Writing
Long-form notes and writing.
· 8 min
Why complex differentiability is so much stronger than real. The Cauchy-Riemann equations, Goursat's vanishing contour integral, the Cauchy integral formula recovering a function from its boundary values, and the analyticity and Liouville theorems that follow.
· 5 min
Evaluating integrals by what a function leaves behind at its poles. The Laurent series and the residue, the residue theorem, and a contour computation of the Cauchy characteristic function.
· 6 min
Geometry on a vector space. Inner products, the Cauchy-Schwarz inequality, the norm they induce, the parallelogram law that characterises inner-product norms, orthogonality with Pythagoras and Bessel's inequality, and the definition of a Hilbert space.
· 5 min
The two model infinite-dimensional Hilbert spaces, the square-integrable functions and the square-summable sequences, and the Riesz-Fischer theorem that makes them complete.
· 5 min
The two theorems that make a Hilbert space usable. The projection theorem that a closed convex set has a unique nearest point, the orthogonal decomposition into a subspace and its complement, and the Riesz representation of every bounded linear functional.
· 6 min
When an orthonormal set spans a Hilbert space. The convergence of orthogonal series, the equivalent conditions for an orthonormal basis with Parseval's identity, the Gram-Schmidt construction, and the isometry of every separable Hilbert space with the sequence space l-squared.
· 6 min
The algebra of operators on a Hilbert space. The operator norm and the completeness of the bounded operators, the adjoint built from the Riesz representation, the C-star identity, and the self-adjoint operators whose norm the quadratic form attains.
· 7 mincornerstone
The infinite-dimensional analogue of a symmetric matrix. Compact operators as norm limits of finite-rank ones, attainment of the norm at an eigenvector, and the spectral theorem diagonalising a compact self-adjoint operator by eigenvectors with eigenvalues tending to zero.
· 9 min
The spectral theorem made explicit for kernels. A continuous positive kernel gives a compact positive integral operator with continuous eigenfunctions, and Mercer's theorem expands it as a uniformly convergent eigenfunction series that builds the reproducing kernel Hilbert space.
· 5 min
How a symmetric matrix is diagonalised. Eigenvalues and eigenvectors, the real spectrum and orthogonal eigenvectors of a symmetric matrix, the spectral theorem diagonalising it by an orthonormal basis, and the variational characterisation behind principal component analysis.
· 5 min
The matrices that act as squared lengths. Positive semidefinite matrices characterised by nonnegative eigenvalues, by a Gram factorisation, and by a square root, the Cholesky factorisation of a positive definite matrix, and the covariance matrix as the canonical example.
· 18 min
How a limit order book works as a mechanism. Bids and asks, the price grid, every order type and time-in-force, price-time and pro-rata priority, matching, crossed and locked books, the opening and closing auction, hidden liquidity, and the message stream, with worked examples.
· 31 min
The models built on the order book mechanism. The efficient price is a martingale, the bounce and adverse selection set spreads, queue position sets fills, imbalance sets the microprice, Kyle's lambda turns information into impact, and Almgren-Chriss solves execution.
· 8 mincornerstone
How size is assigned to sets. Sigma-algebras and measures, continuity along monotone limits, closure of measurable functions under pointwise limits, the Caratheodory extension theorem that turns an outer measure into a measure, and the construction of Lebesgue measure.
· 7 min
The Lebesgue integral built from simple functions, and the three convergence theorems that make it usable. We prove monotone convergence, deduce Fatou and dominated convergence, and record the L^p inequalities.
· 5 min
The Banach spaces of integrable powers. Young's inequality, the Holder and Minkowski inequalities that make the p-norm a norm, and the completeness theorem that promotes every L-p to a Banach space, the family of which L-squared is the one Hilbert member.
· 5 min
The exact condition for convergence in the mean. Uniform integrability, its characterisation by uniform absolute continuity, and the Vitali theorem that convergence in measure upgrades to L-one convergence exactly when the sequence is uniformly integrable.
· 7 min
When a double integral equals an iterated one. The product sigma-algebra and the measurability of sections, the construction of the product measure, and the Tonelli and Fubini theorems that exchange the order of integration for nonnegative and for integrable functions.
· 4 min
Absolute continuity, equivalence, and the density that connects two measures.
· 6 min
The geometry that makes minimisation tractable. Convex sets and functions and their first and second-order characterisations, the separating and supporting hyperplane theorems from projection onto a closed convex set, and the fact that a convex function's local minimum is global.
· 6 min
How a constrained minimum is certified. The Lagrangian and its dual function, weak duality, strong duality under Slater's condition from the supporting hyperplane theorem, and the Karush-Kuhn-Tucker conditions that characterise the optimum of a convex program.
· 6 mincornerstone
Probability as measure theory with total mass one. The probability space, random variables and their laws as pushforward measures, expectation as the integral, the change of variables that computes it from the law, and the Markov, Chebyshev, and Jensen inequalities.
· 8 min
The factorisation that makes randomness combine. Independence of events, sigma-algebras, and random variables, the equivalence with a product law, the factorisation of expectations through Fubini, the Borel-Cantelli lemmas, and the Kolmogorov zero-one law for tail events.
· 7 mincornerstone
The Fourier transform of a law. The characteristic function, factorisation over independent sums, the moment expansion, the inversion formula that recovers the law, and the Levy continuity theorem behind the central limit theorem.
· 6 mincornerstone
The distribution stable under linear maps. The Gaussian characteristic function, the Gaussian vector defined by mean and covariance, the equivalence of uncorrelated and independent in the Gaussian case, and the Gaussian process specified by a mean and covariance function.
· 5 min
The modes of convergence for random variables and the two theorems that govern sample averages. We prove the Markov and Chebyshev inequalities, Borel-Cantelli, a strong law under a fourth moment, and the central limit theorem by characteristic functions.
· 5 min
Conditional expectation defined by its averaging property, shown to exist via Radon-Nikodym, and identified with orthogonal projection in L^2. We prove the tower property, the pull-out rule, and conditional Jensen.
· 7 min
Random functions as curves in a Hilbert space. Second-order processes through the geometry of L-squared, mean-square continuity equivalent to a continuous covariance, the mean-square integral, and the covariance operator that the Karhunen-Loeve expansion diagonalises.
· 13 min
The Karhunen-Loeve expansion writes a stochastic process in the eigenbasis of its covariance operator, coordinates that are uncorrelated and mean-square optimal. We prove the expansion and its optimality, then derive it for Brownian motion and the Brownian bridge.
· 6 mincornerstone
Building the canonical random path. The Levy-Ciesielski construction of Brownian motion as a random series in the Schauder basis, the almost-sure uniform convergence giving continuous paths, and the verification of the defining covariance through Parseval's identity.
· 5 min
How to trade return against risk optimally. The Markowitz problem of minimising variance at a target return, its closed-form solution by Lagrange multipliers, the efficient frontier it traces, and the tangency portfolio maximising the Sharpe ratio.
· 7 min
How an option is priced by hedging away its risk. The Black-Scholes partial differential equation derived from Ito's formula and a delta hedge, its risk-neutral solution as a discounted expected payoff, and the closed-form Black-Scholes formula for a European call.
· 8 min
The completeness of the real numbers and what it proves. The least upper bound axiom, limits of sequences, the monotone convergence theorem, Cauchy sequences, and the Bolzano-Weierstrass theorem, the four equivalent faces of completeness that the rest of analysis rests on.
· 6 min
What it means for a function to have no jumps. Limits of functions and their sequential characterisation, continuity, the intermediate value theorem from completeness, the extreme value theorem from Bolzano-Weierstrass, and uniform continuity on a closed interval.
· 8 min
Distance, abstracted. Metric and normed spaces, open and closed sets, completeness and the Banach fixed-point theorem, compactness and Heine-Borel, the equivalence of all norms in finite dimensions, and that a continuous function on a compact set attains its extremes.
· 5 min
The derivative and what the mean value theorem extracts from it. The derivative as a limit, differentiability implying continuity, Fermat's interior-extremum principle, Rolle's theorem and the mean value theorem, and Taylor's theorem with the Lagrange remainder.
· 6 min
Area as a limit of sums squeezed between over and under estimates. Upper and lower sums, the Riemann criterion for integrability, the integrability of continuous functions through uniform continuity, and the fundamental theorem of calculus tying the integral to the derivative.
· 7 min
Infinite sums and the functions they define. Convergence of series, absolute convergence and the root test, the radius of convergence of a power series by Cauchy-Hadamard, term-by-term differentiation, and the exponential series with its defining identity.
· 5 min
Filtrations, stopping times, and the predictable sigma-algebra that encodes non-anticipation. We prove the basic properties of the stopping-time sigma-algebra and identify predictable processes with the measurable closure of the simple integrands.
· 4 min
The defining fair-game property, optional stopping, and Doob's inequalities. We prove the discrete optional stopping theorem, the maximal and L^p inequalities, and the convergence theorem by upcrossings.
· 6 min
Why Brownian motion needs its own calculus. The quadratic variation of a path, the theorem that Brownian motion accumulates quadratic variation equal to elapsed time, the resulting infinite total variation, and the covariation that gives the chain rule its extra term.
· 5 min
The Ito integral built from simple predictable integrands by the isometry, extended to the full L^2 class, and shown to be a martingale. We prove the isometry and the extension.
· 8 min
The chain rule of stochastic calculus. The second-order Taylor expansion that the quadratic variation of Brownian motion forces, Ito's formula for a function of a Brownian motion and of an Ito process, the integration-by-parts rule, and the solution of geometric Brownian motion.
· 4 min
The density process of an equivalent measure, the Bayes rule for conditional expectation, and the Girsanov theorem that removes a drift. We prove the density is a martingale and the conditional Bayes formula, and state Girsanov with its proof outline.
· 7 min
Equations driven by noise, and the theorem that they have a unique solution. The strong solution, Gronwall's inequality, uniqueness via the Lipschitz condition and the Ito isometry, and existence by Picard iteration with the factorial decay that makes the iterates converge.
· 5 min
The canonical mean-reverting diffusion. We solve the Ornstein-Uhlenbeck SDE in closed form, derive its Gaussian transition and stationary laws, read off the half-life, and reduce it to an exact AR(1) recursion for estimation.