Stochastic Integrals

increments

Stochastic integrals for diffusion tems in SDEs.

For a typical SDE in differential form, \(d\mathbf{X}_t = \mathbf{b}(\mathbf{X}_t,t)dt + \mathbf{\Sigma}(X_t,t)d\mathbf{W}_t\), objects derived from BaseRandomIncrement generate the Wiener process increments \(d\mathbf{W}_t\). In the base class, we assume basically nothing about what this class actually returns in view of the numerical integrator. This means that an implementation could return a simple random increment vector for first order schemes, as well as more complex combinations of Lévy areas. The ABC simply enforces an interface for the sample method.

Classes:

Name	Description
BaseRandomIncrement	ABC of stochastic integrals for diffusion tems in SDEs.
BrownianIncrement	Simple implementation of a Wiener process increment for first order schemes.

pysde.increments.BaseRandomIncrement

Bases: ABC

ABC of stochastic integrals for diffusion tems in SDEs.

Implements a minimal interface for stochastic integrals in numerical integration schemes.

Methods:

Name	Description
sample	Generate a random increment.

__init__

__init__(seed: int = 0) -> None

Initialize a random increment with at least a seed.

Parameters:

Name	Type	Description	Default
seed	int	The seed for the random number generator. Defaults to 0.	0

sample abstractmethod

sample(_dimension: Annotated[int, Is[lambda x: x > 0]], _num_trajectories: Annotated[int, Is[lambda x: x > 0]], _step_size: Real, _dtype: np.float32 | np.float64) -> object

Generate a random increment.

Stochastic integrals take at least a step size \(\Delta t\), a physical dimension \(d_W\), and the number of trajectories \(N\). Generation of the random increment has to be vectorized over \(N\).

Make sure to use the right dtype

For the SDE sample to be floating point agnostic, the sample method in all derived classes needs to return arrays of the correct dtype, as provided through the function argument.

Parameters:

Name	Type	Description	Default
_dimension	int	Physical dimension \(d_W\) of the increment	required
_num_trajectories	int	Number of trajectories \(N\)	required
_step_size	Real	Discrete step size \(\nabla t\) of the integrator	required
_dtype	np.float32 | np.float64	Data type of the random increment, can be 32 or 64 bits	required

Raises: NotImplementedError: Exception indicating that the method needs to be implemented in derived classes.

Returns:

Name	Type	Description
object	object	Random increment

pysde.increments.BrownianIncrement

Bases: BaseRandomIncrement

Simple implementation of a Wiener process increment for first order schemes.

For a step size \(\Delta t\), we approximate \(dW_t\) simply as \(\sqrt{\Delta t}\xi\), where \(\xi\) is a vector of i.i.d. unit normal samples. The sample is implemented to also work with \(\Delta t < 0\) for integration backwards in time.

sample

sample(dimension: Annotated[int, Is[lambda x: x > 0]], num_trajectories: Annotated[int, Is[lambda x: x > 0]], step_size: Real, dtype: object) -> npt.NDArray[np.floating]

Generate a random increment \(dW_t\).

Parameters:

Name	Type	Description	Default
dimension	int	Physical dimension \(d_W\) of the increment	required
num_trajectories	int	Number of trajectories \(N\)	required
step_size	Real	Discrete step size \(\nabla t\) of the integrator	required
dtype	np.float32 | np.float64	Data type of the random increment, can be 32 or 64 bits	required

Returns: npt.NDArray[np.floating]: Sample of dimension \(d_W\times N\)