Tau-Leaping Methods for Jump-Diffusion

Tau-leaping methods approximate jump processes by "leaping" over multiple potential jump events in a single time step. These methods are essential for efficiently simulating systems with jump-diffusion processes.

Tau-Leaping Methods

TauLeaping - Basic Tau-Leaping

StochasticDiffEq.TauLeaping — Type
TauLeaping()

TauLeaping: Basic Tau-Leaping Method (Jump-Diffusion)

Basic tau-leaping method for approximating jump-diffusion processes by "leaping" over multiple potential jump events.

Method Properties

  • Problem type: Jump-diffusion processes
  • Approach: Approximate multiple jumps per time step
  • Time stepping: Fixed tau approach
  • Accuracy: Depends on tau selection

When to Use

  • Jump-diffusion systems with many small jumps
  • When exact jump simulation is computationally prohibitive
  • Chemical reaction networks with fast reactions
  • Population models with high birth-death rates
  • Initial exploration of jump-diffusion problems

Algorithm Description

Approximates Poisson processes by assuming constant propensities over time interval tau, then sampling number of jumps from Poisson distribution.

Tau Selection

Critical parameter: tau should be small enough that jump rates don't change significantly over [t, t+tau].

References

  • Gillespie, D.T., "Approximate accelerated stochastic simulation of chemically reacting systems"

CaoTauLeaping - Cao's Tau-Leaping

StochasticDiffEq.CaoTauLeaping — Type
CaoTauLeaping()

CaoTauLeaping: Cao's Adaptive Tau-Leaping Method (Jump-Diffusion)

Advanced tau-leaping method with adaptive tau selection and improved error control.

Method Properties

  • Problem type: Jump-diffusion processes
  • Approach: Adaptive tau selection with error control
  • Time stepping: Adaptive tau based on error estimates
  • Accuracy: Superior to basic tau-leaping

When to Use

  • Production jump-diffusion simulations requiring reliability
  • When adaptive tau selection is needed
  • Problems where basic TauLeaping gives poor accuracy
  • Chemical reaction networks requiring precise control

Algorithm Features

  • Adaptive tau selection based on error estimates
  • Better stability and accuracy than basic tau-leaping
  • Automatic step size control
  • More sophisticated error estimation

Tau Selection

Automatically adjusts tau based on:

  • Local error estimates
  • Jump rate variations
  • Solution stability requirements

References

  • Cao, Y., Gillespie, D.T., Petzold, L.R., "Efficient step size selection for the tau-leaping method"

Understanding Jump-Diffusion Processes

Jump-diffusion processes combine:

  1. Continuous diffusion: Standard Brownian motion terms
  2. Jump processes: Discontinuous jumps at random times

General form:

dX = μ(X,t)dt + σ(X,t)dW + ∫ h(X-,z)Ñ(dt,dz)

Where:

  • μ(X,t)dt: Drift term
  • σ(X,t)dW: Diffusion term
  • Ñ(dt,dz): Compensated random measure (jumps)

When to Use Tau-Leaping

Appropriate for:

  • Systems with many small jumps
  • When exact jump simulation is computationally prohibitive
  • Chemical reaction networks
  • Population models with birth-death processes
  • Financial models with rare events

Not appropriate for:

  • Systems dominated by large, infrequent jumps
  • When exact jump timing is critical
  • Small systems where exact methods are feasible

Method Characteristics

TauLeaping:

  • Basic tau-leaping approximation
  • Fixed tau approach
  • Good for initial exploration

CaoTauLeaping:

  • Adaptive tau selection
  • More sophisticated error control
  • Better for production simulations

Configuration

Tau-leaping methods require:

  1. Jump rate functions: λ(X,t) for each reaction/jump type
  2. Jump effects: How state changes with each jump
  3. Tau selection: Time step size strategy
# Basic setup
prob = JumpProblem(base_problem, aggregator, jumps...)
sol = solve(prob, TauLeaping())

# With adaptive tau
sol = solve(prob, CaoTauLeaping(), tau_tol=0.01)

Accuracy Considerations

Tau-leaping approximation quality depends on:

  • Jump frequency vs. tau size
  • State change magnitude per jump
  • System stiffness
  • Error tolerance requirements

Rule of thumb: Tau should be small enough that jump rates don't change significantly over [t, t+tau].

Alternative Approaches

If tau-leaping is inadequate:

  1. Exact methods: Gillespie algorithm for small systems
  2. Hybrid methods: Combine exact and approximate regions
  3. Moment closure: For statistical properties only
  4. Piecewise deterministic: For systems with rare jumps

Performance Tips

  1. Vectorize jump computations when possible
  2. Use sparse representations for large systems
  3. Tune tau carefully - too large gives poor accuracy, too small is inefficient
  4. Monitor jump frequencies to validate approximation

Integration with DifferentialEquations.jl

using DifferentialEquations, StochasticDiffEq

# Define base SDE
function drift!(du, u, p, t)
    # Continuous drift
end

function diffusion!(du, u, p, t)  
    # Continuous diffusion
end

# Define jumps
jump1 = ConstantRateJump(rate1, affect1!)
jump2 = VariableRateJump(rate2, affect2!)

# Combine into jump-diffusion problem
sde_prob = SDEProblem(drift!, diffusion!, u0, tspan)
jump_prob = JumpProblem(sde_prob, Direct(), jump1, jump2)

# Solve with tau-leaping
sol = solve(jump_prob, TauLeaping())

References

  • Gillespie, D.T., "Approximate accelerated stochastic simulation of chemically reacting systems"
  • Cao, Y., Gillespie, D.T., Petzold, L.R., "Efficient step size selection for the tau-leaping method"