Arash Mehrjou, PhD

Arash Mehrjou, PhD

Research Scientist in GSK.ai.

Stochastic Calculus

1 minute read

Date:

Stochastic calculus is how we mathematically deal with randomness. It lets us write equations where uncertainty is baked into the dynamics — essential when modeling noisy systems in nature or chaotic movements in financial markets.

What Even Is Randomness?

In deterministic systems, the future is fully determined by the present. But in the real world? Noise, uncertainty, chaos. Enter randomness.

The key object in stochastic calculus is Brownian motion, denoted by $ B_t $. It’s a mathematical model for random movement — think of pollen dancing on water.

The Star: Brownian Motion

Brownian motion $ B_t $ satisfies:

  • $ B_0 = 0 $
  • $ B_t \sim \mathcal{N}(0, t) $: Gaussian with mean 0 and variance $ t $
  • Independent increments: $ B_{t+s} - B_t \sim \mathcal{N}(0, s) $

This process has continuous paths but is nowhere differentiable. Wild, right?

We define the Itô integral:

\[\int_0^t f(s) \, dB_s\]

This is the core tool that makes stochastic calculus tick. It’s like a Riemann integral, but tailored for noise.

Finance: Where It All Took Off

The Black-Scholes model for option pricing is based on the stochastic differential equation (SDE):

\[dS_t = \mu S_t \, dt + \sigma S_t \, dB_t\]

Here, $ S_t $ is the stock price, $ \mu $ the drift, and $ \sigma $ the volatility. This equation captures both expected trends and unpredictable shocks.

Biology and Genomics

Living systems are noisy.

  • Gene expression: Modeled as stochastic processes due to molecular noise.
\[dX_t = a(X_t) \, dt + b(X_t) \, dB_t\]

where $ X_t $ is concentration of mRNA or protein, $ a(X_t) $ the deterministic regulation, and $ b(X_t) $ the stochastic fluctuation.

  • Population dynamics: In small populations, random birth/death events dominate.

  • Neural activity: The timing of neuron firing often follows stochastic models like Poisson or even SDE-driven integrate-and-fire models.

Why It Matters

Stochastic calculus is a powerful lens for seeing the world — not as a set of fixed equations, but as dynamic systems dancing with uncertainty. Whether you’re pricing derivatives or modeling noisy gene circuits, this math gives you the language to describe it.

And it’s beautiful.