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

CaoTauLeaping - Cao's Tau-Leaping

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"