Phase 8 Exam

Stochastic Processes & Brownian Motion - Phase Exam

15 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches

Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.

Q1 Medium

The stationary distribution \(\pi\) of a chain with transition matrix \(P\) satisfies:

\(P\pi=\pi\)

\(\pi P=\pi\) with \(\sum_i\pi_i=1\)

\(\pi P=0\)

\(P^2=P\)

Q2 Easy

For the simple symmetric random walk, \(\mathrm{sd}(S_n)\) grows like:

\(n\)

\(\sqrt n\)

\(\log n\)

constant

Q3 Medium

For standard Brownian motion, \(\Cov(B_s,B_t)\) with \(s\le t\) equals:

\(st\)

\(\min(s,t)=s\)

\(t-s\)

\(0\)

Q4 Medium

The quadratic variation of a continuously differentiable path on \([0,t]\) is:

\(t\)

infinite

equal to its total variation

Q5 Medium

A Gaussian process is completely determined by:

Its sample paths

Its mean function and covariance function

Its maximum over time

Its first four moments

Q6 Medium

For an irreducible, aperiodic finite chain, \(\lim_{n\to\infty}(P^n)_{ij}\) equals:

0 for all j

\(\pi_j\), independent of the start i

\(p_{ij}\)

1 for j=i

Q7 Medium

A trading position \(H_n\) that avoids look-ahead must be:

\(\F_n\)-measurable (adapted)

\(\F_{n-1}\)-measurable (predictable)

independent of the past

constant

Q8 Medium

Brownian sample paths are:

Smooth and differentiable

Continuous everywhere but differentiable nowhere

Discontinuous with jumps

Piecewise linear

Q9 Medium

For standard Brownian motion, \(\sum_k(B_{t_k}-B_{t_{k-1}})^2\) as the mesh \(\to0\) converges to:

\(t\)

\(t^2\)

a random Gaussian

Q10 Medium

Two equivalent measures \(\Prob\sim\Q\) must agree on:

The probability of every event

Which events have probability zero

The mean of every random variable

The variance of every random variable

Q11 Hard

Which property guarantees the n-step matrix \(P^n\) actually converges (not just that \(\pi\) exists)?

Recurrence alone

Aperiodicity (together with irreducibility)

A symmetric \(P\)

Having two states

Q12 Hard

The martingale-transform theorem implies that a predictable betting strategy on a fair game yields wealth with:

Positive expected gain

Zero expected gain (still a martingale)

Negative expected gain

Undefined expectation

Q13 Hard

Donsker’s theorem states that the rescaled random walk \(S_{\lfloor nt\rfloor}/\sqrt n\):

Diverges

Converges in distribution to Brownian motion

Converges to a straight line

Becomes a Poisson process

Q14 Hard

The identity \((dB_t)^2=dt\) is important because it:

Makes Brownian paths differentiable

Produces the extra second-order term in Ito’s formula

Shows BM has finite total variation

Implies \(\Var(B_t)=t^2\)

Q15 Hard

Girsanov’s theorem is used in pricing to:

Change the volatility of the asset

Remove the drift so discounted prices become martingales

Make paths differentiable

Eliminate randomness

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