Partial differential equations

A partial differential equation (PDE) relates an unknown function to its partial derivatives:

Unlike ODEs, several independent variables coexist — typically time and space or market variables (underlying price, rates, etc.).

A solution is generally fixed by boundary conditions (on the domain boundary) and/or initial conditions (at ).

Intuition

A PDE describes a phenomenon varying simultaneously in time and space (or across several factors). The solution is no longer a curve but a surface or field. Elliptic: equilibrium state, no preferred propagation direction (Laplace). Parabolic: diffusion / dissipation over time (heat, Black-Scholes). Hyperbolic: wave propagation at finite speed (wave equation).

Classification (linear, order 2)

For a linear second-order PDE in two variables :

examine the discriminant :

Class	Canonical example	Behavior
Elliptic	(Laplace)	Equilibrium, no privileged time
Parabolic	(heat)	Diffusion, smoothing over time
Hyperbolic	(waves)	Propagation, domains of dependence

Fundamental examples

PDE	Form	Role
Laplace		Equilibrium potential, boundary problems
Heat		Diffusion, Gaussian smoothing
Wave		Propagation without dissipation
Black-Scholes		Option pricing — see Black-Scholes
Transport		Advection, displacement without diffusion

Boundary and initial conditions

Type	Example	Meaning
Dirichlet	on the boundary	Fixed value (e.g. payoff at maturity for an option)
Neumann	on the boundary	Imposed flux or normal derivative
Initial (Cauchy)		Starting state for evolution PDEs
Terminal (finance)		Payoff at maturity; backward time resolution

Solution methods

Separation of variables (Fourier, Sturm-Liouville). - Fourier / Laplace transform for linear equations on simple domains. - Green's functions: solution for a point source. - Black-Scholes: transform to heat equation via change of variables.

Finite differences: discretize , on a grid — see Numerical methods.
Finite elements: variational basis on complex domains.
Monte Carlo: probabilistic alternative via Feynman-Kac (risk-neutral expectation).

Choice depends on domain, dimension, and boundary conditions.

Probability link: Feynman-Kac

For certain parabolic PDEs, the solution is an expectation:

where follows an SDE tied to the PDE coefficients. This bridges:

PDE pricing (Black-Scholes backward);
Monte Carlo pricing (risk-neutral simulation).

Both approaches should yield the same price under the same assumptions — useful for cross-validation.

Applications

Quantitative finance: option pricing (Black-Scholes), local vol models, surface calibration.
Physics: diffusion, fluid mechanics, electromagnetism.
Engineering: heat transfer, shape optimization.
Image / ML: diffusion as noise models (score-based models) — same parabolic formalism.

ODEs are the single independent variable case; PDEs add spatial or multi-factor dimension.

Limits and pitfalls

Situation	Issue	Consequence
Ill-posed (Hadamard)	Solution unstable to initial data	Small I.C. noise → blow-up (e.g. ill-posed backward heat)
High dimension	2D/3D+ grid costly	Curse of dimensionality; Monte Carlo often preferable
Incompatible conditions	Contradictory Dirichlet + Neumann	No solution or non-uniqueness
Numerical stability	Poor explicit scheme	Divergent numerical solution (CFL for parabolic/hyperbolic)
Complex boundaries	Exotic options, barriers	Adapted grids or specialized methods needed