# PRP-37: Jump Diffusion Model Implementation
## Context & Motivation
**Integration Goal**: Implement Merton's jump diffusion model for generating price series with sudden jumps.
**User Requirement**: Generate realistic market data with both continuous movements and discrete jumps (e.g., earnings announcements, news events).
**Technical Challenge**: Combine Brownian motion with Poisson jump process while maintaining realistic dynamics.
## Requirements
### Jump Diffusion Components
1. **Continuous Diffusion**: Standard Brownian motion component
2. **Jump Process**: Poisson-distributed jump arrivals
3. **Jump Size Distribution**: Log-normal jump magnitudes
4. **Compound Process**: Combined continuous and jump components
### Model Parameters
1. **Jump Intensity (λ)**: Average jumps per period
2. **Jump Mean (μ_J)**: Average jump size
3. **Jump Volatility (σ_J)**: Jump size standard deviation
4. **Drift Adjustment**: Compensate for jump effects
## Implementation Blueprint
### Phase 1: Jump Process
1. Create `src/algorithms/jump_diffusion.rs`
2. Define `JumpDiffusionGenerator` struct
3. Implement Poisson jump arrival
4. Add log-normal jump sizes
### Phase 2: Combined Model
1. Implement continuous diffusion component
2. Combine with jump process
3. Add drift compensation
4. Create price path generation
### Phase 3: Advanced Features
1. Add time-varying jump intensity
2. Implement asymmetric jumps
3. Create event-driven jumps
4. Add jump detection analytics
## Success Criteria
### Validation Gates
```bash
# Test jump diffusion
cargo test jump_diffusion
cargo test poisson_jumps
# Validate distributions
cargo test jump_size_distribution
```
### Implementation Metrics
- [ ] Jump frequency matches λ parameter
- [ ] Jump sizes follow specified distribution
- [ ] Price paths exhibit both diffusion and jumps
- [ ] Kurtosis higher than pure diffusion
## Dependencies & References
**Research Sources**:
- Merton's jump diffusion paper
- Option pricing under jump diffusion
- Event-driven price models
**Mathematical Model**:
- dS/S = (μ - λκ)dt + σdW + (Y-1)dN
- N(t) ~ Poisson(λt)
- log(Y) ~ N(μ_J, σ_J²)
- κ = E[Y-1] jump compensator
**Rust Libraries**:
- Use rand_distr for Poisson
- Leverage existing diffusion code
- RustQuant for numerical methods
## Implementation Tasks
### Phase 1: Jump Components (2-3 hours)
1. Implement Poisson process
2. Add jump size generation
3. Create jump timing
4. Test distributions
### Phase 2: Integration (2-3 hours)
1. Combine with diffusion
2. Add drift adjustment
3. Generate price paths
4. Validate dynamics
### Phase 3: Enhancements (2-3 hours)
1. Add event triggers
2. Implement analytics
3. Create examples
4. Document parameters
## Risk Mitigation
- Validate jump parameters (λ > 0)
- Ensure price positivity
- Test extreme jump scenarios
- Provide calibration guidance
## Success Score
**7/10** - More complex than pure diffusion but well-documented model with clear math.