Skip to content

Feynman Kac Running

Let \(X_t\) be a diffusion with generator \(L\). Let

  • \(f(x)\): terminal payoff
  • \(r(x,t)\): discount/killing rate
  • \(g(x,t)\): running payoff (source term)

Statement General

Define

\[ u(x,t)=\mathbb E\!\left[ \exp\!\Big(-\int_t^T r(X_s,s)\,ds\Big)f(X_T) + \int_t^T \exp\!\Big(-\int_t^s r(X_\tau,\tau)\,d\tau\Big)g(X_s,s)\,ds \,\middle|\,X_t=x \right]. \]

Then \(u\) solves

\[ u_t + Lu - r\,u + g = 0, \qquad u(x,T)=f(x). \]

Proof augmented

Let

\[ Z_s=\exp\!\Big(-\int_t^s r(X_u,u)\,du\Big),\qquad dZ_s=-r(X_s,s)Z_s\,ds. \]

Assume \(u\in C^{1,2}\) solves the PDE. Define

\[ Y_s := Z_s u(X_s,s) + \int_t^s Z_\tau\,g(X_\tau,\tau)\,d\tau. \]

Compute \(dY_s\). Using Itô + product rule:

\[ d(Z_su(X_s,s)) = Z_s(u_t+Lu-r\,u)(X_s,s)\,ds + Z_s\sigma u_x\,dW_s. \]

Adding \(d\left(\int_t^s Z_\tau g(X_\tau,\tau)\,d\tau\right)=Z_sg(X_s,s)\,ds\), we get

\[ dY_s = Z_s\big(u_t+Lu-r\,u+g\big)(X_s,s)\,ds + Z_s\sigma u_x\,dW_s. \]

By the PDE the drift is zero, hence \(Y_s\) is a local martingale.

Taking expectations from \(t\) to \(T\) yields

\[ u(x,t)=\mathbb E^{x,t}\!\left[ Z_T f(X_T) + \int_t^T Z_s g(X_s,s)\,ds \right], \]

which is exactly the stated representation.

Interpretation

  • \(g\) acts as a source term (like heat generation).
  • \(r\) acts as discounting (finance) or killing (probability/physics).

Exercises

Exercise 1. Consider the PDE \(u_t + \frac{1}{2}\sigma^2 u_{xx} - ru + g(x) = 0\) with \(u(T, x) = 0\) and constant \(r\), \(g(x) = 1\). Use the Feynman-Kac representation with running payoff to write \(u(t, x)\) as an expectation. Evaluate explicitly for the process \(dX_s = \sigma\,dW_s\).

Solution to Exercise 1

The PDE is \(u_t + \frac{1}{2}\sigma^2 u_{xx} - ru + 1 = 0\) with \(u(T, x) = 0\). By Feynman-Kac with running payoff \(g(x) = 1\), terminal payoff \(f(x) = 0\), and constant discount rate \(r\):

\[ u(t, x) = \mathbb{E}\!\left[\int_t^T e^{-r(s-t)} \cdot 1\,ds \,\Big|\, X_t = x\right] \]

Since \(g = 1\) is constant and \(dX_s = \sigma\,dW_s\) (the process does not appear in the integrand), the expectation simplifies to:

\[ u(t, x) = \int_t^T e^{-r(s-t)}\,ds = \frac{1}{r}\!\left(1 - e^{-r(T-t)}\right) \]

Note that \(u\) does not depend on \(x\), which makes sense because neither the running payoff nor the terminal payoff depends on the spatial variable.

Verification: \(u_t = -e^{-r(T-t)}\). Since \(u\) does not depend on \(x\), \(u_{xx} = 0\). Also \(ru = 1 - e^{-r(T-t)}\). Then:

\[ u_t + \frac{1}{2}\sigma^2 u_{xx} - ru + 1 = -e^{-r(T-t)} + 0 - (1 - e^{-r(T-t)}) + 1 = 0 \;\checkmark \]

Exercise 2. A bond with continuous coupon payments at rate \(c\) has value \(V(t, r) = \mathbb{E}[\int_t^T e^{-\int_t^s r_u\,du}\,c\,ds + e^{-\int_t^T r_u\,du} | r_t = r]\). Write the PDE that \(V\) satisfies. Identify the running payoff \(g = c\), the terminal payoff \(f = 1\), and the discounting rate.

Solution to Exercise 2

The bond value includes both a running payoff (continuous coupons at rate \(c\)) and a terminal payoff (face value \(1\) at maturity). By the Feynman-Kac formula:

\[ V(t, r) = \mathbb{E}\!\left[\int_t^T e^{-\int_t^s r_u\,du}\,c\,ds + e^{-\int_t^T r_u\,du} \,\Big|\, r_t = r\right] \]

Identifying the components:

  • Running payoff: \(g = c\) (the continuous coupon rate)
  • Terminal payoff: \(f = 1\) (the face value at maturity)
  • Discounting rate: \(r_t\) (the short rate, which is the state variable)

The PDE that \(V\) satisfies is:

\[ \frac{\partial V}{\partial t} + \mathcal{L}_r V - r\,V + c = 0, \quad V(T, r) = 1 \]

where \(\mathcal{L}_r\) is the generator of the short rate process. For example, in the Vasicek model \(dr_t = \kappa(\theta - r_t)\,dt + \sigma_r\,dW_t\):

\[ \frac{\partial V}{\partial t} + \kappa(\theta - r)\frac{\partial V}{\partial r} + \frac{1}{2}\sigma_r^2\frac{\partial^2 V}{\partial r^2} - r\,V + c = 0 \]

Exercise 3. In the proof sketch, the process \(Y_s = Z_s u(X_s, s) + \int_t^s Z_\tau g(X_\tau, \tau)\,d\tau\) is claimed to be a local martingale. Show that when the PDE \(u_t + \mathcal{L}u - ru + g = 0\) holds, the drift of \(Y_s\) vanishes. Identify where the running payoff \(g\) cancels.

Solution to Exercise 3

We compute \(dY_s\) where \(Y_s = Z_s\,u(X_s, s) + \int_t^s Z_\tau\,g(X_\tau, \tau)\,d\tau\) and \(Z_s = e^{-\int_t^s r(X_u, u)\,du}\).

First, by the product rule and Ito's lemma:

\[ d(Z_s\,u) = Z_s\,du + u\,dZ_s = Z_s\!\left[u_t + \mathcal{L}u\right]ds + Z_s\,\sigma\,u_x\,dW_s + u\,(-r\,Z_s)\,ds \]
\[ = Z_s\!\left[u_t + \mathcal{L}u - r\,u\right]ds + Z_s\,\sigma\,u_x\,dW_s \]

Second, the integral term contributes: \(d\!\left(\int_t^s Z_\tau\,g\,d\tau\right) = Z_s\,g(X_s, s)\,ds\).

Combining:

\[ dY_s = Z_s\!\left[u_t + \mathcal{L}u - r\,u + g\right]ds + Z_s\,\sigma\,u_x\,dW_s \]

When the PDE \(u_t + \mathcal{L}u - r\,u + g = 0\) holds, the drift vanishes. The cancellation occurs as follows: the PDE gives \(u_t + \mathcal{L}u - r\,u = -g\), so the drift becomes \(Z_s(-g + g) = 0\). The running payoff \(g\) from the integral term exactly cancels the \(-g\) arising from the PDE, leaving only the martingale part \(Z_s\,\sigma\,u_x\,dW_s\).

Exercise 4. A derivative pays a continuous dividend at rate \(q S_t\) (proportional to the stock price) plus a terminal payoff \(g(S_T)\). Write the Feynman-Kac representation with both running and terminal payoffs, and derive the corresponding PDE.

Solution to Exercise 4

A stock paying continuous dividends at rate \(qS_t\) generates a running cash flow. Under \(\mathbb{Q}\), the stock follows \(dS_t = (r - q)S_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}\).

The derivative value with running dividend payments and terminal payoff is:

\[ V(t, S) = \mathbb{E}^{\mathbb{Q}}\!\left[\int_t^T e^{-r(s-t)}\,qS_s\,ds + e^{-r(T-t)}\,g(S_T) \,\Big|\, S_t = S\right] \]

Here the running payoff is \(f(s, S_s) = qS_s\) and the terminal payoff is \(g(S_T)\).

The corresponding PDE is:

\[ \frac{\partial V}{\partial t} + (r-q)S\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2} - rV + qS = 0 \]

with \(V(T, S) = g(S)\). The \(+qS\) term is the source from the running dividend payments.

Exercise 5. Show that the running payoff formula reduces to the discounted Feynman-Kac formula when \(g = 0\) (no running payoff). Show that it reduces to a pure annuity-like formula when \(f = 0\) (no terminal payoff) and \(g\) is constant.

Solution to Exercise 5

Case \(g = 0\) (no running payoff): The general formula becomes:

\[ u(t, x) = \mathbb{E}\!\left[e^{-\int_t^T r(X_s, s)\,ds}\,f(X_T) \,\Big|\, X_t = x\right] \]

with PDE \(u_t + \mathcal{L}u - r\,u = 0\) and \(u(T, x) = f(x)\). This is exactly the discounted Feynman-Kac formula, which applies to standard option pricing with a terminal payoff and no intermediate cash flows.

Case \(f = 0\) (no terminal payoff) with constant \(g\): The formula becomes:

\[ u(t, x) = \mathbb{E}\!\left[\int_t^T e^{-\int_t^s r(X_\tau, \tau)\,d\tau}\,g\,ds \,\Big|\, X_t = x\right] \]

If \(r\) is also constant, then \(e^{-\int_t^s r\,d\tau} = e^{-r(s-t)}\) and:

\[ u(t, x) = g\int_t^T e^{-r(s-t)}\,ds = \frac{g}{r}\!\left(1 - e^{-r(T-t)}\right) \]

This is the present value of a continuous annuity paying \(g\) per unit time for \((T - t)\) years at discount rate \(r\). The formula recovers the standard annuity pricing formula from fixed-income mathematics.

Exercise 6. Consider \(u_t + \mu u_x + \frac{1}{2}\sigma^2 u_{xx} + g(x, t) = 0\) (no discounting, \(r = 0\)) with \(u(T, x) = 0\). Write the Feynman-Kac representation and verify that \(u(t, x) = \mathbb{E}[\int_t^T g(X_s, s)\,ds | X_t = x]\). Compute explicitly for \(g(x, t) = x\) and \(dX_s = \mu\,ds + \sigma\,dW_s\).

Solution to Exercise 6

With \(r = 0\) and \(u(T, x) = 0\), the Feynman-Kac representation gives:

\[ u(t, x) = \mathbb{E}\!\left[\int_t^T g(X_s, s)\,ds \,\Big|\, X_t = x\right] \]

For \(g(x, t) = x\) and \(dX_s = \mu\,ds + \sigma\,dW_s\) with \(X_t = x\):

\[ X_s = x + \mu(s - t) + \sigma(W_s - W_t) \]

Therefore \(\mathbb{E}[X_s \mid X_t = x] = x + \mu(s - t)\), and:

\[ u(t, x) = \int_t^T \!\left[x + \mu(s - t)\right]ds = x(T - t) + \mu\frac{(T - t)^2}{2} \]

Verification: Let \(\tau = T - t\). Then \(u = x\tau + \frac{1}{2}\mu\tau^2\).

\[ u_t = -x - \mu\tau, \quad u_x = \tau, \quad u_{xx} = 0 \]
\[ u_t + \mu u_x + \frac{1}{2}\sigma^2 u_{xx} + x = (-x - \mu\tau) + \mu\tau + 0 + x = 0 \;\checkmark \]

Terminal condition: \(u(T, x) = x \cdot 0 + 0 = 0\). \(\checkmark\)

Exercise 7. In mathematical physics, the source term \(g(x,t)\) represents heat generation at rate \(g\) in a medium with thermal diffusivity \(\sigma^2/2\). The killing term \(-ru\) represents heat loss proportional to temperature. Give the financial analogues of each term and explain why the general Feynman-Kac formula with all terms (\(\mathcal{L}u\), \(-ru\), \(f\), \(g\)) is needed for realistic derivative pricing.

Solution to Exercise 7

Financial analogues of each physics term:

Physics Term Physics Meaning Financial Analogue
\(\mathcal{L}u\) (diffusion/advection) Heat conduction and convection Risk-neutral drift and volatility of the underlying asset
\(-r\,u\) (killing/absorption) Heat loss proportional to temperature Discounting at the risk-free rate (time value of money)
\(g(x, t)\) (source/generation) Internal heat generation Continuous cash flows: dividends, coupons, running costs
\(f(x)\) (terminal condition) Initial/boundary temperature Derivative payoff at maturity (e.g., \((S_T - K)^+\))

Why all terms are needed: Realistic derivatives often involve multiple cash flow components simultaneously:

  • A convertible bond has a terminal payoff (conversion or redemption value), continuous coupons (running payoff \(g\)), and discounting at the risk-free rate (\(-r\,u\)). The underlying stock dynamics enter through \(\mathcal{L}u\).
  • A total return swap involves running payments based on the asset return plus a terminal settlement.
  • Employee stock options may include continuous vesting schedules (running payoff), a terminal exercise payoff, and discounting for the time value of money, all while the stock evolves stochastically.

Omitting any term would restrict the framework to special cases. The full Feynman-Kac formula with \(\mathcal{L}u - r\,u + g = -u_t\) is the minimal PDE structure that captures all economically relevant cash flow patterns in derivatives pricing.