This is the capstone. Six lessons built the machinery for randomness that unfolds over time — starting from the bare definition of a process and ending with the equations quants actually trade on. You learned what a process even is (a whole family of random variables indexed by time, with a sample path running across one universe and a cross-sectional distribution slicing across many); how random walks let variance fan out like √t while martingales refuse to drift; how Markov chains forget their past and settle into a stationary distribution; how Brownian motion is the continuous limit that is everywhere jagged and nowhere differentiable; how Itô’s lemma sneaks in a ½σ² correction because (dW)² = dt refuses to vanish; and how mean reversion and jumps bend plain geometric Brownian motion back toward reality. No formula sheet, no hints, no take-backs: every answer locks the instant you submit, the wrong options are the exact traps that fool real desks, and your score stays hidden until the end.
Big picture
Stochastic Processes — the whole ladder
- Stochastic Processes
- What a process is
- Sample path vs cross-sectional distribution
- Index set, state space, filtration
- Independent vs stationary increments
- Random walks & martingales
- Variance grows like t, std like √t
- Drift nμ vs noise √n
- Martingale: E[X next | info now] = X now
- Markov chains
- Memoryless: only the present matters
- Transition matrix, n-step = matrix power
- Stationary distribution πP = π
- Brownian motion
- Continuous limit of a random walk
- Increments Normal(0, time gap)
- Continuous but nowhere differentiable; (dW)² = dt
- Itô & SDEs
- SDE = drift dt + diffusion dW
- Itô adds ½σ² f″ dt
- Log-price drifts at μ − ½σ²; GBM is lognormal
- Mean reversion & jumps
- OU restoring drift κ(θ − X), half-life ln2/κ
- Finite long-run variance, unlike a random walk
- Jump-diffusion adds Poisson jumps for gaps & fat tails
- What a process is
How this exam works
This is a graded exam. Questions arrive one at a time. Once you submit an answer it is final — there is no going back, no second try, and a wrong answer simply fails that question. Your score stays hidden until the very end, where you need 70% to pass. Read every option before you commit.
What is a stochastic process, precisely?
Select an answer to continue.
Passed? Here's what you now own
You can read a process the way a quant does: tell a sample path from a cross-sectional distribution, spot when variance fans out like √t, recognize a martingale’s missing drift, power a transition matrix to its stationary distribution, write an SDE as drift plus diffusion, and remember that (dW)² = dt is the whole reason Itô calculus exists. Most of all, you know which model bends toward which reality — GBM for prices, OU for spreads and rates, and jumps for the gaps the smooth models pretend away.
That’s the stochastic-processes toolkit, end to end — the continuous-time language under option pricing, term-structure models, and risk engines alike. You now own both the equations and the judgment to know which one your problem actually speaks.