Feynman–Kac with Running Payoff¶

This page isolates what is specific to the source term \(+f\) in the Feynman–Kac formula: the intuition for what it represents and the cancellation that makes the martingale argument go through. The full canonical statement (with all four ingredients \(\mathcal{L}u\), \(-ru\), \(+f\), \(g\)) lives in § The Feynman–Kac Formula; the general martingale derivation lives in § Proof Sketch.

Throughout: \(X_t\) is a diffusion with generator \(\mathcal{L}\), \(g\) is the terminal payoff, \(r\) is the discount/killing rate, and \(f\) is the running payoff (source term).

The Source-Term Specialization¶

Strip the discounting and the terminal payoff to isolate what the source term alone contributes:

\[ u_t + \mathcal{L}u + f = 0, \qquad u(T,x) = 0, \qquad u(t,x) = \mathbb{E}\!\left[\int_t^T f(s, X_s)\,ds \,\Big|\, X_t = x\right]. \]

Read this as "accumulated cashflow (or accumulated heat generation) along the diffusion path." The full canonical formula reinstates discounting on each contribution and adds the terminal payoff back on top.

Why the Source Term Cancels¶

Recall (see § Proof Sketch — Direction 1): the unaugmented process \(D(t,s)u(s,X_s)\) has drift \(D(t,s)[\partial_s u + \mathcal{L}u - r\,u]\,ds\). With a source term in the PDE this drift equals \(-D\,f\,ds\), which is not zero.

The fix is to enrich the candidate martingale with the discounted running payoff already collected:

\[ Y_s := D(t,s)\,u(s,X_s) + \int_t^s D(t,\tau)\,f(\tau,X_\tau)\,d\tau, \qquad D(t,s) = e^{-\int_t^s r\,d\tau}. \]

The integral contributes \(+D(t,s)\,f(s,X_s)\,ds\) to \(dY_s\), exactly cancelling the \(-D\,f\,ds\) from the PDE. The drift of \(Y_s\) vanishes, \(Y_s\) becomes a martingale, and \(Y_t=\mathbb{E}[Y_T\mid\mathcal{F}_t]\) gives the stated formula. \(\square\)

Interpretation¶

\(f\) acts as a source term (heat generation, dividends, coupons).
\(r\) acts as discounting, killing, or absorption — see § Discounted Feynman–Kac § The -ru Term: Three Equivalent Interpretations.

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.