Stochastic Calculus & Brownian Motion

In classical calculus, we deal with functions $f(t)$ that are sufficiently smooth. However, in the physical and financial worlds, many processes are driven by inherent randomness that is everywhere non-differentiable. Stochastic calculus provides the mathematical framework to integrate and differentiate with respect to these “jagged” paths, most notably Brownian Motion.

1. Brownian Motion (The Wiener Process)

A stochastic process $\{W_t{\}}_{t\ge 0}$ is a standard Wiener Process (or Brownian Motion) if it satisfies the following properties:

1. Initial Value: $W_0=0$ almost surely.
2. Independent Increments: For any $0\le s<t$, the increment $W_t-W_s$ is independent of the process history up to time $s$, i.e., $W_t-W_s$ is independent of $\sigma (W_u:u\le s)$.
3. Stationary Gaussian Increments: The increment $W_t-W_s$ is normally distributed with mean 0 and variance $t-s$: $W_t-W_s\sim \mathcal{N}(0,t-s)$
4. Path Continuity: $t\mapsto W_t$ is continuous almost surely.

The Path-Wise Paradox

While Brownian motion is continuous, it is nowhere differentiable and has infinite total variation on any interval $[0,t]$. However, it has a finite and non-zero quadratic variation: $[W,W{]}_t={\lim }_{\Vert \Pi \Vert \to 0}{\sum }_{i=1}^n(W_{t_i}-W_{t_{i-1}}{)}^2=t$ Heuristically, this leads to the fundamental identity of stochastic calculus: $(d{W}_t{)}^2=dt$

2. Martingales and Filtrations

To handle information arriving over time, we define a filtration $\{{\mathcal{F}}_t{\}}_{t\ge 0}$, representing the “information available at time $t$“.

A process $\{M_t\}$ is a Martingale with respect to a filtration ${\mathcal{F}}_t$ if:

1. $M_t$ is ${\mathcal{F}}_t$-measurable for all $t$.
2. $E[|M_t|]<\infty$ for all $t$.
3. Fairness Property: For all $s\le t$, $E[M_t|{\mathcal{F}}_s]=M_s$

Standard Brownian motion $W_t$ is a martingale, as is $W_t^2-t$.

3. The Ito Integral

We wish to define an integral of the form: $I_t={\int }_0^t{X}_{s}d{W}_s$ where $X_s$ is an adapted process (its value at $s$ depends only on ${\mathcal{F}}_s$). Because $W_t$ has infinite variation, the traditional Riemann-Stieltjes integral does not converge.

The Ito Integral is defined as the limit of the sum: ${\int }_0^t{X}_{s}d{W}_s={\lim }_{n\to \infty }{\sum }_{i=1}^n{X}_{t_{i-1}}(W_{t_i}-W_{t_{i-1}})$ Crucially, the integrand $X$ is evaluated at the left-endpoint $t_{i-1}$. This ensures that $I_t$ is a martingale. If we were to evaluate at the midpoint (Stratonovich integral), we would lose the martingale property but gain standard calculus rules.

Ito Isometry

One of the most powerful tools for computing variances: $E\left[{\left({\int }_0^t{X}_{s}d{W}_s\right)}^2\right]=E\left[{\int }_0^t{X}_s^{2}ds\right]$

4. Ito’s Lemma

Ito’s Lemma is the stochastic counterpart to the chain rule. If $f(t,x)$ is a twice-differentiable function and $X_t$ is an Ito process, then: $df(t,X_t)=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}d{X}_t+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}(d{X}_t{)}^2$ Substituting $d{X}_t=\mu dt+\sigma d{W}_t$ and using $(d{W}_t{)}^2=dt$: $df(t,W_t)=\left(\frac{\partial f}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}\right)dt+\frac{\partial f}{\partial x}d{W}_t$

5. Stochastic Differential Equations (SDEs)

A general SDE takes the form: $d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t$ where $\mu$ is the drift and $\sigma$ is the diffusion (volatility).

Geometric Brownian Motion (GBM)

In finance, the price of an asset $S_t$ is often modeled by: $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$ Using Ito’s Lemma on $f(S_t)=\ln S_t$:

1. $\frac{\partial f}{\partial S}=\frac{1}{S}$
2. $\frac{{\partial }^{2}f}{\partial S^2}=-\frac{1}{S^2}$
3. $d(\ln S_t)=\frac{1}{S_t}d{S}_t+\frac{1}{2}(-\frac{1}{S_t^2})(d{S}_t{)}^2$
4. $d(\ln S_t)=\frac{1}{S_t}(\mu S_{t}dt+\sigma S_{t}d{W}_t)-\frac{1}{2S_t^2}({\sigma }^2{S}_t^{2}dt)$
5. $d(\ln S_t)=(\mu -\frac{1}{2}{\sigma }^2)dt+\sigma d{W}_t$

Integrating both sides gives the closed-form solution: $S_t=S_0\exp \left((\mu -\frac{1}{2}{\sigma }^2)t+\sigma W_t\right)$

6. Feynman-Kac Formula

The Feynman-Kac formula establishes a link between SDEs and second-order linear PDEs. It states that the solution to the PDE: $\frac{\partial u}{\partial t}+\mu (x,t)\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2(x,t)\frac{{\partial }^{2}u}{\partial x^2}-r(x,t)u=0$ with terminal condition $u(x,T)=\psi (x)$, can be represented as an expectation of the stochastic process $X_t$: $u(x,t)=E\left[e^{-{\int }_t^{T}r(X_{\tau },\tau )d\tau }\psi (X_T)\mid X_t=x\right]$ This is fundamental to the Black-Scholes model and many fields of physics.

7. Python Implementation: Simulating SDEs

We can use the Euler-Maruyama method to simulate paths of Brownian Motion and GBM.

import numpy as np
import matplotlib.pyplot as plt

def simulate_paths(S0, mu, sigma, T, dt, N_paths):
    N_steps = int(T / dt)
    t = np.linspace(0, T, N_steps)
    
    # Generate random increments
    dW = np.random.normal(0, np.sqrt(dt), (N_paths, N_steps))
    W = np.cumsum(dW, axis=1)
    
    # 1. Standard Brownian Motion
    # W contains N_paths of BM
    
    # 2. Geometric Brownian Motion
    # Solution: S_t = S0 * exp((mu - 0.5*sigma**2)*t + sigma*W_t)
    S = S0 * np.exp((mu - 0.5 * sigma**2) * t + sigma * W)
    
    return t, W, S

# Parameters
T = 1.0; dt = 0.001; N_paths = 5
t, W, S = simulate_paths(S0=100, mu=0.05, sigma=0.2, T=T, dt=dt, N_paths=N_paths)

plt.figure(figsize=(12, 5))
plt.subplot(1, 2, 1)
for i in range(N_paths):
    plt.plot(t, W[i, :])
plt.title("Standard Brownian Motion $W_t$")
plt.grid(True)

plt.subplot(1, 2, 2)
for i in range(N_paths):
    plt.plot(t, S[i, :])
plt.title("Geometric Brownian Motion $S_t$")
plt.grid(True)
plt.show()