Function sample

tf_quant_finance/models/milstein_sampling.py:35–255

r"""Returns a sample paths from the process using the Milstein method. For an Ito process, ``` dX = a(t, X_t) dt + b(t, X_t) dW_t ``` given drift `a`, volatility `b` and derivative of volatility `b'`, the Milstein method generates a sequence {Y_n} approximating X ``` Y_{n+1} =

(
    *,
    dim: int,
    drift_fn: Callable[..., types.RealTensor],
    volatility_fn: Callable[..., types.RealTensor],
    times: types.RealTensor,
    time_step: Optional[types.RealTensor] = None,
    num_time_steps: Optional[types.IntTensor] = None,
    num_samples: types.IntTensor = 1,
    initial_state: Optional[types.RealTensor] = None,
    grad_volatility_fn: Optional[Callable[..., List[types.RealTensor]]] = None,
    random_type: Optional[random.RandomType] = None,
    seed: Optional[types.IntTensor] = None,
    swap_memory: bool = True,
    skip: types.IntTensor = 0,
    precompute_normal_draws: bool = True,
    watch_params: Optional[List[types.RealTensor]] = None,
    stratonovich_order: int = 5,
    dtype: Optional[tf.DType] = None,
    name: Optional[str] = None)

def sample(
    *,
    dim: int,
    drift_fn: Callable[..., types.RealTensor],
    volatility_fn: Callable[..., types.RealTensor],
    times: types.RealTensor,
    time_step: Optional[types.RealTensor] = None,
    num_time_steps: Optional[types.IntTensor] = None,
    num_samples: types.IntTensor = 1,
    initial_state: Optional[types.RealTensor] = None,
    grad_volatility_fn: Optional[Callable[..., List[types.RealTensor]]] = None,
    random_type: Optional[random.RandomType] = None,
    seed: Optional[types.IntTensor] = None,
    swap_memory: bool = True,
    skip: types.IntTensor = 0,
    precompute_normal_draws: bool = True,
    watch_params: Optional[List[types.RealTensor]] = None,
    stratonovich_order: int = 5,
    dtype: Optional[tf.DType] = None,
    name: Optional[str] = None) -> types.RealTensor:
  r"""Returns a sample paths from the process using the Milstein method.

  For an Ito process,

  ```
    dX = a(t, X_t) dt + b(t, X_t) dW_t
  ```
  given drift `a`, volatility `b` and derivative of volatility `b'`, the
  Milstein method generates a
  sequence {Y_n} approximating X

  ```
  Y_{n+1} = Y_n + a(t_n, Y_n) dt + b(t_n, Y_n) dW_n + \frac{1}{2} b(t_n, Y_n)
  b'(t_n, Y_n) ((dW_n)^2 - dt)
  ```
  where `dt = t_{n+1} - t_n`, `dW_n = (N(0, t_{n+1}) - N(0, t_n))` and `N` is a
  sample from the Normal distribution.

  In higher dimensions, when `a(t, X_t)` is a d-dimensional vector valued
  function and `W_t` is a d-dimensional Wiener process, we have for the kth
  element of the expansion:

  ```
  Y_{n+1}[k] = Y_n[k] + a(t_n, Y_n)[k] dt + \sum_{j=1}^d b(t_n, Y_n)[k, j]
  dW_n[j] + \sum_{j_1=1}^d \sum_{j_2=1}^d L_{j_1} b(t_n, Y_n)[k, j_2] I(j_1,
  j_2)
  ```
  where `L_{j} = \sum_{i=1}^d b(t_n, Y_n)[i, j] \frac{\partial}{\partial x^i}`
  is an operator and `I(j_1, j_2) = \int_{t_n}^{t_{n+1}} \int_{t_n}^{s_1}
  dW_{s_2}[j_1] dW_{s_1}[j_2]` is a multiple Ito integral.


  See [1] and [2] for details.

  #### References
  [1]: Wikipedia. Milstein method:
  https://en.wikipedia.org/wiki/Milstein_method
  [2]: Peter E. Kloeden,  Eckhard Platen. Numerical Solution of Stochastic