Optiver·Software Engineer·Technical Phone Screen·Intermediate

Intermediate

Apr 2026

Summary

Quant Engineer interview at Optiver with a probability/stochastic processes question that sounds deceptively simple but has a non-obvious answer if you haven't seen it before.

Questions Asked (1)

Consider a 2D random walk where a particle moves up, down, left, or right each with equal probability. What is the probability that the particle eventually returns to the origin?

Algorithms & Data StructuresTechnical Trade-offs

Author's notes

The answer is 1, it returns with certainty, same as 1D.

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AI HintsAI Generated

Suggested Approach

Recognize this as a classic problem in probability theory: a 2D simple symmetric random walk is recurrent, so the probability of eventual return to the origin is 1. Explain the intuition and, if possible, sketch a proof using the divergence of the sum of return probabilities or electrical network theory. Be prepared to contrast with the 1D and 3D cases to demonstrate deeper understanding.

Pro tip: Mention that while the return probability is 1, the expected return time is infinite—this subtlety often impresses interviewers and shows you understand the difference between recurrence and positive recurrence.

1. Clarify the problem

Confirm that the walk is simple, symmetric, and on the infinite 2D integer lattice, starting at the origin. Define 'eventually returns' as the probability that the particle hits the origin at some time n > 0.

2. Recall known results

State that for a simple symmetric random walk, the return probability is 1 in 1D and 2D (recurrent), but less than 1 in 3D and higher (transient).

3. Provide intuition or proof sketch

Explain that in 2D, the number of possible paths grows as 4^n while the probability of being at the origin at time 2n decays as 1/n, leading to a divergent sum of return probabilities, which implies recurrence.

4. Address potential follow-ups

Be ready to discuss the expected return time (infinite) and the contrast with 3D, where the return probability is about 0.34.

Key Points to Mention

Definition of simple symmetric random walk on Z^2.
Recurrence of 2D random walk: probability of return is 1.
Difference between recurrence and positive recurrence (expected return time is infinite).
Contrast with 1D (also recurrent) and 3D (transient, return probability < 1).
Proof sketch using sum of return probabilities diverging or electrical networks.
Implications: the walk visits every point infinitely often, but the time between visits has infinite expectation.

AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.