This module is a toolbox for generating and calibrating Diffusion Models. A model is derived from the ItoProcess class. The model is described in terms of the underying stochastic differential equation (SDE):

$$dX_i = a_i(t, X) dt + \sum_{i, j = 1}^n S_{ij}(t, X) dW_j,$$

where $$n$$ is the dimensionality of the process $$a_i$$ and $$S_{ij}$$ are the drift and volatility terms of the SDE. $${ W_i }_{i=1}^n$$ is an $$n$$-dimensional Wiener process.

For example, the Geometric Brownian Motion is an underlying of the Black-Scholes model.

A minimal description of the model class should contain the following methods:

• dim - dimensionality of the underlying SDE;
• dtype - dtype of the model coefficients;
• name - name of the model class;
• drift_fn - drift rate of the process expressed as a callable which maps time and position to a vector. (corresponds to the $$a_i$$ above);
• volatility_fn - volatility of the process expressed as a callable which maps time and position to a volatility matrix(corresponds to the $$S_{ij}$$ above);
• sample_paths - return sample paths of the process at specified time points. The base class provides Euler scheme sampling if the drift and volatility functions are defined;
• fd_solver_backward - returns a finite difference method for solving the Feynman Kac PDE associated with the SDE. This equation is a slight generalization of the Kolmogorov backward equation with the inclusion of a discounting function. The Feynman Kac PDE is:

$$V_t + Sum[mu_i(t, x) V_i, 1<=i<=n] + \frac{1}{2} Sum[ D_{ij} V_{ij}, 1 <= i,j <= n] - r(t, x) V = 0$$

with the final value condition $$V(T, x) = u(x)$$.

The corresponding Kolmogorov forward/Fokker Plank equation is

$$\frac{\partial}{\partial t} V(t, x) + \sum_{i=1}^n \frac{\partial}{\partial x_i} a_i(t, X) V(t, x) + \sum_{i, j, k = 1}^n \frac{\partial^2}{\partial x_i \partial x_j} S_{ik} S_{jk} V(t, x) = 0.$$

with the initial value condition $$V(0, x) = u(x)$$.

A minimum description of the model should only include dim, dtype, and name. In order to use the provided sample_paths method, both drift_fn and volatility_fn should be defined.

TODO(b/140290854): Provide description of model calibration procedure. TODO(b/140313472): Provide description of Ito process algebra API.