Source Terms in Parabolic Pricing Equations as Ongoing Cash Flows

Summary

The answer connects a parabolic partial differential equation with a source term to a contingent claim that pays continuously according to the state of an underlying Brownian motion and makes a terminal payment. Its value is represented as the expected terminal payoff plus the accumulated expected source payments, conditional on the current state. This gives a financial interpretation for the source term: it represents an ongoing cash flow during the contract’s life, while the terminal function represents the payoff at maturity.

The equation shown uses a half coefficient on the second spatial derivative, matching the generator of standard Brownian motion. The question initially writes a different coefficient, so the precise diffusion scale must be specified for the PDE and expectation representation to agree. The short exchange offers a basic Feynman-Kac interpretation, but it does not discuss discounting, risk-neutral valuation, or more general underlying dynamics.

Key ideas

• A source term in a parabolic pricing PDE can represent continuous state-dependent payments over the contract’s life.
• The value representation combines the expected terminal payoff with accumulated expected cash flows.
• The diffusion coefficient in the PDE must match the dynamics assumed for the underlying process.
• The example uses Brownian motion and omits discounting and broader valuation assumptions.

Tags

# parabolic pde with source term


# parabolic pde with source term












I was wondering if someone is aware of the application when pdes of the form arise $$u_t+u_{xx}+g=0$$ i.e. there is a source term now. Any financial instruments that have this type of pde?

## Answer by M. Jeunesse (score 2, accepted)

https://quant.stackexchange.com/a/29817

$$u(t,x)=\mathbb{E}\left[h(B_T)+\int_t^Tg(B_s)ds|B_t=x\right]$$ where $B$ is a brownian motion.

So if you enter a contract whose underlying asset is $B$, such that you pay every day $t$, $-g(B_t)dt$ up to time $T$ where you receive $h(B_T)$, then the value of this contract is $u$

$$\partial_t u + \frac{1}{2}\partial_{xx}u + g = 0$$ there is $\frac{1}{2}$ in front of $\partial_{xx}u$ same for my comment below.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.