src/risk_math.py

"""VSCE continuous first-passage analysis for a planar isotropic OU process.

Model: dZ = -kappa*Z dt + sqrt(2*diffusion) dB, exit when |Z| >= radius.
Analytic bounds concern CONTINUOUS monitoring. Finite-volume probabilities are
numerical approximations, not interval-certified values. Classical first-passage
identities are rederived in the manuscript; none is claimed as newly discovered.
"""
from __future__ import annotations

import math
import numpy as np
from scipy.integrate import quad
from scipy.linalg import eigh_tridiagonal, solve_banded
from scipy.optimize import minimize_scalar, brentq
from scipy.special import hyp1f1

MAX_ANALYTIC_BETA = 650.0
MAX_FV_BETA = 15.0
FV_EXIT_RESOLUTION = 1e-11


def _positive_upper_from_log(logvalue: float) -> float:
    """Cap at one and round positive underflow upward, never to false zero."""
    return max(float(np.nextafter(0.0,1.0)),math.exp(min(0.0,logvalue)))


def _parameters(radius: float, kappa: float, diffusion: float,
                initial_radius: float = 0.0) -> tuple[float, float]:
    if not all(math.isfinite(v) for v in (radius, kappa, diffusion, initial_radius)):
        raise ValueError("parameters must be finite")
    if radius <= 0 or kappa <= 0 or diffusion <= 0:
        raise ValueError("radius, kappa and diffusion must be positive")
    if initial_radius < 0 or initial_radius > radius:
        raise ValueError("initial_radius must be in [0,radius]")
    beta = kappa * radius * radius / (2 * diffusion)
    x0 = kappa * initial_radius * initial_radius / (2 * diffusion)
    if not math.isfinite(beta) or beta <= 0 or beta > MAX_ANALYTIC_BETA:
        raise ValueError("dimensionless beta exceeds supported range (beta<=650)")
    return beta,x0


def _h_integral(x: float) -> float:
    """Integral_0^x expm1(u)/u du using a strictly positive series."""
    if x < 0:
        raise ValueError("x must be nonnegative")
    if x == 0:
        return 0.0
    if x > MAX_ANALYTIC_BETA:
        raise ValueError("Poisson integral supports x<=650; use log/arbitrary precision")
    term = x
    total = term
    for n in range(1, 100000):
        term *= x * n / (n + 1) ** 2
        total += term
        if n > x and term <= total * 2e-16:
            return total
    raise RuntimeError("mean-exit positive series did not converge")


def mean_exit_time(radius: float, kappa: float, diffusion: float,
                   initial_radius: float = 0.0) -> float:
    """Exact mean exit time (special-function identity evaluated numerically)."""
    beta, x0 = _parameters(radius, kappa, diffusion, initial_radius)
    if initial_radius == radius:
        return 0.0
    if x0 == 0:
        return _h_integral(beta) / kappa / 2
    # Direct integration prevents cancellation when the initial point is close
    # to the absorbing boundary; the integrand has removable value one at zero.
    val, _ = quad(lambda x: math.expm1(x) / x if x else 1.0,
                  x0, beta, epsabs=1e-12, epsrel=2e-12)
    return val / kappa / 2


def mean_exit_upper(radius: float, T: float, kappa: float, diffusion: float,
                    initial_radius: float = 0.0) -> float:
    """Stopped positive-Poisson certificate, not generic Markov on a mean.

    P(tau <= T) <= [h(x0)+T]/h(beta), Lh=1, h(0)=0.
    """
    if T < 0 or not math.isfinite(T):
        raise ValueError("T must be finite and nonnegative")
    beta, x0 = _parameters(radius, kappa, diffusion, initial_radius)
    if x0 == beta:
        return 1.0
    if T == 0:
        return 0.0
    hx0 = _h_integral(x0)
    log_initial = math.log(hx0) if hx0 else -math.inf
    log_time = math.log(2.0)+math.log(kappa)+math.log(T)
    log_numerator = float(np.logaddexp(log_initial,log_time))
    return _positive_upper_from_log(log_numerator-math.log(_h_integral(beta)))


def _log_kummer_positive(a: float, x: float) -> float:
    """log M(a,1,x), a,x >= 0, with a positive-series overflow fallback."""
    if a < 0 or x < 0:
        raise ValueError("positive arguments required")
    if a == 0 or x == 0:
        return 0.0
    direct = float(hyp1f1(a, 1.0, x))
    if math.isfinite(direct) and direct > 0:
        return math.log(direct)
    logterm = 0.0
    logsum = 0.0
    safe_after = math.ceil(x + math.sqrt(a*x) + 2)
    for n in range(1, 1000000):
        logterm += math.log(a+n-1) + math.log(x) - 2*math.log(n)
        logsum = float(np.logaddexp(logsum, logterm))
        if n > safe_after and logterm - logsum < math.log(2e-16):
            return logsum
    raise RuntimeError("Kummer positive series did not converge")


def log_laplace_exit(s: float, radius: float, kappa: float, diffusion: float,
                     initial_radius: float = 0.0) -> float:
    """log E[exp(-s*tau)] from the regular Kummer boundary-value solution."""
    if s < 0 or not math.isfinite(s):
        raise ValueError("s must be finite and nonnegative")
    beta, x0 = _parameters(radius, kappa, diffusion, initial_radius)
    a = s/(2*kappa)
    return _log_kummer_positive(a, x0) - _log_kummer_positive(a, beta)


def laplace_exit(s: float, radius: float, kappa: float, diffusion: float,
                 initial_radius: float = 0.0) -> float:
    return math.exp(log_laplace_exit(s, radius, kappa, diffusion, initial_radius))


def laplace_chernoff_upper(radius: float, T: float, kappa: float,
                          diffusion: float, initial_radius: float = 0.0) -> dict:
    """Optimize the valid bound exp(s*T)*E exp(-s*tau), s>=0.

    The analytic expression is a rigorous upper bound for every s; double
    precision evaluation/optimization is not an outward-rounded certificate.
    No reliance on the optimizer finding the global minimum is needed for bound
    validity. Returns the selected s and its log bound for rare-event reporting.
    """
    if T < 0 or not math.isfinite(T):
        raise ValueError("T must be finite and nonnegative")
    beta, x0 = _parameters(radius, kappa, diffusion, initial_radius)
    if x0 == beta:
        return {"upper":1.0,"log_upper":0.0,"s":0.0}
    if T == 0:
        return {"upper":0.0,"log_upper":-math.inf,"s":math.inf}
    if T >= mean_exit_time(radius,kappa,diffusion,initial_radius):
        return {"upper":1.0,"log_upper":0.0,"s":0.0}
    def objective(a):
        return (2*kappa*T*a + _log_kummer_positive(a,x0)
                - _log_kummer_positive(a,beta))
    high = 1.0
    old = objective(high)
    for _ in range(50):
        current = objective(2*high)
        if current >= old:
            high *= 2
            break
        high *= 2
        old = current
    result = minimize_scalar(objective,bounds=(0.0,high),method="bounded",
                             options={"xatol":1e-10})
    candidates = [(0.0,0.0),(float(result.fun),float(result.x)),
                  (objective(high),high)]
    logupper,a = min(candidates)
    min_positive = float(np.nextafter(0.0,1.0))
    underflow = logupper < math.log(min_positive)
    # Preserve a positive, outward-underflow-rounded value instead of returning
    # a false zero probability certificate at positive time and diffusion.
    upper = min_positive if underflow else math.exp(logupper)
    return {"upper":upper,"log_upper":logupper,"s":2*kappa*a,
            "underflow_rounded_up":underflow}


def additive_exponential_upper(radius: float, T: float, kappa: float,
                               diffusion: float,
                               initial_radius: float = 0.0) -> dict:
    """Optimize positive Lyapunov functions exp(a*x)-1, 0<a<1.

    Useful even when an exact Poisson solution is unavailable in extensions.
    """
    if T < 0 or not math.isfinite(T):
        raise ValueError("T must be finite and nonnegative")
    beta, x0 = _parameters(radius,kappa,diffusion,initial_radius)
    if x0 == beta:
        return {"upper":1.0,"a":0.5,"compensator":kappa}
    if T == 0:
        return {"upper":0.0,"a":0.5,"compensator":kappa}
    def bound(a):
        xstar = max(0.0,min(beta,(2*a-1)/(a*(1-a))))
        gstar = math.exp(a*xstar)*(1-(1-a)*xstar)
        c = 2*kappa*a*max(1.0,gstar)
        numerator = math.expm1(a*x0)+c*T
        if numerator == 0:
            upper = float(np.nextafter(0.0,1.0))
        else:
            upper = _positive_upper_from_log(math.log(numerator)-math.log(math.expm1(a*beta)))
        return upper,c
    opt = minimize_scalar(lambda a: bound(a)[0],bounds=(1e-8,1-1e-8),
                          method="bounded",options={"xatol":1e-12})
    upper,c = bound(float(opt.x))
    return {"upper":upper,"a":float(opt.x),"compensator":c}


def ou_continuous_bounds(radius: float, T: float, kappa: float,
                         diffusion: float) -> dict:
    """Bounds from a centered start; diffusion here is d, not D before q."""
    if T <= 0:
        raise ValueError("T must be positive")
    beta,_ = _parameters(radius,kappa,diffusion)
    endpoint = math.exp(-beta/(-math.expm1(-2*kappa*T)))
    direct = math.exp(min(0.0,2*kappa*T-beta))
    blocks = max(1,math.ceil(kappa*T))
    h = T/blocks
    block = min(1.0,blocks*(1+math.exp(2*kappa*h))*math.exp(-beta/4))
    mean = mean_exit_upper(radius,T,kappa,diffusion)
    chernoff = laplace_chernoff_upper(radius,T,kappa,diffusion)
    additive = additive_exponential_upper(radius,T,kappa,diffusion)
    return {"beta":beta,"endpoint_lower":endpoint,"v5_direct_upper":direct,
            "v5_block_upper":block,"poisson_upper":mean,
            "laplace_upper":chernoff["upper"],"laplace_s":chernoff["s"],
            "laplace_log_upper":chernoff["log_upper"],
            "additive_upper":additive["upper"],
            "upper":min(direct,block,mean,chernoff["upper"],additive["upper"]),
            "mean_exit_time":mean_exit_time(radius,kappa,diffusion)}


def poisson_clearance(T: float, kappa: float, diffusion: float,
                       error_budget: float, events: int = 1) -> float:
    """Choose common radius so the union of ``events`` exits has risk <= budget.

    Numerical root of the exact monotone sufficient inequality; independent
    events are not assumed. The returned radius is rounded slightly upward.
    """
    if T<=0 or kappa<=0 or diffusion<=0 or events<1 or int(events)!=events:
        raise ValueError("positive T,kappa,diffusion and integer events required")
    if not 0<error_budget<1:
        raise ValueError("error_budget must be between zero and one")
    target=2*kappa*events*T/error_budget
    if not math.isfinite(target) or target > _h_integral(MAX_ANALYTIC_BETA):
        raise ValueError("clearance target exceeds supported beta<=650")
    high=1.0
    while _h_integral(high)<target:
        high=min(MAX_ANALYTIC_BETA,high*2)
    beta=brentq(lambda b:_h_integral(b)-target,0.0,high,xtol=1e-13)
    return math.sqrt(2*diffusion*beta/kappa)*(1+2e-14)


def _fv_generator(beta: float, kappa: float, cells: int):
    """Weighted nodal finite-volume generator, absorbing node x=beta.

    Nodes i*beta/cells, i=0,...,cells-1, plus Dirichlet node beta. The
    regular origin has zero weighted flux. Off-diagonal rates are positive.
    """
    if not (0 < beta <= MAX_FV_BETA) or not 8 <= cells <= 2000 or kappa <= 0:
        raise ValueError("validated FV support: 0<beta<=15, 8<=cells<=2000, kappa>0")
    h = beta/cells
    x = np.arange(cells)*h
    left = np.maximum(0.0,x-h/2)
    right = x+h/2
    weights = np.exp(-left)*(-np.expm1(-(right-left)))
    face_x = (np.arange(cells)+0.5)*h
    conductance = 2*kappa*face_x*np.exp(-face_x)/h
    upper = conductance/weights
    lower = np.zeros(cells)
    lower[1:] = conductance[:-1]/weights[1:]
    diagonal = -(upper+lower)
    return x,weights,diagonal,upper,lower


def survival_fv(radius: float, times, kappa: float, diffusion: float,
                 cells: int = 400) -> dict:
    """Continuous-time killed-process PDE estimate from a centered start.

    Positive spatial discretization; exact matrix exponential in time (to
    floating precision). No Monte Carlo or sample monitoring is involved.
    Grid convergence and independent transform/MFPT checks remain necessary.
    Positive-time exits below 1e-11 are marked NaN/unresolved rather than zero.
    """
    beta,_ = _parameters(radius,kappa,diffusion)
    times = np.atleast_1d(np.asarray(times,dtype=float))
    if np.any(~np.isfinite(times)) or np.any(times<0):
        raise ValueError("times must be finite and nonnegative")
    x,w,diag,upper,lower = _fv_generator(beta,kappa,cells)
    offdiag = np.sqrt(upper[:-1]*lower[1:])
    eig,vec = eigh_tridiagonal(diag,offdiag)
    if eig[-1]>=0:
        raise ArithmeticError("principal eigenvalue unresolved; FV estimate rejected")
    coefficients = vec[0,:]*(vec.T@np.sqrt(w))/math.sqrt(w[0])
    raw = np.exp(np.outer(times,eig))@coefficients
    # Tiny clipping corrects roundoff, never supplies a certified interval.
    survival = np.clip(raw,0.0,1.0)
    survival[times==0] = 1.0
    probability = 1-survival
    resolved = (times==0)|(probability>FV_EXIT_RESOLUTION)
    probability[~resolved] = np.nan
    return {"times":times,"survival":survival,"exit_probability":probability,
            "cells":cells,"beta":beta,"principal_eigenvalue":float(-eig[-1]),
            "roundoff_clip_max":float(np.max(np.abs(raw-survival))),
            "resolved":resolved,"exit_resolution_floor":FV_EXIT_RESOLUTION,
            "max_supported_beta":MAX_FV_BETA,
            "interpretation":"PDE estimate; NaN means exit probability below numerical resolution"}


def mean_exit_fv(radius: float, kappa: float, diffusion: float,
                 cells: int = 400) -> float:
    beta,_ = _parameters(radius,kappa,diffusion)
    _,_,diag,upper,lower = _fv_generator(beta,kappa,cells)
    band = np.zeros((3,cells))
    band[0,1:] = -upper[:-1]
    band[1,:] = -diag
    band[2,:-1] = -lower[1:]
    return float(solve_banded((1,1),band,np.ones(cells))[0])


def laplace_exit_fv(s: float, radius: float, kappa: float, diffusion: float,
                    cells: int = 400) -> float:
    if s<0:
        raise ValueError("s must be nonnegative")
    beta,_ = _parameters(radius,kappa,diffusion)
    _,_,diag,upper,lower = _fv_generator(beta,kappa,cells)
    band = np.zeros((3,cells))
    band[0,1:] = -upper[:-1]
    band[1,:] = s-diag
    band[2,:-1] = -lower[1:]
    rhs = np.zeros(cells)
    rhs[-1] = upper[-1]  # absorbing boundary has Laplace value one
    return float(solve_banded((1,1),band,rhs)[0])


def simulate_nested_sampled_exit(radius: float, T: float, kappa: float,
                                  diffusion: float, fine_dt: float,
                                  sample_strides=(1,2,4,10), trials=20000,
                                  seed=20261013) -> dict:
    """Exact OU transitions, nested observation grids; every rate is a LOWER
    approximation to continuous monitoring, including the finest grid.
    """
    _parameters(radius,kappa,diffusion)
    steps = round(T/fine_dt)
    if steps<1 or abs(steps*fine_dt-T)>1e-10:
        raise ValueError("fine_dt must divide positive T")
    strides = sorted(set(map(int,sample_strides)))
    if any(s<1 or steps%s for s in strides) or trials<1:
        raise ValueError("positive strides must divide number of steps")
    rng = np.random.default_rng(seed)
    z = np.zeros((trials,2))
    hit = {s:np.zeros(trials,dtype=bool) for s in strides}
    decay = math.exp(-kappa*fine_dt)
    sigma = math.sqrt(diffusion/kappa*(-math.expm1(-2*kappa*fine_dt)))
    for step in range(1,steps+1):
        z = decay*z+sigma*rng.standard_normal((trials,2))
        outside = np.sum(z*z,axis=1)>=radius*radius
        for s in strides:
            if step%s==0:
                hit[s] |= outside
    return {"trials":trials,"seed":seed,"fine_dt":fine_dt,
            "T":T,"radius":radius,"kappa":kappa,"diffusion":diffusion,
            "counts":{str(s):int(hit[s].sum()) for s in strides},
            "rates":{str(s):float(hit[s].mean()) for s in strides},
            "interpretation":"sampled first exits; all rates underdetect continuous first exits"}