When Exponentially Affine Conditional Moments Imply the Markov Property

Summary

This question asks whether an exponentially affine conditional transform of a process, together with Riccati equations for its coefficients, is enough to establish that the process is Markov. It distinguishes that transform property from the definition of an affine process, which already includes Markovianity. The question seeks a proof or reference connecting the two ideas.

The document provides no answer or supporting example, so it does not establish conditions under which the stated transform implies the Markov property. The formula concerns conditional expectations given the filtration, while the Markov property requires the conditional distribution of future states to depend on the past only through the current state. Assessing whether the transform is sufficient would require details such as the class of test vectors, integrability, and whether the transform uniquely determines the conditional law. It is a useful conceptual question for researchers studying affine stochastic models, but it is not a complete derivation or practical trading method.

Key ideas

• The question concerns whether an exponentially affine conditional transform implies the Markov property.
• The displayed coefficient functions are specified through Riccati equations.
• The document notes that the definition of an affine process already assumes Markovianity.
• No answer is provided to establish when the transform condition is sufficient.

Tags

# If a process satisfy the "affine property" is it a Markov processes?


# If a process satisfy the "affine property" is it a Markov processes?












Let $X$ be a stochastic process valued in $\mathbf{R}^d$ that, given a $d$-dimensional row vector $u$ and under suitable integrability conditions, satisfies: \begin{equation} E[\exp(uX_T)|\mathcal{F}_t] = \exp(\phi(T-t)+\psi(T-t)X_t) \end{equation} where the real-valued function $\phi$ and row-vector-valued function $\psi$ satisfy the Riccati equations: $$ \phi(t) =\int_0^t (\psi(s)b_0+ \frac{1}{2} \psi(s) A_0 \psi(s)^{\mathsf{T}}) ds, $$ $$ \psi = u + \int_0^t (\psi(s)B + \frac{1}{2} A (\psi(s))) ds, $$ with $A(u) =(uA^1u^{\mathsf{T}}, \ldots, u A^du^{\mathsf{T}})$ and $B=(b^1, \ldots, b^d)$.

Can I conclude that $X$ is Markov according to The Markov property for Ito diffusions? (see e.g. Oksendal-Partial Differential Equations Thm 7.1.2)

The definition of an affine process (see e.g., Definition 2.1 in https://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0142.pdf) requires the process $X$ to be Markovian. Therefore, I am inquiring whether a process whose characteristic function is exponentially affine with respect to the state vector is a Markov process.

Any reference to a paper or book is extremely welcomed. Thank you!

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.