Chapter 5: Partial Differential Equations in Finance¶
This chapter develops the fundamental connection between stochastic differential equations and partial differential equations -- the insight that expected values of diffusion processes satisfy deterministic PDEs. Beginning with the classification of second-order PDEs and the motivation for PDEs in quantitative finance, we study the heat equation as the prototype parabolic PDE, construct Green's functions and transition densities on free and bounded domains, derive the Kolmogorov forward and backward equations as dual descriptions of diffusion processes, and culminate in the Feynman--Kac formula that provides the mathematical foundation for risk-neutral pricing. The infinitesimal generator serves as the bridge throughout: it transforms questions about random processes into deterministic equations.
Key Concepts¶
Overview: Why PDEs Arise in Finance and the SDE--PDE Bridge¶
PDEs appear naturally in finance because option prices, as conditional expectations under the risk-neutral measure, satisfy deterministic equations governed by the infinitesimal generator of the underlying diffusion. For a diffusion \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\), the generator \(\mathcal{L} = \mu(x)\partial_x + \frac{1}{2}\sigma^2(x)\partial_{xx}\) encodes the local dynamics -- drift through its first-order term and volatility through its second-order term. The conditional expectation \(u(t,x) = \mathbb{E}[g(X_T) \mid X_t = x]\) is a deterministic function of the starting point that satisfies a PDE involving \(\mathcal{L}\). Second-order linear PDEs are classified by the discriminant \(B^2 - AC\) of the general form \(Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots = 0\) into elliptic (\(B^2 - AC < 0\), equilibrium), parabolic (\(B^2 - AC = 0\), diffusion), and hyperbolic (\(B^2 - AC > 0\), wave propagation); pricing PDEs in finance are parabolic, reflecting irreversible time evolution and diffusive asset dynamics. Boundary value problems -- Dirichlet (absorbing), Neumann (reflecting), and Robin (partial absorption) -- encode contract features such as payoff conditions, barriers, and exercise boundaries. The SDE--PDE hierarchy unfolds from the generator through the Kolmogorov backward equation \(\partial_t u = \mathcal{L}u\), the Kolmogorov forward equation \(\partial_t p = \mathcal{L}^* p\), and ultimately the Feynman--Kac formula \(\partial_t u + \mathcal{L}u - ru = 0\) with its discounted expectation representation.
The Heat Equation and Its Properties¶
The heat equation \(\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}\) is the canonical parabolic PDE and the analytical counterpart of Brownian motion. Its fundamental solution (heat kernel) is the Gaussian density
which is simultaneously the transition density of standard Brownian motion \(B_t \sim N(0,t)\). General solutions are obtained by convolution: \(u(t,x) = \int_{\mathbb{R}} G(t,x-y)\,f(y)\,dy = \mathbb{E}[f(x + B_t)]\), derived systematically via the Fourier transform where the PDE reduces to the ODE \(\partial_t \hat{u} = -\frac{\xi^2}{2}\hat{u}\). The heat equation exhibits parabolic scaling invariance -- if \(u(t,x)\) is a solution, so is \(u(\lambda^2 t, \lambda x)\) -- reflecting the fundamental \(x \sim \sqrt{t}\) relationship of diffusion and the identical scaling law \((B_t)_{t \geq 0} \overset{d}{=} (\lambda^{-1} B_{\lambda^2 t})_{t \geq 0}\) of Brownian motion. Self-similar solutions of the form \(u(t,x) = t^{-1/2}F(x/\sqrt{t})\) reduce the PDE to an ODE, recovering the heat kernel. The maximum principle states that for subsolutions (\(u_t \leq \frac{1}{2}u_{xx}\)), the maximum is attained on the parabolic boundary \(\Gamma_T\), not in the interior; the strong form implies that an interior maximum forces the solution to be constant. This yields uniqueness, comparison results, stability, and positivity preservation. The probabilistic interpretation is immediate: \(u(t,x) = \mathbb{E}[f(x + B_t)] \leq \sup f\), since expected values of bounded functions are bounded. Energy methods provide an alternative route to uniqueness through the monotonicity of \(\int u^2\,dx\). Key qualitative properties include instantaneous smoothing of rough initial data, infinite speed of propagation, conservation of total mass, and decay of the maximum. In higher dimensions the heat kernel becomes \(G(t,x) = (2\pi t)^{-d/2}\exp(-|x|^2/2t)\), the density of \(d\)-dimensional Brownian motion. The connection between the heat equation and Brownian motion is formalized through the generator: the heat operator \(\partial_t - \frac{1}{2}\partial_{xx}\) corresponds to the generator \(\mathcal{L} = \frac{1}{2}\frac{d^2}{dx^2}\), martingale characterizations (if \(u\) solves the heat equation then \(u(T-t, B_t)\) is a martingale by Ito's lemma), exit problems (Kakutani's theorem connecting the Dirichlet problem to \(\mathbb{E}_x[f(B_\tau)]\)), and Monte Carlo solution methods.
Green's Functions and Transition Densities¶
The Green's function \(G(x,t; y,s)\) for a parabolic PDE is the response to a point source -- the solution with initial condition \(\delta(x-y)\) at time \(s\). For diffusion processes, the transition density \(p(x,t \mid x_0,t_0)\) is precisely the Green's function of the associated Kolmogorov equation, establishing a fundamental link between PDE theory and stochastic processes. On free-space (unbounded) domains, the Green's function is the heat kernel itself, and solutions follow by direct convolution. On bounded domains \([a,b]\) with boundary conditions (absorbing barriers, reflecting walls), the Green's function must be constructed differently -- via the method of images or eigenfunction expansions. Spectral decomposition expands the Green's function in eigenfunctions of the spatial operator:
where \(\mathcal{L}^* \phi_n = -\lambda_n \phi_n\) with eigenvalues \(0 = \lambda_0 < \lambda_1 < \lambda_2 < \cdots\). The eigenfunction \(\phi_0\) corresponding to \(\lambda_0 = 0\) is the stationary distribution (if it exists). For the Ornstein--Uhlenbeck process, the eigenfunctions are Hermite polynomials with eigenvalues \(\lambda_n = n\kappa\), and the spectral expansion gives the full transition density. The distinction between free and bounded domains is critical in finance: vanilla options live on unbounded domains, while barrier options, lookbacks, and knock-in/knock-out contracts require bounded-domain Green's functions that incorporate absorption or reflection at barriers.
Kolmogorov Equations: Forward and Backward¶
The Kolmogorov equations provide two dual descriptions of the same diffusion process \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\). The backward equation \(\frac{\partial u}{\partial t} = \mathcal{L}u\) (initial value form) or equivalently \(\frac{\partial v}{\partial t} + \mathcal{L}v = 0\) (terminal value form) acts on the initial point via the generator \(\mathcal{L} = \mu(x)\partial_x + \frac{1}{2}\sigma^2(x)\partial_{xx}\), and its solution \(u(t,x) = \mathbb{E}_x[g(X_t)]\) describes how expected values depend on starting position. The derivation proceeds from the definition of the generator combined with the Markov property, and is verified via Ito's lemma: the process \(v(t, X_t)\) is a martingale precisely because \(v\) solves the PDE. The forward equation (Fokker--Planck)
acts on the current state via the adjoint operator \(\mathcal{L}^*\) and describes the evolution of the probability density. It can be written as a continuity equation \(\partial_t p + \partial_x J = 0\) with probability current \(J = \mu p - \frac{1}{2}\partial_x[\sigma^2 p]\), expressing conservation of probability mass. The forward and backward equations are connected through adjoint duality: \(\int f(\mathcal{L}g)\,dx = \int (\mathcal{L}^* f)g\,dx\). The transition density \(p(x,t \mid x_0,t_0)\) simultaneously satisfies the backward equation in \((x_0,t_0)\) and the forward equation in \((x,t)\) -- the same object obeys two different PDEs in different variables. Stationary distributions \(p_\infty\) satisfy \(\mathcal{L}^* p_\infty = 0\) with the general one-dimensional formula \(p_\infty(x) \propto \sigma^{-2}(x)\exp(\int^x 2\mu(z)/\sigma^2(z)\,dz)\). For canonical financial models, the Fokker--Planck equation yields known transition densities: Gaussian for Brownian motion with drift, log-normal for geometric Brownian motion (\(dS_t = \mu S_t\,dt + \sigma S_t\,dW_t\)), mean-reverting Gaussian for the Ornstein--Uhlenbeck process (\(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\) with variance \(v(\tau) = \frac{\sigma^2}{2\kappa}(1 - e^{-2\kappa\tau})\)), and non-central chi-square for the CIR model (\(dr_t = \kappa(\theta - r_t)\,dt + \sqrt{\gamma r_t}\,dB_t\)). The forward equation also connects to modern machine learning: in score-based diffusion models, the forward SDE adds noise with density evolving by Fokker--Planck, and the reverse-time SDE \(dX_t = [f - g^2 \nabla_x \log p]\,dt + g\,d\bar{W}_t\) generates samples by learning the score function \(\nabla_x \log p\).
The Feynman--Kac Formula¶
The Feynman--Kac formula is the central result connecting stochastic analysis to partial differential equations. For the SDE \(dX_s = \mu(s,X_s)\,ds + \sigma(s,X_s)\,dW_s\) with \(X_t = x\), the solution to the parabolic PDE
has the probabilistic representation
The proof defines the augmented process \(Y_s = e^{-\int_t^s r\,d\tau}\,u(s,X_s) + \int_t^s e^{-\int_t^\tau r\,d\xi}\,f(\tau,X_\tau)\,d\tau\), applies Ito's lemma, and uses the PDE to show the drift vanishes -- making \(Y_s\) a martingale whose initial and terminal values yield the representation. The discounted Feynman--Kac formula (special case \(f = 0\)) gives \(u(t,x) = \mathbb{E}[e^{-\int_t^T r\,ds}\,g(X_T) \mid X_t = x]\); with constant rate this simplifies to \(u = e^{-r(T-t)}\mathbb{E}[g(X_T)]\), where \(r\) acts as a killing or discounting term. The running payoff extension (\(f \neq 0\)) adds continuous cash flows, with \(f\) as a source term (like heat generation in physics or coupon payments in finance). The converse also holds: defining \(u\) as the expectation yields a viscosity solution of the PDE. In finance, the Feynman--Kac formula is the mathematical justification for risk-neutral pricing. The Black--Scholes PDE
is exactly the Feynman--Kac equation for geometric Brownian motion under the risk-neutral measure, with solution \(V(t,S) = e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}[\Phi(S_T) \mid S_t = S]\). Bond pricing under stochastic rates (e.g., the Vasicek model \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\)) follows the same pattern, yielding affine term structure solutions \(P(t,r) = A(t,T)\,e^{-B(t,T)r}\). The formula also handles expected exit times (\(\mathcal{L}u = -1\) with Dirichlet boundary conditions). This duality between PDE and expectation enables a choice of numerical method: Monte Carlo simulation of SDE paths (advantageous in high dimensions and for path-dependent payoffs) versus finite-difference solution of the PDE (efficient in low dimensions, providing the full solution surface at once).
Role in the Book
The PDE framework developed here is applied directly in Chapter 6, where the Black--Scholes PDE is derived and solved via heat equation reduction and the Feynman--Kac representation. The Kolmogorov forward equation reappears in calibration (Chapter 17) through the Dupire equation, which extracts local volatility from observed option prices. The SDE--PDE bridge established by the infinitesimal generator connects back to the diffusion theory of Chapter 3, while the risk-neutral expectations interpreted through Feynman--Kac rely on the measure change machinery of Chapter 4.