Positive Semidefiniteness of Diffusion Terms in Affine Process Generators

Summary

The document asks why the state-dependent matrix multiplying the second derivatives in a regular affine process generator must be positive semidefinite. It defines the matrix through constant and state-dependent coefficients and asks whether a general theorem about stochastic-process generators guarantees this property. The mathematical issue connects the generator’s second-order term to the covariance structure of the process’s diffusion component.

No answer or proof is included, so the document does not establish the required conditions or explain how positivity follows from a particular affine-process definition. It is a focused theoretical question rather than a trading method, empirical analysis, or complete derivation. Readers would need additional material on diffusion generators and affine process admissibility to resolve it.

Key ideas

• The second-order coefficients in an affine generator form a state-dependent matrix.
• The question asks whether this matrix must be positive semidefinite for a stochastic process.
• The issue concerns the relationship between a generator’s second-order term and diffusion covariance.
• The document supplies no proof, theorem, or admissibility conditions that settle the question.

Tags

# For an affine process, how do we know the second order term is positive definite?


# For an affine process, how do we know the second order term is positive definite?












A regular affine process is defined to have the generator

$Af(x) = \sum_{k,l=1}^d(a_{kl}+\langle a_{I,kl},y\rangle)\frac{\partial^2f(x)}{\partial x_k\partial x_l}+\langle b+\beta x,\nabla f(x)\rangle - ...$

see the bottom of page 8 of this following file:

http://web.stanford.edu/~duffie/affine.pdf

My question is, how do we know the matrix $M$, defined by $M_{kl} = (a_{kl}+\langle a_{I,kl},y\rangle)$ is always positive semidefinite? (I assume this must the be the case, but how does one show this? is there a general theorem which tells you this must be the case for the generator of a stochastic process?)

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