Transition Density as Green's Function¶
The transition density of a diffusion and the Green's function of its generator are the same mathematical object viewed through two lenses. This page is the probability lens: it establishes the equivalence, derives the forward/backward equations, and develops the financial interpretation (state-price density, Arrow-Debreu price, Breeden-Litzenberger). The PDE lens -- operator definition, superposition principle, and smoothing -- is in Green's Function for Parabolic PDEs.
The Two Perspectives¶
Given the SDE \(dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\) with generator \(\mathcal{L}\), define:
- Probabilistic. The transition density \(p(t, x \mid s, y)\) by \(\mathbb{P}(X_t \in A \mid X_s = y) = \int_A p(t, x \mid s, y)\,dx\).
- Analytical. The Green's function \(G(t, x; s, y)\) as the fundamental solution of \(\partial_t - \mathcal{L}\) (see Green's Function for Parabolic PDEs).
The central claim:
The transition density of a diffusion is the Green's function of its generator.
Proof of the Equivalence¶
Let \(g\) be a bounded continuous test function and set
The Kolmogorov backward equation (derived from Itô's formula applied to \(u(s, X_s)\)) gives
The Green's function representation of the same backward problem gives
Both representations hold for every bounded continuous \(g\). By the density of such \(g\) in the space of test functions, the kernels must agree:
The delta-function initial condition of \(G\) reflects the point-mass initial law \(X_s = y\) a.s.: both encode "the system starts with certainty at \(y\)".
Forward and Backward Equations¶
Recall (see § Kolmogorov Forward Equation and § Kolmogorov Backward Equation): \(p(t,x\mid s,y)\) satisfies \(\partial_t p = \mathcal{L}_x^* p\) in the destination \((t,x)\) and \(-\partial_s p = \mathcal{L}_y p\) in the origin \((s,y)\), each with the delta initial/terminal condition. The forward operator \(\mathcal{L}^* p = -\partial_x[\mu p] + \tfrac12\partial_{xx}[\sigma^2 p]\) is the formal adjoint of the generator; this adjoint structure is what enforces conservation of probability.
| Equation | Varies | Fixed | Operator | Role |
|---|---|---|---|---|
| Forward (Fokker-Planck) | \((t, x)\) destination | \((s, y)\) origin | \(\mathcal{L}_x^*\) | Density evolution |
| Backward (Kolmogorov) | \((s, y)\) origin | \((t, x)\) destination | \(\mathcal{L}_y\) | Expectation / pricing |
Here the Green's-function lens adds one observation: the same kernel \(G = p\) solves both PDEs simultaneously -- forward in \((t,x)\), backward in \((s,y)\) -- because the delta-source condition is symmetric in the two variable pairs. The two Kolmogorov equations are thus two views of one object, not two unrelated facts.
Chapman-Kolmogorov Equation¶
For \(s < r < t\):
Probabilistic derivation. By the Markov property and total probability,
Dividing by \(dx\) gives the displayed equation.
PDE interpretation. Solve the point-source problem in two stages: first evolve from \(s\) to \(r\), obtaining \(p(r, z \mid s, y)\); then use this as initial data for the segment \(r \to t\). By uniqueness, the two-stage result equals the direct transition \(p(t, x \mid s, y)\). This is the semigroup property of \(G\) from Green's Function for Parabolic PDEs.
The pricing analogue is the law of iterated expectations: \(\mathbb{E}[\,\cdot \mid \mathcal{F}_s] = \mathbb{E}[\mathbb{E}[\,\cdot \mid \mathcal{F}_r] \mid \mathcal{F}_s]\).
Brownian Motion with Drift¶
Recall (see § Fundamental Solution and § Transition Densities for Standard SDEs): for \(dX_t = \mu\,dt + \sigma\,dW_t\) with \(X_t - X_s \sim N(\mu(t-s), \sigma^2(t-s))\),
is the heat kernel translated by the drift -- and identifying it as \(G\) makes the Black-Scholes operator's fundamental solution an Arrow-Debreu kernel.
Ornstein-Uhlenbeck¶
Recall (see § Transition Densities for Standard SDEs): for \(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\), \(p(t,x\mid s,y)\) is Gaussian with mean \(ye^{-\kappa\tau}\) and variance \(v(\tau) = \sigma^2(1-e^{-2\kappa\tau})/(2\kappa)\), \(\tau = t-s\).
As \(\tau \to \infty\), \(p \to N(0, \sigma^2/(2\kappa))\) independent of \(y\): the process forgets its initial condition and relaxes to the stationary density. The same phenomenon appears in Spectral Decomposition as the decay of all non-zero eigenmodes -- and in Free vs Bounded Domains as the contrast between free-space dissipation and bounded-domain equilibration.
Financial Interpretation¶
State-Price Density¶
Under the risk-neutral measure \(\mathbb{Q}\), the transition density of the asset price is the state-price density (Arrow-Debreu kernel):
For each terminal state \(S_T\), the quantity \(e^{-r(T-t)}p^{\mathbb{Q}}(T, S_T \mid t, S)\,dS_T\) is the price today of an Arrow-Debreu security paying $1 in that state and nothing otherwise. Knowing \(p^{\mathbb{Q}}\) is equivalent to knowing all European option prices simultaneously. The integral-against-\(G\) formulation is developed in Green's Function for Parabolic PDEs.
Breeden-Litzenberger¶
The second derivative of call prices with respect to strike recovers the risk-neutral density:
This is the basis of implied-density estimation and non-parametric calibration from option quotes.
Black-Scholes¶
Recall (see § Geometric Brownian Motion and § Black-Scholes via Heat Equation): the risk-neutral GBM transition density is the lognormal \(p^{\mathbb{Q}}(T,S_T\mid t,S)\), and integrating any payoff against \(e^{-r(T-t)}p^{\mathbb{Q}}\) reproduces the Black-Scholes price. The Green's-function lens identifies this kernel as the Arrow-Debreu state-price density for the Black-Scholes operator.
Dictionary¶
| Object | Probability | PDE | Finance |
|---|---|---|---|
| \(p(t, x \mid s, y) = G\) | Transition density | Green's function | State-price density |
| Action \(\int G\cdot f\) | Expected value | Solution of IVP | Option price |
| Semigroup / CK | Markov property | Uniqueness / superposition | Iterated expectations |
| \(\delta(x - y)\) initial | \(X_s = y\) a.s. | Point source | Unit Arrow-Debreu |
Summary¶
The identification collapses two parallel theories into one. Probabilistic questions become PDE problems; PDE problems become stochastic computations. The forward equation tracks density evolution; the backward equation prices claims. The Chapman-Kolmogorov / semigroup property expresses the Markov property in analytic form. And in finance, the Green's function is the Arrow-Debreu pricing kernel.
See Also¶
- Green's Function for Parabolic PDEs -- operator definition, superposition, parametrix
- Spectral Decomposition -- eigenfunction expansion of \(p\), modal convergence to stationarity
- Free vs Bounded Domains -- boundary effects and killed transition densities
- Kolmogorov Forward Equation
- Kolmogorov Backward Equation
- Feynman-Kac Formula
Exercises¶
Exercise 1. The direct verification of the forward equation \(\partial_t p = -\mu\partial_x p + \tfrac12\sigma^2\partial_{xx}p\) for the Gaussian density of \(dX_t = \mu\,dt + \sigma\,dW_t\) is carried out in § Kolmogorov Forward Equation. Explain why this same Gaussian must also solve the backward equation \(-\partial_s p = \mu\partial_y p + \tfrac12\sigma^2\partial_{yy}p\), without redoing the differentiation.
Solution to Exercise 1
The Gaussian depends on \((t,x,s,y)\) only through \(\tau = t-s\) and \(w = x - y - \mu\tau\). Under the symmetry \((t,x) \leftrightarrow (s,y)\), \(\tau \to -\tau\) and \(w \to -w\), but the density depends only on \(\tau\) and \(w^2\), so the forward operator in \((t,x)\) and the backward operator in \((s,y)\) produce identical actions on \(p\). This is the Green's-function identity at work: one kernel, two PDEs.
Exercise 2. Verify the Chapman-Kolmogorov equation for standard Brownian motion by explicit Gaussian convolution.
Solution to Exercise 2
With \(\tau_1 = r - s\), \(\tau_2 = t - r\), combine exponents:
with \(z^* = (\tau_1 x + \tau_2 y)/(\tau_1 + \tau_2)\). The \(z\)-integral gives \(\sqrt{2\pi\tau_1\tau_2/(\tau_1+\tau_2)}\). Combining prefactors:
The result says: adding independent Gaussian increments adds variances -- the defining property of Brownian motion.
Exercise 3. State the forward and backward equations for the Ornstein-Uhlenbeck process \(dX_t = -\kappa X_t\,dt + \sigma\,dW_t\) in the two pairs of variables, and identify which PDE governs the option price \(V(s, y) = \mathbb{E}[g(X_t) \mid X_s = y]\).
Solution to Exercise 3
With \(\mathcal{L} = -\kappa x\,\partial_x + \tfrac{1}{2}\sigma^2\partial_{xx}\), the adjoint is \(\mathcal{L}^* p = \kappa\,\partial_x(x p) + \tfrac{1}{2}\sigma^2\partial_{xx}p\).
Forward (in \((t, x)\)): \(\partial_t p = \kappa\,\partial_x(x p) + \tfrac{1}{2}\sigma^2\partial_{xx}p\).
Backward (in \((s, y)\)): \(-\partial_s p = -\kappa y\,\partial_y p + \tfrac{1}{2}\sigma^2\partial_{yy}p\).
The option price \(V(s, y) = \mathbb{E}[g(X_t) \mid X_s = y]\) depends on \((s, y)\) and satisfies the backward equation \(-\partial_s V = \mathcal{L}_y V\) with \(V(t, y) = g(y)\). The backward equation is the pricing equation; the forward equation governs implied-distribution and risk analysis.
Exercise 4. The full derivation of \(V = S\mathcal{N}(d_1) - Ke^{-r(T-t)}\mathcal{N}(d_2)\) from the lognormal transition density of geometric Brownian motion is carried out in § Black-Scholes PDE: Analytic Solutions. In the Green's-function language of this page, state which object \(G\) is being integrated against and identify the financial role of the prefactor \(e^{-r(T-t)}\).
Solution to Exercise 4
\(G\) is the risk-neutral lognormal transition density \(p^{\mathbb{Q}}(T,S_T \mid t,S)\) for GBM under \(\mathbb{Q}\); the pricing integral \(V(t,S) = e^{-r(T-t)}\int (S_T - K)^+ G\,dS_T\) integrates the payoff against \(G\) over the in-the-money region.
The prefactor \(e^{-r(T-t)}\) converts the kernel into the Arrow-Debreu state-price density: \(e^{-r(T-t)}G(T,S_T;t,S)\,dS_T\) is the price today of $1 delivered if \(S_T \in [S_T, S_T+dS_T]\). The Black-Scholes formula is then the integral of the call payoff against this state-price density.
Exercise 5. Explain Breeden-Litzenberger: derive \(p^{\mathbb{Q}}(T, K \mid t, S) = e^{r(T-t)}\partial_K^2 C(t, S; T, K)\) by differentiating the pricing integral twice.
Solution to Exercise 5
\(C(t, S; T, K) = e^{-r(T-t)}\int_K^\infty (S_T - K)\,p^{\mathbb{Q}}(T, S_T \mid t, S)\,dS_T\).
Differentiate once in \(K\): the boundary term \((S_T - K)p^{\mathbb{Q}}\) at \(S_T = K\) vanishes, leaving
Differentiate again, applying Leibniz to the moving lower limit:
Rearranging gives the Breeden-Litzenberger formula. Model-free content: given a dense set of call quotes in \(K\), the risk-neutral marginal density at maturity \(T\) is recovered by numerical second differencing.
Exercise 6. For a diffusion killed at an absorbing barrier \(B\), the killed density \(p_B(t, x \mid s, y) \le p(t, x \mid s, y)\) integrates to less than \(1\). Relate the deficit to the first-passage probability \(\mathbb{P}(\tau_B \le t \mid X_s = y)\) and derive the flux formula for the rate at which mass leaks out.
Solution to Exercise 6
The killed density is
Integrating over \(x\):
so the deficit is \(1 - \int p_B\,dx = \mathbb{P}(\tau_B \le t \mid X_s = y)\).
The leakage rate follows from the Fokker-Planck equation with absorbing condition \(p_B = 0\) at \(x = B\):
(sign determined by whether \(B\) is upper or lower barrier). In barrier-option pricing, this deficit is the knock-out probability; see Free vs Bounded Domains for the full boundary-value-problem treatment.