Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · First-year graduate

Stochastic processes

Name: ______________________

Date: ______________________

You will learn to

Construct the Poisson process from exponential waiting times and derive its Poisson counts, independent increments and memorylessness
Describe a continuous-time Markov chain by its generator and solve Kolmogorov's forward and backward equations in simple cases
State the defining properties of Brownian motion and prove its basic path properties — scaling, martingales, quadratic variation, infinite variation and nowhere differentiability
Prove Itô's isometry, compute the integral of B dB, and use Itô's formula to solve the equations of geometric Brownian motion and the Ornstein–Uhlenbeck process

1.
Let $B$ be a standard Brownian motion. Compute $\operatorname{Var}(B_3 - B_1) + \mathbb E[B_2 B_5]$ .
2.
Requests reach a server in Nairobi as a Poisson process of rate $2$ per minute. Given that exactly $10$ requests arrived in the first $5$ minutes, what is the expected number of requests in the first $8$ minutes?
3.
Which of these processes are martingales with respect to the natural filtration of a standard Brownian motion $B$ ? Select all that apply.
- A. $B_t^2 - t$
- B. $B_t^3 - 3 t B_t$
- C. $\exp(B_t - t/2)$
- D. $\exp(B_t)$
- E. $B_t^2$
- F. $B_t^3$
4.
Compute $\mathbb E\Big[\Big(\int_0^4 B_s \, dB_s\Big)^2\Big]$ .
5.
Which equation is correct for $X_t = B_t^2$ ?
- A. $dX_t = 2 B_t \, dB_t$
- B. $dX_t = 2 B_t \, dB_t + dt$
- C. $dX_t = 2 B_t \, dB_t + 2 \, dt$
- D. $dX_t = dt$
6.
A geometric Brownian motion has $\mu = 0.05$ and $\sigma = 0.4$ , so $S_t = S_0 \exp\big((\mu - \sigma^2/2) t + \sigma B_t\big)$ . Which statement is true as $t \to \infty$ ?
- A. $\mathbb E S_t \to \infty$ , and yet $S_t \to 0$ almost surely.
- B. $S_t \to \infty$ almost surely, because $\mu > 0$ .
- C. $\mathbb E S_t \to 0$ , because typical paths tend to $0$ .
- D. $S_t$ oscillates between $0$ and $\infty$ forever.
7.
The proof that Brownian paths have infinite variation uses $\sum_k (\Delta B_k)^2 \le \max_k |\Delta B_k| \cdot \sum_k |\Delta B_k|$ along dyadic partitions of $[0, t]$ . Which property of Brownian motion guarantees $\max_k |\Delta B_k| \to 0$ for each path?
- A.Continuity of the paths (hence uniform continuity on $[0, t]$ ).
- B.Independence of the increments.
- C.The Gaussian distribution of the increments.
- D.The martingale property.
8.
For a real sequence S : ℕ → ℝ, which Lean statement is the discrete Itô formula proved in the lesson?
- A.∑ k ∈ Finset.range n, S k * (S (k + 1) - S k) = (S n ^ 2 - S 0 ^ 2) / 2 - (∑ k ∈ Finset.range n, (S (k + 1) - S k) ^ 2) / 2
- B.∑ k ∈ Finset.range n, S k * (S (k + 1) - S k) = (S n ^ 2 - S 0 ^ 2) / 2
- C.∑ k ∈ Finset.range n, S k * (S (k + 1) - S k) = (S n ^ 2 - S 0 ^ 2) / 2 + (∑ k ∈ Finset.range n, (S (k + 1) - S k) ^ 2) / 2
- D.∑ k ∈ Finset.range n, S (k + 1) * (S (k + 1) - S k) = (S n ^ 2 - S 0 ^ 2) / 2 - (∑ k ∈ Finset.range n, (S (k + 1) - S k) ^ 2) / 2
9.
A pure birth chain on $\{0, 1, 2, \dots\}$ jumps from $n$ to $n + 1$ at rate $\lambda_n$ (so the holding time in state $n$ is exponential with mean $1/\lambda_n$ ). For which rate sequences does the chain make infinitely many jumps in finite time (explode) with probability one? Select all that apply.
- A. $\lambda_n = 3$ (a Poisson process)
- B. $\lambda_n = n + 1$
- C. $\lambda_n = (n + 1)^2$
- D. $\lambda_n = 2^n$
10.
A path is given by its values path = [S_0, S_1, ..., S_n] on a partition. Write ito_sum(path) (left-point sum of $S \, dS$ ), stratonovich_sum(path) (midpoint sum, with $\frac12(S_k + S_{k+1})$ ) and quadratic_variation(path). The tests check them on exact rational paths and verify the discrete Itô and Stratonovich formulas.
Computational — Practice

Answer key — Stochastic processes

1.
$\operatorname{Var}(B_3 - B_1) = 3 - 1 = 2$ , and $\mathbb E[B_2 B_5] = \mathbb E[B_2 (B_2 + (B_5 - B_2))] = \mathbb E B_2^2 + 0 = 2 = \min(2, 5)$ . Total: $4$ .
2.
$\mathbb E[N_8 \mid N_5 = 10] = 10 + \mathbb E[N_8 - N_5] = 10 + 2 \cdot 3 = 16$ , using independent increments. (Coincidentally $\mathbb E N_8 = 16$ too, because $10 = \mathbb E N_5$ .)
3.
For $f(t, x)$ with $\partial_t f + \frac12 \partial_{xx} f = 0$ (and enough integrability), $f(t, B_t)$ is a martingale. (a) $x^2 - t$ : $-1 + 1 = 0$ . (b) $x^3 - 3tx$ : $-3x + 3x = 0$ . (c) $e^{x - t/2}$ : $-\frac12 f + \frac12 f = 0$ . (d) $e^x$ : $\frac12 e^x \ne 0$ . (e) $x^2$ : $1$ . (f) $x^3$ : $3x$ . So (a), (b), (c) are martingales; (d), (e) are submartingales and (f) is neither.
4.
By the Itô isometry and Fubini, $\mathbb E (\int_0^4 B \, dB)^2 = \int_0^4 \mathbb E B_s^2 \, ds = \int_0^4 s \, ds = 8$ . Check with $\int_0^4 B \, dB = \frac12 (B_4^2 - 4)$ : since $B_4^2 = 4Z^2$ with $Z$ standard normal, this is $2(Z^2 - 1)$ , whose second moment is $4 \operatorname{Var}(Z^2) = 4 \cdot 2 = 8$ .
5.
With $f(x) = x^2$ : $d(B_t^2) = 2B_t \, dB_t + \frac12 \cdot 2 \, dt = 2 B_t \, dB_t + dt$ . Integrating, $\int_0^t B \, dB = \frac12 B_t^2 - \frac12 t$ , as proved in the lesson by the discrete Itô formula and the quadratic variation.
6.
$\mu - \sigma^2/2 = 0.05 - 0.08 = -0.03$ . Since $B_t/t \to 0$ a.s., $\log S_t = \log S_0 - 0.03 t + 0.4 B_t \to -\infty$ a.s., so $S_t \to 0$ almost surely. But $\mathbb E S_t = S_0 e^{0.05 t} \to \infty$ : the expectation is dominated by rare paths on which $B_t$ is exceptionally large.
7.
(a). Each path is continuous on $[0, t]$ , hence uniformly continuous, so $\max_k |B_{t_{k+1}} - B_{t_k}| \to 0$ as the mesh tends to $0$ . If the total variation $V$ were finite, the quadratic variation sums would tend to $0$ ; but they tend to $t > 0$ almost surely. The same argument shows that any continuous path of finite variation has zero quadratic variation.
8.
(a). For one step, $a(b - a) = \frac12 (b^2 - a^2) - \frac12 (b - a)^2$ ; summing gives (a). Right-point sums use $b(b - a) = \frac12(b^2 - a^2) + \frac12 (b - a)^2$ , so (c) would be correct for them, and (d) mixes the two. (b) is the Stratonovich (midpoint) identity applied to the wrong sum. In Lean the proof is induction on n with Finset.sum_range_succ and ring.
9.
The time of the $n$ -th jump is $E_0 + \dots + E_{n-1}$ with $E_k \sim \mathrm{Exp}(\lambda_k)$ independent, so the chain explodes iff $\zeta = \sum_k E_k < \infty$ . If $\sum 1/\lambda_k < \infty$ , then $\mathbb E \zeta < \infty$ and $\zeta < \infty$ a.s.: (c) and (d). If $\sum 1/\lambda_k = \infty$ , then $\mathbb E e^{-\zeta} = \prod_k \frac{\lambda_k}{\lambda_k + 1} = \prod_k (1 + 1/\lambda_k)^{-1} = 0$ , so $\zeta = \infty$ a.s.: (a) and (b) do not explode.
10.
```
from fractions import Fraction

def ito_sum(path):
    return sum(a * (b - a) for a, b in zip(path, path[1:]))

def stratonovich_sum(path):
    return sum(Fraction(a + b) / 2 * (b - a) for a, b in zip(path, path[1:]))

def quadratic_variation(path):
    return sum((b - a) ** 2 for a, b in zip(path, path[1:]))
```
For each step $a(b - a) = \frac12(b^2 - a^2) - \frac12 (b - a)^2$ and $\frac{a + b}2 (b - a) = \frac12 (b^2 - a^2)$ ; summing gives the discrete Itô and Stratonovich formulas (formalised in Lean as discrete_ito and discrete_stratonovich). For a $\pm1$ walk the quadratic variation is the number of steps, so the Itô sum is $(S_n^2 - n)/2$ — the discrete $\int B \, dB = \frac12 (B_t^2 - t)$ .