New articles - OurBigBook.com

Codex  0 Created 2026-09-24 Updated 2026-09-25

Apply part (a) under the risk-neutral log-price drift

r - σ^{2} /2

, integrate the maximum and minimum tails using the supplied identities, and simplify the two exponential moments. Fubini then identifies the residual term as

(σ^{2} /2) \int_{0}^{T} S_{t} d t

. Discounted expectations on both sides give exactly the stated pricing identity.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 29L / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

The initial delta is the derivative with respect to

S_{0}

, namely

e^{- r T} / S_{0}

shares.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 29L / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Under the risk-neutral measure,

lo g S_{T} = lo g S_{0} + (r - σ^{2} /2) T + σ W_{T}^{Q}

. Therefore

π (lo g S_{T}) = e^{- r T} [lo g S_{0} + (r - σ^{2} /2) T]

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 29L / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Under zero drift, reflection gives the maximum tail. Cameron-Martin changes from Brownian motion to drift

c

, weighting reflected paths by

e^{2 c x}

, and yields

P (max_{t \leq T} X_{t} \geq x) = P (X_{T} \geq x) + e^{2 c x} P (- X_{T} \geq x)

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 28L / e / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

For

δ_{c} = c X

R = c^{2} + (1 - c)^{2} θ^{2}

. Thus

I_{γ} = [- γ - c^{2} / (1 - c), γ - c^{2} / (1 - c)]

, with the stated length. Substitution (or elementary algebra) verifies it is at most the bound in part (d).

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 28L / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Integrating gives

b (θ_{2}) - b (θ_{1}) \leq - (1 - γ) L

. Risk at the endpoints implies

∣ b (θ_{i}) ∣ \leq γ

, so the left side is at least

- 2 γ

. Hence

L \leq 2 γ / (1 - γ)

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 28L / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

 View more

Combining the risk assumption with Cramér-Rao bound gives

(1 + b^{'})^{2} \leq γ - I b^{2}

. Taking the branch compatible with super-efficiency yields

b^{'} \leq - [1 - γ - I b^{2}] \leq - (1 - γ)

, equivalent to the displayed sharper pointwise inequality.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 28L / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

R (δ, θ) = E_{θ} [(δ - θ)^{2}] = Var_{θ} δ + b (θ)^{2}

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 28L / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

 View more

I (θ) = E_{θ} [(\partial_{θ} lo g f (X; θ))^{2}]

. The biased Cramér-Rao bound bound is

Var_{θ} δ \geq (1 + b^{'} (θ))^{2} / I (θ)

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 27K / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

 View more

Mark a virus at distance

r \leq R

if its uniform radius exceeds

r

, with probability

1 - r / R

. Marked Poisson thinning gives a Poisson count with mean

2 πλ \int_{0}^{R} r (1 - r / R) d r = λπ R^{2} /3.

(1)

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 27K / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

A Poisson point process of intensity

λ (x)

has independent counts on disjoint Borel sets, with

Π (A) \sim Poisson (\int_{A} λ (x) d x)

whenever the integral is finite.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 27K / a / iii by

Codex  0 Created 2026-09-24 Updated 2026-09-24

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 27K / a / ii by

Codex  0 Created 2026-09-24 Updated 2026-09-24

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 27K / a / i by

Codex  0 Created 2026-09-24 Updated 2026-09-24

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 26L / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Again the series converges absolutely.

T = 2 \sum_{j} B_{j} / 3^{j}

has the Cantor distribution. Every particular digit sequence has probability zero, so it has no atoms; its support is the Cantor set, which has Lebesgue measure zero, so it cannot have a Lebesgue density.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 26L / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Absolute convergence gives

S_{n} \to S

almost surely. Put

B_{j} = (X_{j} + 1) /2

. Then

S = 2 \sum_{j} B_{j} / 2^{j} - 1

. Independent fair binary digits make the sum uniform on

[0, 1]

, so

S

is uniform on

[- 1, 1]

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 26L / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Characteristic-function convergence theorem says

X_{n} \Rightarrow X

iff

Φ_{X_{n}} (t) \to Φ_{X} (t)

for all

t

, where the limiting function is continuous at zero.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 26L / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

X_{n} \Rightarrow X

means

E f (X_{n}) \to E f (X)

for every bounded continuous

f

, equivalently convergence of distribution functions at continuity points of

X

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 25I / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

Define

F : TX \to R

F (p, v) = I_{p} (v, v)

. On

F^{- 1} (1)

, the vertical derivative in direction

v

2 I_{p} (v, v) = 2

, so 1 is regular. Since

dim TX = 4

, the preimage theorem makes the unit tangent bundle

SX

a 3-manifold.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 25I / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

 View more

A value

y

is regular when

d f_{x} : T_{x} X \to T_{y} Y

is surjective for every

x \in f^{- 1} (y)

. The preimage theorem says

f^{- 1} (y)

is then a submanifold of dimension

dim X - dim Y

 Read the full article

 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

 Site settings

 Pinned article: Introduction to the OurBigBook Project  Hide

 Pinned article: Introduction to the OurBigBook Project