In this blog post, we will explore the essential concepts of Itō calculus and how they apply to solving stochastic differential equations (SDEs). We will also see how these concepts are implemented in Python code to simulate and filter stochastic processes.

What is Itō Calculus?

Itō calculus extends classical calculus to stochastic processes, particularly those driven by Brownian motion. It includes key tools like the Itō integral and Itō's Lemma.

Brownian Motion

Brownian motion $B_t$ is a continuous-time stochastic process with:

1. $B_0 = 0$.
2. Independent increments.
3. Normally distributed increments: $B_{t+s} - B_t \sim \mathcal{N}(0, s)$.

Itō Integral

The Itō integral of a process $X_t$ with respect to Brownian motion $B_t$ is denoted: $$\int_0^t X_s , dB_s$$

Itō's Lemma

Itō's Lemma is the stochastic counterpart of the chain rule. For a function $f(t, X_t)$ where $X_t$ follows: $$dX_t = \mu(t, X_t) , dt + \sigma(t, X_t) , dB_t,$$ Itō's Lemma states: $$df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial X} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial X^2} \right) dt + \sigma \frac{\partial f}{\partial X} , dB_t.$$

Solving SDEs with Itō's Lemma

Consider the SDE: $$dX_t = \mu X_t , dt + \sigma X_t , dB_t$$

Applying Itō's Lemma to $Y_t = \ln(X_t)$: $$dY_t = \left( \mu - \frac{1}{2} \sigma^2 \right) dt + \sigma , dB_t$$

Solving this: $$Y_t = \ln(X_t) = \ln(X_0) + \left( \mu - \frac{1}{2} \sigma^2 \right) t + \sigma B_t$$

Exponentiating both sides: $$X_t = X_0 \exp \left( \left( \mu - \frac{1}{2} \sigma^2 \right) t + \sigma B_t \right)$$

Implementation in Python

Here's how this is implemented in the context of a particle filter simulation:

import numpy as np

# Parameters
y0 = 1.0
mu = 0.1
sigma = 0.2
T = 1.0
N = 100
M = 1000
dt = T / N

# Placeholder for observations and time
time = np.linspace(0, T, N+1)
true_state = np.exp((mu - 0.5 * sigma**2) * time + sigma * np.random.normal(0, np.sqrt(time)))

# Simulate and plot the true state
import matplotlib.pyplot as plt

plt.figure(figsize=(10, 6))
plt.plot(time, true_state, label='True State')
plt.xlabel('Time')
plt.ylabel('State')
plt.title('True State Evolution Using Itō Calculus')
plt.legend()
plt.show()

In the code:

• true_state is calculated using the derived formula from Itō's Lemma.
• np.random.normal(0, np.sqrt(time)) simulates the Brownian motion increments. This demonstrates how Itō calculus is applied to model and simulate stochastic processes, providing a powerful toolset for understanding systems influenced by randomness.

Missing:

• Sequential Monte Carlo
• Geometric Brownian motion
• Stochastic differential equations definition
• full demo
• Applications in finance