TREATED_AS_INDEPENDENT_ACROSS_TIME

  RULE_6:
    STOCHASTIC_DIFFERENTIALS:
      MUST_BE:
        INTERPRETED_USING_STOCHASTIC_CALCULUS

  RULE_7:
    SIMULATION:
      MUST_SCALE_INCREMENT_STANDARD_DEVIATION_BY:
        sqrt(Delta_t)

CONFIDENCE:
  ontology_identity: 0.99
  defining_properties: 0.99
  gaussian_structure: 0.99
  covariance_structure: 0.99
  markov_martingale_structure: 0.99
  quadratic_variation_structure: 0.99
  stochastic_calculus_relationship: 0.99

UPDATE_MODEL:
  IF:
    NEW_TIME_POINT_ADDED

  THEN:
    GENERATE_OR_OBSERVE:
      NEW_INCREMENT

    UPDATE:
      SAMPLE_PATH
      FILTRATION

  IF:
    DRIFT_OR_SCALE_PARAMETER_ADDED

  THEN:
    TRANSFORM:
      STANDARD_BROWNIAN_MOTION
      INTO:
        DRIFTED_OR_SCALED_PROCESS

  IF:
    MULTIDIMENSIONAL_DEPENDENCE_ADDED

  THEN:
    DEFINE:
      COVARIANCE_MATRIX
      LINEAR_TRANSFORMATION

  PRESERVE:
    INITIAL_STATE
    TIME_ORDER
    INCREMENT_STRUCTURE
    FILTRATION
    RANDOM_SEED_WHEN_RECORDED
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_STANDARD_OR_TRANSFORMED_BROWNIAN_PROCESS
    - VERIFY_INITIAL_STATE
    - IDENTIFY_TIME_INTERVALS
    - VERIFY_GAUSSIAN_INCREMENT_STRUCTURE
    - VERIFY_INDEPENDENT_INCREMENTS
    - COMPUTE_MEAN_VARIANCE_AND_COVARIANCE
    - IDENTIFY_FILTRATION
    - APPLY_MARKOV_OR_MARTINGALE_PROPERTIES_IF_REQUIRED
    - APPLY_ITO_CALCULUS_IF_REQUIRED
    - COMPUTE_HITTING_OR_MAXIMUM_QUANTITIES_IF_REQUIRED
    - SIMULATE_SAMPLE_PATH_IF_REQUIRED
    - RETURN_STOCHASTIC_INTERPRETATION

  constraints:
    - PRESERVE_CONTINUOUS_TIME_STRUCTURE
    - PRESERVE_INDEPENDENT_INCREMENT_STRUCTURE
    - DO_NOT_TREAT_PROCESS_VALUES_AS_INDEPENDENT
    - DISTINGUISH_STATIONARY_INCREMENTS_FROM_STATIONARY_PROCESS
    - DISTINGUISH_BROWNIAN_MOTION_FROM_RANDOM_WALK
    - USE_STOCHASTIC_CALCULUS_FOR_DIFFERENTIAL_REASONING
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "brownian-motion"
  namespace: "probability-statistics/stochastic-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - GAUSSIAN_INCREMENTS
    - INDEPENDENT_INCREMENTS
    - CONTINUOUS_SAMPLE_PATHS

END_FLOW