svStochasticIntegratorW2Ito

struct W2ItoCoefficients

#include <svStochasticIntegratorW2Ito.h>

Extended Butcher tableau for the Tang & Xiao weak-order-2 SRK family.

The coefficients characterise a specific member of the general weak-order-2 stochastic Runge-Kutta scheme for Ito SDEs. The number of stages \(s\) is inferred from the length of alpha. The tableau is

\[\begin{split} \begin{array}{c|c|c} & A^{(0)} & B^{(0)} \\ \hline & A^{(1)} & B^{(1)}\; B^{(2)} \\ \hline & \alpha & \beta^{(0)}\; \beta^{(1)} \end{array} \end{split}\]

with \(A^{(0)}, A^{(1)}, B^{(0)}, B^{(1)}, B^{(2)}\) strictly lower-triangular \(s\times s\) matrices (explicit method) and \(\alpha, \beta^{(0)}, \beta^{(1)}\) length- \(s\) weight vectors.

Public Functions

inline size_t numStages() const: Number of stages, inferred from the length of alpha.

inline double c0(size_t i) const: Drift stage node \(c^{(0)}_i = \sum_j A^{(0)}_{ij}\) (row sum of A0).

inline double c1(size_t i) const: Diffusion stage node \(c^{(1)}_i = \sum_j A^{(1)}_{ij}\) (row sum of A1).

Public Members

std::vector<double> alpha: drift weights, length s

std::vector<double> beta0: diffusion weights on the three-point increment, length s

std::vector<double> beta1: diffusion weights on the diagonal iterated integral, length s

std::vector<std::vector<double>> A0: drift-to-drift stage matrix, s x s

std::vector<std::vector<double>> B0: diffusion-to-drift stage matrix, s x s

std::vector<std::vector<double>> A1: drift-to-diffusion stage matrix, s x s

std::vector<std::vector<double>> B1: self-noise diffusion-to-diffusion stage matrix, s x s

std::vector<std::vector<double>> B2: cross-noise diffusion-to-diffusion stage matrix, s x s

class svStochasticIntegratorW2Ito : public StochasticRKIntegratorBase 

#include <svStochasticIntegratorW2Ito.h>

The svStochasticIntegratorW2Ito class implements the Tang & Xiao efficient weak second-order (deterministic order 3+) Runge-Kutta family for Ito SDEs.

Tang, X., Xiao, A. "Efficient weak second-order stochastic Runge-Kutta methods
for Ito stochastic differential equations", BIT Numer. Math. 57, 241-260 (2017).
https://doi.org/10.1007/s10543-016-0618-9

This is the shared base for the concrete W2Ito1 and W2Ito2 methods, which differ only in their coefficient tableau (see svStochasticIntegratorW2Ito1 and svStochasticIntegratorW2Ito2). It shares the pluggable noise-generator architecture of the other native stochastic integrators, so the same prescribed-noise test harness drives every method unchanged.

For an \(s\)-stage tableau the step (paper eq. 3.1) is

\[ y_{n+1} = y_n + \sum_i \alpha_i f(H^{(0)}_i)\,h + \sum_i \sum_k \beta^{(0)}_i g_k(H^{(k)}_i)\,\hat I_k + \sum_i \sum_k \beta^{(1)}_i g_k(H^{(k)}_i)\,\hat I_{(k,k)} \]

with drift stages \(H^{(0)}_i\) and per-noise-source diffusion stages \(H^{(k)}_i\). The driving random variables are a three-point increment \(\hat I_k \in \{\pm\sqrt{3h}, 0\}\) (from the Gaussian \(\Delta W_k\)), a two-point scale \(\xi = \eta_1\sqrt h\) (from the Gaussian \(\Delta Z_0\)), and the mixed iterated integrals \(\hat I_{(k,l)} = \tfrac12(\hat I_l \mp \eta_2 \hat I_l)\) for \(k\neq l\) (using a second two-point sign \(\eta_2\) from \(\Delta Z_1\)) and \(\hat I_{(k,k)} = \tfrac12(\hat I_k^2/\xi - \xi)\) on the diagonal. The step is explicit and derivative-free.

Warning

Stochastic integration is in beta.

Subclassed by svStochasticIntegratorW2Ito1, svStochasticIntegratorW2Ito2

Public Functions

virtual void integrate(double currentTime, double timeStep) override: Performs the integration of the associated dynamic objects up to time currentTime+timeStep

Protected Functions

svStochasticIntegratorW2Ito(DynamicObject *dyn, const W2ItoCoefficients &coefficients): Constructor used by the concrete W2Ito methods, which supply their tableau.

ExtendedStateVector scaledSum(const std::vector<double> &factors, const std::vector<ExtendedStateVector> &vectors, size_t length): Returns sum_{j<length} factors[j]*vectors[j] (skipping zero factors).

Protected Attributes

const W2ItoCoefficients coefficients: The extended Butcher tableau of the specific W2Ito method.