OU Processes

These are the classes that implement gaussian and non-gaussian Ornstein-Uhlenbeck process.

class quantflow.sp.ou.Vasicek(*, rate: ~typing.Annotated[float, ~annotated_types.Gt(gt=0)] = 1.0, kappa: ~typing.Annotated[float, ~annotated_types.Gt(gt=0)] = 1.0, bdlp: ~quantflow.sp.weiner.WeinerProcess = <factory>, theta: ~typing.Annotated[float, ~annotated_types.Gt(gt=0)] = 1.0)

Gaussian OU process, also know as the Vasiceck model.

Historically, the Vasicek model was used to model the short rate, but it can be used to model any process that reverts to a mean level at a rate proportional to the difference between the current level and the mean level.

\[dx_t = \kappa (\theta - x_t) dt + \sigma dw_t\]

It derives from IntensityProcess, although, it is not strictly an intensity process since it is not positive.

Methods:

analytical_cdf	Analytical cdf of the process at time t
analytical_mean	Analytical mean of the process at time t
analytical_pdf	Analytical pdf of the process at time t
analytical_variance	Analytical variance of the process at time t
characteristic_exponent	Characteristic exponent at time t for a given input parameter
integrated_log_laplace	The log-Laplace transform of the cumulative process:
sample	Generate random Paths from the process.
sample_from_draws	Sample Paths from the process given a set of draws

Attributes:

bdlp	Background driving Weiner process
model_config	Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].
theta	Mean rate \(\theta\)

analytical_cdf(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, x: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical cdf of the process at time t

Implement if available

analytical_mean(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical mean of the process at time t

Implement if available

analytical_pdf(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, x: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical pdf of the process at time t

Implement if available

analytical_variance(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical variance of the process at time t

Implement if available

characteristic_exponent(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Characteristic exponent at time t for a given input parameter

integrated_log_laplace(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

The log-Laplace transform of the cumulative process:

\[e^{\phi_{t, u}} = {\mathbb E} \left[e^{i u \int_0^t x_s ds}\right]\]

Parameters:

• t – time horizon

• u – frequency

sample(n: int, time_horizon: float = 1, time_steps: int = 100) → Paths

Generate random Paths from the process.

Parameters:

• n – number of paths

• time_horizon – time horizon

• time_steps – number of time steps to arrive at horizon

sample_from_draws(draws: Paths, *args: Paths) → Paths

Sample Paths from the process given a set of draws

bdlp: WeinerProcess

Background driving Weiner process

kappa: float

Mean reversion speed \(\kappa\)

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

rate: float

Instantaneous initial rate \(r_0\)

theta: float

Mean rate \(\theta\)

class quantflow.sp.ou.GammaOU(*, rate: Annotated[float, Gt(gt=0)] = 1.0, kappa: Annotated[float, Gt(gt=0)] = 1.0, bdlp: CompoundPoissonProcess[Exponential])

Methods:

analytical_mean	Analytical mean of the process at time t
analytical_pdf	Analytical pdf of the process at time t
analytical_variance	Analytical variance of the process at time t
characteristic_exponent	Characteristic exponent at time t for a given input parameter
cumulative_characteristic2	Formula from a paper
integrated_log_laplace	The log-Laplace transform of the cumulative process:
sample	Generate random Paths from the process.

Attributes:

model_config

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

analytical_mean(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical mean of the process at time t

Implement if available

analytical_pdf(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, x: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical pdf of the process at time t

Implement if available

analytical_variance(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical variance of the process at time t

Implement if available

characteristic_exponent(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Characteristic exponent at time t for a given input parameter

cumulative_characteristic2(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Formula from a paper

integrated_log_laplace(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

The log-Laplace transform of the cumulative process:

\[e^{\phi_{t, u}} = {\mathbb E} \left[e^{i u \int_0^t x_s ds}\right]\]

Parameters:

• t – time horizon

• u – frequency

sample(n: int, time_horizon: float = 1, time_steps: int = 100) → Paths

Generate random Paths from the process.

Parameters:

• n – number of paths

• time_horizon – time horizon

• time_steps – number of time steps to arrive at horizon

kappa: float

Mean reversion speed \(\kappa\)

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

rate: float

Instantaneous initial rate \(r_0\)