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Jump Problem and Jump Diffusion Solvers

solve(prob::JumpProblem, alg; kwargs)

JumpProblems can be solved with two classes of methods, exact and inexact. Exact algorithms currently sample realizations of the jump processes in chronological order, executing individual jumps sequentially at randomly sampled times. In contrast, inexact (τ-leaping) methods are time-step based, executing multiple occurrences of jumps during each time-step. These methods can be much faster as they only simulate the total number of jumps over each leap interval, and thus do not need to simulate the realization of every single jump. Jumps for use with exact simulation methods can be defined as ConstantRateJumps, MassActionJumps, and/or VariableRateJump. Jumps for use with inexact τ-leaping methods should be defined as RegularJumps.

There are special algorithms available for efficiently simulating an exact, pure JumpProblem (i.e., a JumpProblem over a DiscreteProblem). SSAStepper() is an efficient streamlined integrator for time stepping such problems from individual jump to jump. This integrator is named after Stochastic Simulation Algorithms (SSAs), commonly used naming in chemistry and biology applications for the class of exact jump process simulation algorithms. In turn, we denote by "aggregators" the algorithms that SSAStepper calls to calculate the next jump time and to execute a jump (i.e., change the system state appropriately). All JumpProcesses aggregators can be used with ConstantRateJumps and MassActionJumps, with a subset of aggregators also working with bounded VariableRateJumps (see the first tutorial for the definition of bounded VariableRateJumps). Although SSAStepper() is usually faster, it only supports discrete events (DiscreteCallbacks), for pure jump problems requiring continuous events (ContinuousCallbacks) the less performant FunctionMap time-stepper can be used.

If there is a RegularJump, then inexact τ-leaping methods must be used. The current recommended method is TauLeaping if one needs adaptivity, events, etc. If one only needs the most barebones fixed time-step leaping method, then SimpleTauLeaping can have performance benefits.

Special Methods for Pure Jump Problems

If you are using jumps with a differential equation, use the same methods as in the case of the differential equation solving. However, the following algorithms are optimized for pure jump problems.

JumpProcesses.jl

  • SSAStepper: a stepping integrator for JumpProblems defined over DiscreteProblems involving ConstantRateJumps, MassActionJumps, and/or bounded VariableRateJumps . Supports handling of DiscreteCallbacks and saving controls like saveat. Note that DifferentialEquations.jl treats jumps as similar to callbacks, and hence SSAStepper only implements a subset of ODE/SDE solver saving controls. In particular, save_everystep is not supported as saving jumps each step is controlled via the save_positions argument to JumpProblems. Note, in contrast to when JumpProblems are coupled with ODE and SDE timesteppers, with SSAStepper, setting save_positions = (false, true), save_positions = (true, false) or save_positions = (true, true) are equivalent and save only after the jump has occurred (as opposed to saving the state both before and after a jump). This is because the underlying SSAStepper generated-solution uses piecewise constant interpolation, and can therefore exactly evaluate the sampled solution path at any time when only saving the post-jump state for each jump.

RegularJump Compatible Methods

StochasticDiffEq.jl

These methods support mixing with event handling, other jump types, and all of the features of the normal differential equation solvers.

  • TauLeaping: an adaptive tau-leaping algorithm with post-leap estimates.

JumpProcesses.jl

  • SimpleTauLeaping: a tau-leaping algorithm for pure RegularJumpJumpProblems. Requires a choice of dt.
  • RegularSSA: a version of SSA for pure RegularJumpJumpProblems.

Regular Jump Diffusion Compatible Methods

Regular jump diffusions are JumpProblems where the internal problem is an SDEProblem and the jump process has designed a regular jump.

StochasticDiffEq.jl

  • EM: Explicit Euler-Maruyama.
  • ImplicitEM: Implicit Euler-Maruyama. See the SDE solvers page for more details.