tf_quant_finance.models.milstein_sampling.sample

Contents

Last updated: 2023-03-16.

tf_quant_finance.models.milstein_sampling.sample#

View source

Returns a sample paths from the process using the Milstein method.

tf_quant_finance.models.milstein_sampling.sample(
    *, dim, drift_fn, volatility_fn, times, time_step=None, num_time_steps=None,
    num_samples=1, initial_state=None, grad_volatility_fn=None, random_type=None,
    seed=None, swap_memory=True, skip=0, precompute_normal_draws=True,
    watch_params=None, stratonovich_order=5, dtype=None, name=None
)

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 Differential Equations. Springer. 1992

Args:#

  • dim: Python int greater than or equal to 1. The dimension of the Ito Process.

  • drift_fn: A Python callable to compute the drift of the process. 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]. The result is 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].

  • volatility_fn: A Python callable to compute the volatility of the process. The callable should accept two real Tensor arguments of the same dtype as times. The first argument is the scalar time t, the second argument is the value of Ito process X - tensor of shape batch_shape + [dim]. The result is value of volatility b(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].

  • times: Rank 1 Tensor of increasing positive real values. The times at which the path points are to be evaluated.

  • time_step: An optional scalar real Tensor - maximal distance between points in grid in Milstein schema. Either this or num_time_steps should be supplied. Default value: None.

  • num_time_steps: An optional Scalar integer Tensor - a total number of time steps performed by the algorithm. The maximal distance between points in grid is bounded by times[-1] / (num_time_steps - times.shape[0]). Either this or time_step should be supplied. Default value: None.

  • num_samples: Positive scalar int. The number of paths to draw. Default value: 1.

  • initial_state: Tensor of shape [dim]. The initial state of the process. Default value: None which maps to a zero initial state.

  • grad_volatility_fn: An optional python callable to compute the gradient of volatility_fn. The callable should accept three real Tensor arguments of the same dtype as times. The first argument is the scalar time t. The second argument is the value of Ito process X - tensor of shape batch_shape + [dim]. The third argument is a tensor of input gradients of shape batch_shape + [dim] to pass to gradient.fwd_gradient. The result is a list of values corresponding to the forward gradient of volatility b(t, X) with respect to X. The return value of the callable is a list of size dim containing real Tensors of the same dtype as the input arguments and of shape batch_shape + [dim, dim]. Each index of the list corresponds to a dimension of the state. If None, the gradient is computed from volatility_fn using forward differentiation.

  • 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 a 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.

  • swap_memory: A Python bool. Whether GPU-CPU memory swap is enabled for this op. See an equivalent flag in tf.while_loop documentation for more details. Useful when computing a gradient of the op since tf.while_loop is used to propagate stochastic process in time. Default value: True.

  • skip: int32 0-d Tensor. The number of initial points of the Sobol or Halton sequence to skip. Used only when random_type is ‘SOBOL’, ‘HALTON’, or ‘HALTON_RANDOMIZED’, otherwise ignored. Default value: 0.

  • precompute_normal_draws: Python bool. Indicates whether the noise increments N(0, t_{n+1}) - N(0, t_n) are precomputed. For HALTON and SOBOL random types the increments are always precomputed. While the resulting graph consumes more memory, the performance gains might be significant. Default value: True.

  • watch_params: An optional list of zero-dimensional Tensors of the same dtype as initial_state. If provided, specifies Tensors with respect to which the differentiation of the sampling function will happen. A more efficient algorithm is used when watch_params are specified. Note the the function becomes differentiable only wrt to these Tensors and the initial_state. The gradient wrt any other Tensor is set to be zero.

  • stratonovich_order: A positive integer. The number of terms to use when calculating the approximate Stratonovich integrals in the multidimensional scheme. Stratonovich integrals are an alternative to Ito integrals, and can be used interchangeably when defining the higher order terms in the update equation. We use Stratonovich integrals here because they have a convenient approximation scheme for calculating cross terms involving different components of the Wiener process. See Eq. 8.10 in Section 5.8 of [2]. Default value: 5.

  • dtype: tf.Dtype. If supplied the dtype for the input and output Tensors. Default value: None which means that the dtype implied by times is used.

  • name: Python string. The name to give this op. Default value: None which maps to milstein_sample.

Contents