Stochastic Calculus: Conceptual Theory, FAQs, and Quantitative Finance Interview Problems

In quantitative finance, theoretical physics, and academic screens, stochastic calculus is both the ultimate language of continuous-time modeling and the primary filter in technical interviews. Questions on stochastic calculus generally split into two distinct categories:

1. →Conceptual Theory: Why classical calculus fails, how quadratic variation fundamentally changes integration, the difference between Itô and Stratonovich, and the machinery of measure changes.
2. →Practical Problem-Solving & Interview Puzzles: Direct calculations involving Itô's Lemma, solving Stochastic Differential Equations (SDEs), and evaluating stochastic integrals and expectations under pressure.

This guide provides a rigorous yet intuitive reference covering both dimensions.

Part I: Conceptual Theory & Foundational FAQs

1. Why does standard Newtonian calculus fail for Brownian motion?

The Core Reason: Standard Riemann-Stieltjes integration $\int f(t) dg(t)$ requires the integrator $g(t)$ to have bounded (finite) total variation.

A standard Brownian motion (or Wiener process) $W_t$:

• →Is almost surely continuous everywhere, but
• →Is almost surely nowhere differentiable, and
• →Has unbounded (infinite) total variation over any finite time interval $[0, t]$: $\lim_{\Delta t \to 0} \sum_{i=1}^n |W_{t_i} - W_{t_{i-1}}| = \infty \quad \text{a.s.}$

Because Brownian motion oscillates infinitely wildly at infinitesimal scales, the ordinary chain rule: $\frac{df}{dt} = f'(x) \frac{dx}{dt}$ cannot be evaluated because $\frac{dW_t}{dt}$ does not exist in the classical sense.

2. What is Quadratic Variation, and why is $(dW_t)^2 = dt$?

While the total variation of Brownian motion is infinite, its quadratic variation is finite and deterministic.

Over a partition $\Pi = \{0 = t_0 < t_1 < \dots < t_n = t\}$ with mesh size $\|\Pi\| \to 0$: $[W, W]_t = \lim_{\|\Pi\| \to 0} \sum_{i=1}^n (W_{t_i} - W_{t_{i-1}})^2 = t \quad \text{in } L^2$

In differential notation, this yields the fundamental multiplication rule of stochastic calculus: $(dW_t)^2 = dt$

Intuition: Brownian increments scale with the square root of time: $\Delta W_t \sim \mathcal{N}(0, \Delta t) \implies \Delta W_t = \mathcal{O}(\sqrt{\Delta t})$ In classical calculus, terms of order $(\Delta t)^2$ vanish when taking limits. But for Brownian motion: $(\Delta W_t)^2 \sim \mathcal{O}(\Delta t)$ Because $(\Delta W_t)^2$ is of the first order in time, it cannot be discarded in a Taylor expansion.

Multiplication Rules of Stochastic Calculus:
  dt   × dt   = 0
  dt   × dW_t = 0
  dW_t × dt   = 0
  dW_t × dW_t = dt

3. What is Itô's Lemma, and where does the extra term come from?

Itô's Lemma is the stochastic counterpart of the chain rule. If $X_t$ follows the Itô drift-diffusion process: $dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dW_t$

And $f(t, X_t)$ is a twice continuously differentiable function $C^{1,2}$, then the differential $df(t, X_t)$ is:

$df(t, X_t) = \left( \frac{\partial f}{\partial t} + \mu \frac{\partial f}{\partial x} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2} \right) dt + \sigma \frac{\partial f}{\partial x} dW_t$

Where does the extra $\frac{1}{2}\sigma^2 \frac{\partial^2 f}{\partial x^2} dt$ come from? It arises directly from the second-order term of the Taylor expansion: $\Delta f = \frac{\partial f}{\partial t} \Delta t + \frac{\partial f}{\partial x} \Delta X_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} (\Delta X_t)^2 + \dots$ Substituting $\Delta X_t = \mu \Delta t + \sigma \Delta W_t$: $(\Delta X_t)^2 = \sigma^2 (\Delta W_t)^2 + \mathcal{O}((\Delta t)^{3/2}) = \sigma^2 \Delta t$ This is Itô's correction term (often called the convexity or Jensen's inequality correction). If $f$ is convex ($\frac{\partial^2 f}{\partial x^2} > 0$), randomness inherently drifts the expected value upward.

4. What is the difference between Itô and Stratonovich integration?

The distinction lies in the evaluation point chosen when defining the Riemann-type sum: $\int_0^t Y_s \circ dW_s = \lim_{\|\Pi\| \to 0} \sum_{i=0}^{n-1} Y_{\tau_i} (W_{t_{i+1}} - W_{t_i})$

1. →

   Itô Integral ($\tau_i = t_i$, left endpoint):

   • →Definition: Evaluated at the beginning of each interval: $\sum Y_{t_i} \Delta W_i$.
   • →Key Property: It is non-anticipative (causal). The integrator does not look into the future.
   • →Martingale Property: If $Y_t$ is square-integrable, $\mathbb{E}\left[ \int_0^t Y_s dW_s \right] = 0$.
   • →Consequence: The classical chain rule breaks; requires Itô's Lemma.
   • →Domain: Used universally in quantitative finance and economics, where agents cannot see future prices.
2. →

   Stratonovich Integral ($\tau_i = \frac{t_i + t_{i+1}}{2}$, midpoint):

   • →Definition: Evaluated at the midpoint: $\sum Y_{\frac{t_i + t_{i+1}}{2}} \Delta W_i$, denoted with $\circ$.
   • →Key Property: Obeys standard Newtonian chain rule without second-order correction terms!
   • →Consequence: Not a martingale; $\mathbb{E}\left[ \int_0^t Y_s \circ dW_s \right] \ne 0$.
   • →Domain: Used in physics and engineering (e.g., Langevin equations with colored physical noise tending to white noise).

5. What is Girsanov's Theorem, and why is it essential in pricing?

Girsanov's Theorem describes how the drift of a stochastic process changes when switching from one equivalent probability measure to another.

In quantitative finance:

• →Under the physical (real-world) measure $\mathbb{P}$, an asset drifts at its expected return $\mu$: $dS_t = \mu S_t dt + \sigma S_t dW_t^{\mathbb{P}}$
• →Under the risk-neutral measure $\mathbb{Q}$, discounted asset prices must be martingales. Therefore, the asset must drift at the risk-free rate $r$: $dS_t = r S_t dt + \sigma S_t dW_t^{\mathbb{Q}}$

Girsanov's theorem defines the new Brownian motion $W_t^{\mathbb{Q}}$ via the Radon-Nikodym derivative process $\frac{d\mathbb{Q}}{d\mathbb{P}}$: $W_t^{\mathbb{Q}} = W_t^{\mathbb{P}} + \int_0^t \frac{\mu - r}{\sigma} ds$

Crucial Insight: Changing measures alters the drift ($\mu \to r$), but leaves volatility $\sigma$ and quadratic variation completely invariant.

6. What is the Feynman-Kac Formula?

The Feynman-Kac Theorem bridges the gap between stochastic differential equations and deterministic partial differential equations (PDEs).

If a function $V(t, x)$ solves the parabolic terminal-value PDE: $\frac{\partial V}{\partial t} + \mu(t, x) \frac{\partial V}{\partial x} + \frac{1}{2} \sigma^2(t, x) \frac{\partial^2 V}{\partial x^2} - r V = 0, \quad V(T, x) = g(x)$

Then $V(t, x)$ can be represented as the conditional expectation under the risk-neutral measure: $V(t, x) = \mathbb{E}^{\mathbb{Q}} \left[ e^{-r(T - t)} g(X_T) \mid X_t = x \right]$ where $dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dW_t$.

This theorem proves that pricing a derivative via a Black-Scholes PDE or via Monte Carlo expectation are mathematically identical.

Part II: Practical Interview Problems & Quant Puzzles

Problem 1: Solve Geometric Brownian Motion (GBM)

Problem: Find the analytical solution to the SDE: $dS_t = \mu S_t dt + \sigma S_t dW_t, \quad S_0 > 0$

Solution:

1. →Define the transformation $f(S_t) = \ln S_t$.
2. →Compute partial derivatives: $\frac{\partial f}{\partial S} = \frac{1}{S}, \quad \frac{\partial^2 f}{\partial S^2} = -\frac{1}{S^2}, \quad \frac{\partial f}{\partial t} = 0$
3. →Apply Itô's Lemma: $d(\ln S_t) = \frac{1}{S_t} dS_t + \frac{1}{2} \left( -\frac{1}{S_t^2} \right) (dS_t)^2$
4. →Substitute $dS_t = \mu S_t dt + \sigma S_t dW_t$ and $(dS_t)^2 = \sigma^2 S_t^2 dt$: $d(\ln S_t) = \frac{1}{S_t} (\mu S_t dt + \sigma S_t dW_t) - \frac{1}{2 S_t^2} (\sigma^2 S_t^2 dt)$ $d(\ln S_t) = \left( \mu - \frac{1}{2}\sigma^2 \right) dt + \sigma dW_t$
5. →Integrate both sides from $0$ to $t$: $\ln S_t - \ln S_0 = \left( \mu - \frac{1}{2}\sigma^2 \right) t + \sigma W_t$
6. →Exponentiate: $\mathbf{S_t = S_0 \exp\left( \left( \mu - \frac{1}{2}\sigma^2 \right) t + \sigma W_t \right)}$

Problem 2: Evaluate the Stochastic Integral $\int_0^t W_s dW_s$

Problem: Compute $\int_0^t W_s dW_s$ and find its expectation.

Solution: In classical calculus, $\int x dx = \frac{1}{2}x^2$. Let's test $f(W_t) = W_t^2$ using Itô's Lemma:

1. →$f(x) = x^2 \implies f'(x) = 2x, \quad f''(x) = 2$.
2. →Applying Itô's Lemma to $W_t^2$: $d(W_t^2) = 2 W_t dW_t + \frac{1}{2}(2) dt = 2 W_t dW_t + dt$
3. →Rearranging: $W_t dW_t = \frac{1}{2} d(W_t^2) - \frac{1}{2} dt$
4. →Integrating from $0$ to $t$ (noting $W_0 = 0$): $\mathbf{\int_0^t W_s dW_s = \frac{1}{2} W_t^2 - \frac{1}{2} t}$

Expectation Verification: $\mathbb{E}\left[ \int_0^t W_s dW_s \right] = \mathbb{E}\left[ \frac{1}{2} W_t^2 - \frac{1}{2} t \right] = \frac{1}{2} \mathbb{E}[W_t^2] - \frac{1}{2}t$ Since $\mathbb{E}[W_t^2] = \text{Var}(W_t) = t$: $\frac{1}{2}(t) - \frac{1}{2}(t) = 0$ This perfectly confirms the Itô isometry property that the expectation of an Itô integral with respect to Brownian motion is zero.

Problem 3: Find the Fourth Moment $\mathbb{E}[W_t^4]$

Problem: Calculate $\mathbb{E}[W_t^4]$ using Itô's Lemma.

Solution:

1. →Let $f(x) = x^4 \implies f'(x) = 4x^3, \quad f''(x) = 12x^2$.
2. →By Itô's Lemma: $d(W_t^4) = 4 W_t^3 dW_t + 6 W_t^2 dt$
3. →Write in integral form: $W_t^4 = 4 \int_0^t W_s^3 dW_s + 6 \int_0^t W_s^2 ds$
4. →Take expectations of both sides. Since $\mathbb{E}\left[\int_0^t W_s^3 dW_s\right] = 0$: $\mathbb{E}[W_t^4] = 6 \int_0^t \mathbb{E}[W_s^2] ds$
5. →Knowing $\mathbb{E}[W_s^2] = s$: $\mathbb{E}[W_t^4] = 6 \int_0^t s \, ds = 6 \left[ \frac{t^2}{2} \right] = \mathbf{3t^2}$

This matches the standard normal kurtosis formula: for $Z \sim \mathcal{N}(0, 1)$, $\mathbb{E}[Z^4] = 3 \implies \mathbb{E}[(\sqrt{t}Z)^4] = 3t^2$.

Problem 4: The Ornstein-Uhlenbeck (Mean-Reverting) Process

Problem: Solve the Vasicek / Ornstein-Uhlenbeck SDE: $dX_t = \theta(\mu - X_t) dt + \sigma dW_t, \quad \theta > 0$ Find the conditional expectation $\mathbb{E}[X_t \mid X_0]$ and conditional variance $\text{Var}(X_t \mid X_0)$.

Solution:

1. →Multiply by the integrating factor $e^{\theta t}$: $d(e^{\theta t} X_t) = \theta e^{\theta t} X_t dt + e^{\theta t} dX_t$
2. →Substitute $dX_t$: $d(e^{\theta t} X_t) = \theta e^{\theta t} X_t dt + e^{\theta t} (\theta(\mu - X_t) dt + \sigma dW_t)$ $d(e^{\theta t} X_t) = \theta \mu e^{\theta t} dt + \sigma e^{\theta t} dW_t$
3. →Integrate from $0$ to $t$: $e^{\theta t} X_t - X_0 = \theta \mu \int_0^t e^{\theta s} ds + \sigma \int_0^t e^{\theta s} dW_s$ $e^{\theta t} X_t - X_0 = \theta \mu \left( \frac{e^{\theta t} - 1}{\theta} \right) + \sigma \int_0^t e^{\theta s} dW_s$
4. →Multiply through by $e^{-\theta t}$: $\mathbf{X_t = X_0 e^{-\theta t} + \mu (1 - e^{-\theta t}) + \sigma \int_0^t e^{-\theta(t - s)} dW_s}$

Moments:

• →Mean: $\mathbb{E}[X_t] = X_0 e^{-\theta t} + \mu (1 - e^{-\theta t}) \xrightarrow{t \to \infty} \mu$.
• →Variance (via Itô Isometry): $\text{Var}(X_t) = \mathbb{E}\left[ \left( \sigma \int_0^t e^{-\theta(t - s)} dW_s \right)^2 \right] = \sigma^2 \int_0^t e^{-2\theta(t - s)} ds = \mathbf{\frac{\sigma^2}{2\theta} (1 - e^{-2\theta t})}$
• →Steady-State Variance: As $t \to \infty$, $\text{Var}(X_\infty) = \frac{\sigma^2}{2\theta}$.

Summary Cheat Sheet for Interviews