*Last updated: 2023-03-16.*
# tf_quant_finance.models.ItoProcess
View source
Interface for specifying Ito processes.
Interface for defining stochastic process defined by the Ito SDE:
```None
dX_i = a_i(t, X) dt + Sum(S_{ij}(t, X) dW_j for 1 <= j <= n), 1 <= i <= n
```
The vector coefficient `a_i` is referred to as the drift of the process and
the matrix `S_{ij}` as the volatility of the process. For the process to be
well defined, these coefficients need to satisfy certain technical conditions
which may be found in Ref. [1]. The vector `dW_j` represents independent
Brownian increments.
For a simple and instructive example of the implementation of this interface,
see models.GenericItoProcess.
#### References
[1]: Brent Oksendal. Stochastic Differential Equations: An Introduction with
Applications. Springer. 2010.
## Methods
dim
View source
```python
dim()
```
The dimension of the process. A positive python integer.
drift_fn
View source
```python
drift_fn()
```
Python callable calculating instantaneous drift.
The callable should accept two real `Tensor` arguments of the same dtype.
The first argument is the scalar time t, the second argument is the value of
Ito process X - `Tensor` of shape `batch_shape + [dim]`. Here `batch_shape`
is an arbitrary shape. The result is the value of drift a(t, X). The return
value of the callable is a real `Tensor` of the same dtype as the input
arguments and of shape `batch_shape + [dim]`.
#### Returns:
The instantaneous drift rate callable.
dtype
View source
```python
dtype()
```
The data type of process realizations.
fd_solver_backward
View source
```python
fd_solver_backward(
start_time, end_time, coord_grid, values_grid, discounting=None,
one_step_fn=None, boundary_conditions=None, start_step_count=0, num_steps=None,
time_step=None, values_transform_fn=None, dtype=None, **kwargs
)
```
Returns a solver for Feynman-Kac PDE associated to the process.
This method applies a finite difference method to solve the final value
problem as it appears in the Feynman-Kac formula associated to this Ito
process. The Feynman-Kac PDE is closely related to the backward Kolomogorov
equation associated to the stochastic process and allows for the inclusion
of a discounting function.
For more details of the Feynman-Kac theorem see [1]. The PDE solved by this
method is:
```None
V_t + Sum[mu_i(t, x) V_i, 1<=i<=n] +
(1/2) Sum[ D_{ij} V_{ij}, 1 <= i,j <= n] - r(t, x) V = 0
```
In the above, `V_t` is the derivative of `V` with respect to `t`,
`V_i` is the partial derivative with respect to `x_i` and `V_{ij}` the
(mixed) partial derivative with respect to `x_i` and `x_j`. `mu_i` is the
drift of this process and `D_{ij}` are the components of the diffusion
tensor:
```None
D_{ij}(t,x) = (Sigma(t,x) . Transpose[Sigma(t,x)])_{ij}
```
This method evolves a spatially discretized solution of the above PDE from
time `t0` to time `t1 < t0` (i.e. backwards in time).
The solution `V(t,x)` is assumed to be discretized on an `n`-dimensional
rectangular grid. A rectangular grid, G, in n-dimensions may be described
by specifying the coordinates of the points along each axis. For example,
a 2 x 4 grid in two dimensions can be specified by taking the cartesian
product of [1, 3] and [5, 6, 7, 8] to yield the grid points with
coordinates: `[(1, 5), (1, 6), (1, 7), (1, 8), (3, 5) ... (3, 8)]`.
This method allows batching of solutions. In this context, batching means
the ability to represent and evolve multiple independent functions `V`
(e.g. V1, V2 ...) simultaneously. A single discretized solution is specified
by stating its values at each grid point. This can be represented as a
`Tensor` of shape [d1, d2, ... dn] where di is the grid size along the `i`th
axis. A batch of such solutions is represented by a `Tensor` of shape:
[K, d1, d2, ... dn] where `K` is the batch size. This method only requires
that the input parameter `values_grid` be broadcastable with shape
[K, d1, ... dn].
The evolution of the solution from `t0` to `t1` is often done by
discretizing the differential equation to a difference equation along
the spatial and temporal axes. The temporal discretization is given by a
(sequence of) time steps [dt_1, dt_2, ... dt_k] such that the sum of the
time steps is equal to the total time step `t0 - t1`. If a uniform time
step is used, it may equivalently be specified by stating the number of
steps (n_steps) to take. This method provides both options via the
`time_step` and `num_steps` parameters. However, not all methods need
discretization along time direction (e.g. method of lines) so this argument
may not be applicable to some implementations.
The workhorse of this method is the `one_step_fn`. For the commonly used
methods, see functions in math.pde.steppers module.
The mapping between the arguments of this method and the above
equation are described in the Args section below.
For a simple instructive example of implementation of this method, see
models.GenericItoProcess.fd_solver_backward.
TODO(b/142309558): Complete documentation.
#### Args:
* `start_time`: Real positive scalar `Tensor`. The start time of the grid.
Corresponds to time `t0` above.
* `end_time`: Real scalar `Tensor` smaller than the `start_time` and greater
than zero. The time to step back to. Corresponds to time `t1` above.
* `coord_grid`: List of `n` rank 1 real `Tensor`s. `n` is the dimension of the
domain. The i-th `Tensor` has shape, `[d_i]` where `d_i` is the size of
the grid along axis `i`. The coordinates of the grid points. Corresponds
to the spatial grid `G` above.
* `values_grid`: Real `Tensor` containing the function values at time
`start_time` which have to be stepped back to time `end_time`. The shape
of the `Tensor` must broadcast with `[K, d_1, d_2, ..., d_n]`. The first
axis of size `K` is the values batch dimension and allows multiple
functions (with potentially different boundary/final conditions) to be
stepped back simultaneously.
* `discounting`: Callable corresponding to `r(t,x)` above. If not supplied,
zero discounting is assumed.
* `one_step_fn`: The transition kernel. A callable that consumes the following
arguments by keyword:
1. 'time': Current time
2. 'next_time': The next time to step to. For the backwards in time
evolution, this time will be smaller than the current time.
3. 'coord_grid': The coordinate grid.
4. 'values_grid': The values grid.
5. 'boundary_conditions': The boundary conditions.
6. 'quadratic_coeff': A callable returning the quadratic coefficients
of the PDE (i.e. `(1/2)D_{ij}(t, x)` above). The callable accepts
the time and coordinate grid as keyword arguments and returns a
`Tensor` with shape that broadcasts with `[dim, dim]`.
7. 'linear_coeff': A callable returning the linear coefficients of the
PDE (i.e. `mu_i(t, x)` above). Accepts time and coordinate grid as
keyword arguments and returns a `Tensor` with shape that broadcasts
with `[dim]`.
8. 'constant_coeff': A callable returning the coefficient of the
linear homogenous term (i.e. `r(t,x)` above). Same spec as above.
The `one_step_fn` callable returns a 2-tuple containing the next
coordinate grid, next values grid.
* `boundary_conditions`: A list of size `dim` containing boundary conditions.
The i'th element of the list is a 2-tuple containing the lower and upper
boundary condition for the boundary along the i`th axis.
* `start_step_count`: Scalar integer `Tensor`. Initial value for the number of
time steps performed.
Default value: 0 (i.e. no previous steps performed).
* `num_steps`: Positive int scalar `Tensor`. The number of time steps to take
when moving from `start_time` to `end_time`. Either this argument or the
`time_step` argument must be supplied (but not both). If num steps is
`k>=1`, uniform time steps of size `(t0 - t1)/k` are taken to evolve the
solution from `t0` to `t1`. Corresponds to the `n_steps` parameter
above.
* `time_step`: The time step to take. Either this argument or the `num_steps`
argument must be supplied (but not both). The type of this argument may
be one of the following (in order of generality): (a) None in which case
`num_steps` must be supplied. (b) A positive real scalar `Tensor`. The
maximum time step to take. If the value of this argument is `dt`, then
the total number of steps taken is N = (t0 - t1) / dt rounded up to
the nearest integer. The first N-1 steps are of size dt and the last
step is of size `t0 - t1 - (N-1) * dt`. (c) A callable accepting the
current time and returning the size of the step to take. The input and
the output are real scalar `Tensor`s.
* `values_transform_fn`: An optional callable applied to transform the
solution values at each time step. The callable is invoked after the
time step has been performed. The callable should accept the time of the
grid, the coordinate grid and the values grid and should return the
values grid. All input arguments to be passed by keyword.
* `dtype`: The dtype to use.
* `**kwargs`: Any other keyword args needed by an implementation.
#### Returns:
A tuple object containing at least the following attributes:
final_values_grid: A `Tensor` of same shape and dtype as `values_grid`.
Contains the final state of the values grid at time `end_time`.
final_coord_grid: A list of `Tensor`s of the same specification as
the input `coord_grid`. Final state of the coordinate grid at time
`end_time`.
step_count: The total step count (i.e. the sum of the `start_step_count`
and the number of steps performed in this call.).
final_time: The final time at which the evolution stopped. This value
is given by `max(min(end_time, start_time), 0)`.
fd_solver_forward
View source
```python
fd_solver_forward(
start_time, end_time, coord_grid, values_grid, one_step_fn=None,
boundary_conditions=None, start_step_count=0, num_steps=None, time_step=None,
values_transform_fn=None, dtype=None, **kwargs
)
```
Returns a solver for the Fokker Plank equation of this process.
The Fokker Plank equation (also known as the Kolmogorov Forward equation)
associated to this Ito process is given by:
```None
V_t + Sum[(mu_i(t, x) V)_i, 1<=i<=n]
- (1/2) Sum[ (D_{ij} V)_{ij}, 1 <= i,j <= n] = 0
```
with the initial value condition $$V(0, x) = u(x)$$.
This method evolves a spatially discretized solution of the above PDE from
time `t0` to time `t1 > t0` (i.e. forwards in time).
The solution `V(t,x)` is assumed to be discretized on an `n`-dimensional
rectangular grid. A rectangular grid, G, in n-dimensions may be described
by specifying the coordinates of the points along each axis. For example,
a 2 x 4 grid in two dimensions can be specified by taking the cartesian
product of [1, 3] and [5, 6, 7, 8] to yield the grid points with
coordinates: `[(1, 5), (1, 6), (1, 7), (1, 8), (3, 5) ... (3, 8)]`.
Batching of solutions is supported. In this context, batching means
the ability to represent and evolve multiple independent functions `V`
(e.g. V1, V2 ...) simultaneously. A single discretized solution is specified
by stating its values at each grid point. This can be represented as a
`Tensor` of shape [d1, d2, ... dn] where di is the grid size along the `i`th
axis. A batch of such solutions is represented by a `Tensor` of shape:
[K, d1, d2, ... dn] where `K` is the batch size. This method only requires
that the input parameter `values_grid` be broadcastable with shape
[K, d1, ... dn].
The evolution of the solution from `t0` to `t1` is often done by
discretizing the differential equation to a difference equation along
the spatial and temporal axes. The temporal discretization is given by a
(sequence of) time steps [dt_1, dt_2, ... dt_k] such that the sum of the
time steps is equal to the total time step `t1 - t0`. If a uniform time
step is used, it may equivalently be specified by stating the number of
steps (n_steps) to take. This method provides both options via the
`time_step` and `num_steps` parameters. However, not all methods need
discretization along time direction (e.g. method of lines) so this argument
may not be applicable to some implementations.
The workhorse of this method is the `one_step_fn`. For the commonly used
methods, see functions in math.pde.steppers module.
The mapping between the arguments of this method and the above
equation are described in the Args section below.
For a simple instructive example of implementation of this method, see
models.GenericItoProcess.fd_solver_forward.
TODO(b/142309558): Complete documentation.
#### Args:
* `start_time`: Real positive scalar `Tensor`. The start time of the grid.
Corresponds to time `t0` above.
* `end_time`: Real scalar `Tensor` smaller than the `start_time` and greater
than zero. The time to step back to. Corresponds to time `t1` above.
* `coord_grid`: List of `n` rank 1 real `Tensor`s. `n` is the dimension of the
domain. The i-th `Tensor` has shape, `[d_i]` where `d_i` is the size of
the grid along axis `i`. The coordinates of the grid points. Corresponds
to the spatial grid `G` above.
* `values_grid`: Real `Tensor` containing the function values at time
`start_time` which have to be stepped back to time `end_time`. The shape
of the `Tensor` must broadcast with `[K, d_1, d_2, ..., d_n]`. The first
axis of size `K` is the values batch dimension and allows multiple
functions (with potentially different boundary/final conditions) to be
stepped back simultaneously.
* `one_step_fn`: The transition kernel. A callable that consumes the following
arguments by keyword:
1. 'time': Current time
2. 'next_time': The next time to step to. For the backwards in time
evolution, this time will be smaller than the current time.
3. 'coord_grid': The coordinate grid.
4. 'values_grid': The values grid.
5. 'quadratic_coeff': A callable returning the quadratic coefficients
of the PDE (i.e. `(1/2)D_{ij}(t, x)` above). The callable accepts
the time and coordinate grid as keyword arguments and returns a
`Tensor` with shape that broadcasts with `[dim, dim]`.
6. 'linear_coeff': A callable returning the linear coefficients of the
PDE (i.e. `mu_i(t, x)` above). Accepts time and coordinate grid as
keyword arguments and returns a `Tensor` with shape that broadcasts
with `[dim]`.
7. 'constant_coeff': A callable returning the coefficient of the
linear homogenous term (i.e. `r(t,x)` above). Same spec as above.
The `one_step_fn` callable returns a 2-tuple containing the next
coordinate grid, next values grid.
* `boundary_conditions`: A list of size `dim` containing boundary conditions.
The i'th element of the list is a 2-tuple containing the lower and upper
boundary condition for the boundary along the i`th axis.
* `start_step_count`: Scalar integer `Tensor`. Initial value for the number of
time steps performed.
Default value: 0 (i.e. no previous steps performed).
* `num_steps`: Positive int scalar `Tensor`. The number of time steps to take
when moving from `start_time` to `end_time`. Either this argument or the
`time_step` argument must be supplied (but not both). If num steps is
`k>=1`, uniform time steps of size `(t0 - t1)/k` are taken to evolve the
solution from `t0` to `t1`. Corresponds to the `n_steps` parameter
above.
* `time_step`: The time step to take. Either this argument or the `num_steps`
argument must be supplied (but not both). The type of this argument may
be one of the following (in order of generality): (a) None in which case
`num_steps` must be supplied. (b) A positive real scalar `Tensor`. The
maximum time step to take. If the value of this argument is `dt`, then
the total number of steps taken is N = (t1 - t0) / dt rounded up to
the nearest integer. The first N-1 steps are of size dt and the last
step is of size `t1 - t0 - (N-1) * dt`. (c) A callable accepting the
current time and returning the size of the step to take. The input and
the output are real scalar `Tensor`s.
* `values_transform_fn`: An optional callable applied to transform the
solution values at each time step. The callable is invoked after the
time step has been performed. The callable should accept the time of the
grid, the coordinate grid and the values grid and should return the
values grid. All input arguments to be passed by keyword.
* `dtype`: The dtype to use.
* `**kwargs`: Any other keyword args needed by an implementation.
#### Returns:
A tuple object containing at least the following attributes:
final_values_grid: A `Tensor` of same shape and dtype as `values_grid`.
Contains the final state of the values grid at time `end_time`.
final_coord_grid: A list of `Tensor`s of the same specification as
the input `coord_grid`. Final state of the coordinate grid at time
`end_time`.
step_count: The total step count (i.e. the sum of the `start_step_count`
and the number of steps performed in this call.).
final_time: The final time at which the evolution stopped. This value
is given by `max(min(end_time, start_time), 0)`.
name
View source
```python
name()
```
The name to give to ops created by this class.
sample_paths
View source
```python
sample_paths(
times, num_samples=1, initial_state=None, random_type=None, seed=None, **kwargs
)
```
Returns a sample of paths from the process.
#### Args:
* `times`: A `Tensor` of positive real values of a shape `[T, k]`, where
`T` is either empty or a shape which is broadcastable to `batch_shape`
(as defined when this ItoProcess was initialised) and `k` is the
number of time points. The definition of `batch_shape` and how it is set
is dictated by the individual child classes. Typically `batch_shape`
is set based on the shapes of the input parameters that define the
ItoProcess, see `GeometricBrownianMotion` for an example. The times at
which the path points are to be evaluated.
* `num_samples`: Positive scalar `int`. The number of paths to draw.
* `initial_state`: `Tensor` of shape `[dim]`. The initial state of the
process.
Default value: None which maps to a zero initial state.
* `random_type`: Enum value of `RandomType`. The type of (quasi)-random number
generator to use to generate the paths.
Default value: None which maps to the standard pseudo-random numbers.
* `seed`: Seed for the random number generator. The seed is
only relevant if `random_type` is one of
`[STATELESS, PSEUDO, HALTON_RANDOMIZED, PSEUDO_ANTITHETIC,
STATELESS_ANTITHETIC]`. For `PSEUDO`, `PSEUDO_ANTITHETIC` and
`HALTON_RANDOMIZED` the seed should be an Python integer. For
`STATELESS` and `STATELESS_ANTITHETIC `must be supplied as an integer
`Tensor` of shape `[2]`.
Default value: `None` which means no seed is set.
* `**kwargs`: Any other keyword args needed by an implementation.
#### Returns:
A real `Tensor` of shape [`batch_shape`, num_samples, k, n] where `k` is
the size of `times`, `n` is the dimension of the process and `batch_shape`
is the batch dimensions of the ItoProcess.
volatility_fn
View source
```python
volatility_fn()
```
Python callable calculating the instantaneous volatility matrix.
The callable should accept two real `Tensor` arguments of the same dtype and
shape `times_shape`. The first argument is the scalar time t, the second
argument is the value of Ito process X - `Tensor` of shape `batch_shape +
[dim]`. Here `batch_shape` is an arbitrary shape. The result is value of
volatility `S_ij`(t, X). The return value of the callable is a real `Tensor`
of the same dtype as the input arguments and of shape
`batch_shape + [dim, dim]`.
#### Returns:
The instantaneous volatility callable.