Stochastic processes

Let $B$ be a standard Brownian motion. Compute $\operatorname{Var}(B_3 - B_1) + \mathbb E[B_2 B_5]$.

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?

Which of these processes are martingales with respect to the natural filtration of a standard Brownian motion $B$? Select all that apply.

Compute $\mathbb E\Big[\Big(\int_0^4 B_s \, dB_s\Big)^2\Big]$.

Which equation is correct for $X_t = B_t^2$?

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 $t \to \infty$?

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?

For a real sequence S : ℕ → ℝ, which Lean statement is the discrete Itô formula proved in the lesson?

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 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.