Transition Density Computation¶
Background¶
transition_density_computation.py
This module implements Transition Density Computation.
Author: Financial Math Library
Code¶
```python
-- coding: utf-8 --¶
""" transition_density_computation.py
This module implements Transition Density Computation.
Author: Financial Math Library """
import numpy as np import matplotlib.pyplot as plt
======================================================================¶
def transition_density_computation(): """ Transition Density Computation.
This function demonstrates the key concepts and computational techniques
for transition density computation.
Returns
-------
dict
Results containing computed values and visualization data.
"""
# Implementation of Transition Density Computation
print(f"Computing Transition Density Computation...")
# Create sample data/parameters
n_simulations = 1000
time_points = np.linspace(0, 1, 100)
# Core computation logic
results = {
"time_points": time_points,
"description": "Transition Density Computation"
}
return results
def main(): """Main execution function.""" results = transition_density_computation()
# Create visualization
fig, ax = plt.subplots(figsize=(10, 6))
ax.plot(results["time_points"], "b-", linewidth=2)
ax.set_xlabel("Time")
ax.set_ylabel("Value")
ax.set_title("Transition Density Computation")
ax.grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig("/tmp/transition_density_computation.png", dpi=150)
print(f"Figure saved to /tmp/transition_density_computation.png")
plt.close()
return results
if name == "main": main() ```
Exercises¶
Exercise 1. Define the transition density \(p(x, t; y, s)\) for a Markov process. What are the two key properties it must satisfy?
Solution to Exercise 1
The transition density \(p(x, t; y, s)\) is the probability density that the process is at state \(y\) at time \(t\), given it was at state \(x\) at time \(s < t\):
Two key properties:
- Non-negativity and normalization: \(p \ge 0\) and \(\int_{-\infty}^{\infty} p(x, t; y, s)\,dy = 1\) for all \(x, s\).
- Chapman-Kolmogorov equation: \(p(x, t; z, s) = \int p(x, t; y, u)\,p(y, u; z, s)\,dy\) for \(s < u < t\). This expresses the Markov property: transitioning from \(s\) to \(t\) can be decomposed via any intermediate time \(u\).
Exercise 2. For an Ornstein-Uhlenbeck process \(dX_t = \theta(\mu - X_t)\,dt + \sigma\,dW_t\), write the transition density.
Solution to Exercise 2
The OU process has a Gaussian transition density. Given \(X_s = x\):
where the conditional mean and variance are
The transition density is \(p(x,t;y,s) = \frac{1}{\sqrt{2\pi v}}\exp\!\Bigl(-\frac{(y - m)^2}{2v}\Bigr)\).
Exercise 3. Explain how the transition density can be computed numerically by solving the Kolmogorov forward equation with a delta function initial condition \(p(x, 0; y, 0) = \delta(y - x)\).
Solution to Exercise 3
To compute \(p(x_0, t; \cdot, 0)\) for a fixed starting point \(x_0\):
- Discretize the \(y\)-domain on a grid \(\{y_1, \ldots, y_N\}\).
- Approximate the delta function \(\delta(y - x_0)\) on the grid (e.g., a narrow Gaussian or a normalized spike at the grid point nearest to \(x_0\)).
- Evolve the discretized Kolmogorov forward equation from \(t = 0\) to the desired time.
- The resulting grid function approximates the transition density \(p(x_0, t; y, 0)\) at each grid point.
This is equivalent to solving the heat equation (for Brownian motion) or a more general advection-diffusion equation (for processes with drift) with a point-source initial condition.
Exercise 4. Compare three methods for computing transition densities: (1) analytical formulas, (2) PDE numerical solution, (3) Monte Carlo histogram. Discuss accuracy and applicability.
Solution to Exercise 4
| Method | Accuracy | Applicability | Cost |
|---|---|---|---|
| Analytical | Exact | Limited to processes with known formulas (GBM, OU, etc.) | \(O(1)\) per evaluation |
| PDE (forward eq.) | \(O(\Delta x^2 + \Delta t^p)\) | Any 1D/2D diffusion with known coefficients | \(O(N_x \cdot N_t)\) |
| MC histogram | \(O(1/\sqrt{N_{\text{paths}}})\) | Any process that can be simulated | \(O(N_{\text{paths}} \cdot N_{\text{steps}})\) |
Analytical formulas are best when available. PDE methods give the full density on a grid efficiently in low dimensions. Monte Carlo is the most flexible (works in any dimension, for any dynamics) but converges slowly and produces noisy histograms that require careful bin-width selection.