The Black-Scholes equation is a conservation law: the total probability of the stock price being in any region changes at the rate determined by the probability flux through the region's boundary. This is the Fokker-Planck equation, and its derivation rests on the Divergence theorem (also called Gauss's theorem): the net flux of a vector field outward through a closed surface equals the total divergence of the field inside the enclosed volume. The Divergence theorem is the three-dimensional version of Green's theorem — the ultimate "boundary = interior derivative" identity in this dimension — and it is the key step in deriving every PDE that models diffusive or conservative flows.
Learning objectives. By the end of this note you should be able to:
- state the Divergence theorem and apply it to evaluate surface flux integrals;
- compute surface integrals over closed surfaces by converting to easier volume integrals;
- give the physical interpretation (net outflow = total source strength);
- sketch the derivation for a box region.
In this lesson
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- Statement of the Divergence Theorem
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- Computing Surface Integrals via the Divergence Theorem
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- Sketch of the Proof (Box Region)
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- Consequences
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- Summary
Before you start
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