Jump-diffusion timeseries generator

JumpDiff.jdprocess.jdprocess(time: float, delta_t: float, a: callable, b: callable, xi: float, lamb: float, init: float = None, solver: str = 'Euler', b_prime: callable = None) → numpy.ndarray

Integrates a jump-diffusion process with drift a(x), diffusion b(x), jump amplitude xi (\(\xi\)), and jump rate lamb (\(\lambda\)).

\[\mathrm{d} X(t) = a(x,t)\;\mathrm{d} t + b(x,t)\;\mathrm{d} W(t) + \xi\;\mathrm{d} J(t),\]

with \(J\) Poisson with jump rate \(\lambda\).

This integrator has both an Euler─Mayurama and a Milstein method of integration. For Milstein one has to introduce the derivative of the diffusion term b, denoted b_prime.

Parameters:

• time (float > 0) – Total integration time. Positive float or int.
• delta_t (float > 0) – Time sampling, the smaller the better.
• a (callable) –

  The drift function. Can be a function of a lambda. For an Ornstein─Uhlenbeck process with drift -2x, a takes the form

  a =  lambda x: -2x.

• b (callable) –

  The diffusion function. Can be a function of a lambda. For an Ornstein─Uhlenbeck process with diffusion 1, a takes the form

  b =  lambda x: 1.

• xi (float > 0) – Variance of the jump amplitude, which will be turned into a normal distribution like \(\mathcal{N}\)(0,√xi).
• lamb (float > 0) – Jump rate of the Poissonian jumps. This is implemented as the numpy function np.random.poisson(lam = lamb * delta_t).
• init (float (defaul None)) – Initial conditions. If None given, generates a random value from a normal distribution ~ \(\mathcal{N}\)(0,√delta_t).
• solver ('Euler' or 'Milstein' (defaul 'Euler')) – The regular Euler─Maruyama solver ‘Euler’ is the default, with an order of √delta_t. To employ a state-dependent diffusion, i.e., b(x) as a function of x, the Milstein scheme has an order of delta_t. You must introduce as well the derivative of b(x), i.e., b’(x), as the argument b_prime.

Returns:

X – Timeseries of size int(time/delta_t)

Return type:

np.array