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Book 6D
Diffusions, Finance and Computation
The heat equation at work
Brownian motion, martingales and Itô calculus, stochastic differential equations and Feynman–Kac, Fokker–Planck equations, time reversal and diffusion models, Brownian motion on manifolds and curvature, Black–Scholes, American options as free boundary problems, stochastic control, numerical methods for PDE, and computing geometric flows. Written in full detail, to learn from directly.
- Main companionPartial Differential Equations II: Qualitative Studies of Linear Equations · Michael E. Taylor
- Second voicePartial Differential Equations · Jürgen Jost
- ReferenceStochastic Calculus for Finance II: Continuous-Time Models · Steven E. Shreve
- ReferenceStochastic Differential Equations · Bernt Øksendal
Contents
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1
Brownian Motion
Construction, scaling, quadratic variation, the Markov property, the reflection principle and harmonic measure.
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2
Martingales and the Itô Integral
Conditional expectation, martingales and optional stopping, the Itô integral and Itô’s formula.
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3
Stochastic Differential Equations and Feynman–Kac
Existence and uniqueness, generators, the Kolmogorov equations, Fokker–Planck and Feynman–Kac.
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4
Fokker–Planck, Time Reversal and Diffusion Models
How densities of diffusions evolve, relative entropy and convergence to equilibrium, reversing a diffusion in time with the score, and the generative models built on it, with Perelman's conjugate heat equation as the geometric twin.
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5
Diffusions on Manifolds and Curvature
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.
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6
Black–Scholes
Replication, the Black–Scholes equation, its reduction to the heat equation, risk-neutral pricing and the Greeks.
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7
American Options and Free Boundaries
Optimal stopping, the obstacle problem, smooth fit and the perpetual put.
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8
Stochastic Control and the HJB Equation
Dynamic programming, the Hamilton–Jacobi–Bellman equation, verification and Merton’s portfolio problem.
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9
Finite Difference Methods
Explicit, implicit and Crank–Nicolson schemes, consistency, stability and the Lax equivalence theorem.
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10
Monte Carlo, Fourier and Finite Element Methods
Monte Carlo error and variance reduction, Euler–Maruyama, Fourier pricing, and finite element error estimates.
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11
Computing Geometric Flows
Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.
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