Domain and Growth Conditions for the Black–Scholes Operator
Summary
The document asks what function space forms the domain of the Black–Scholes differential operator, which combines second-order price derivatives with drift and discount terms. It distinguishes the operator's basic smoothness needs from the additional conditions required when the operator is used in an option-pricing PDE. The question points to Feynman–Kac representations, uniqueness of PDE solutions, growth restrictions, and a paper that uses Schwartz space.
The response gives a brief characterization: functions should have two price derivatives and one time derivative on positive asset prices and times within a finite horizon. For uniqueness after boundary conditions are imposed, growth as the asset price tends to infinity must be restricted in a way related to uniqueness conditions for the heat equation. It does not settle a formal operator domain for every application, or reconcile the cited Schwartz-space choice with other solution classes; those choices depend on the specific problem and boundary conditions.
Key ideas
- The Black–Scholes pricing PDE needs functions with two price derivatives and one time derivative.
- The stated domain uses positive asset prices and times inside a finite horizon.
- Uniqueness with boundary conditions requires limiting growth as the asset price becomes large.
- The document does not establish one universal functional-analytic domain for every application.
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Full text
# What is the domain of the Black-Scholes operator?
# What is the domain of the Black-Scholes operator?
By the Black-Scholes operator I mean the following.
$$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)$$
Obviously, the domain of $L_{BS}$ must be a subset of twice continuously differentiable functions on some interval but making a definitive statement about the domain requires something specific about the application at hand (option pricing in this instance) and knowledge about the origin of the operator. So what is what I am looking for.
Feynman-Kac says something about which solutions of the Black-Scholes PDE satisfy the pricing formula, that is in the form of conditional expectation. The solution class is the set of twice continuously differentiable functions that are continuous at the boundary and whose growth is bounded by a function of the form $e^{\alpha x^2}$.
There is also the uniqueness class of the Black-Scholes PDE, which has a similar growth characteristic.
This paper mentions the Schwartz space as the domain of Black-Scholes operator but I have never seen this anywhere else.
I am looking for an answer that summarizes all the considerations that go into this, preferably with references. Thank you.
## Answer by P. Carr (score 2)
https://quant.stackexchange.com/a/40041
The domain would be twice differentiable in S, once differentiable in t functions of S and t defined on the Cartesian product of S in (0,infty) and t in (0,T) for finite T >0. To obtain a unique solution when boundary conditions are added, one must curtail the growth as S goes to infinity by adapting the corresponding condition for uniqueness of solutions to the heat equation.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.