General Affine Diffusions

A jump-diffusion process is a Markov process solving the stochastic differential equationd

\[Y_{t}=\mu\left(Y_{t},\theta_{0}\right)dt +\sigma\left(Y_{t},\theta_{0}\right)dW_{t}.\]

A discount-rate function \(R:D\to\mathbb{R}\) is an affine function of the state

\[R\left(Y\right)=\rho_{0}+\rho_{1}\cdot Y,\]

for \(\rho=\left(\rho_{0},\rho_{1}\right)\in\mathbb{R} \times\mathbb{R}^{N}\).

The affine dependence of the drift and diffusion coefficients of \(Y\) are determined by coefficients \(\left(K,H\right)\) defined by:

\[\mu\left(Y\right)=K_{0}+K_{1}Y,\]

for \(K=\left(K_{0},K_{1}\right)\in\mathbb{R}^{N}\times\mathbb{R}^{N\times N}\),

and

\[\left[\sigma\left(Y\right)\sigma\left(Y\right)^{\prime}\right]_{ij} =\left[H_{0}\right]_{ij}+\left[H_{1}\right]_{ij}\cdot Y,\]

for \(H=\left(H_{0},H_{1}\right)\in\mathbb{R}^{N\times N} \times\mathbb{R}^{N\times N\times N}\).

Here

\[\left[H_{1}\right]_{ij}\cdot Y=\sum_{k=1}^{N}\left[H_{1}\right]_{ijk}Y_{k}.\]

A characteristic \(\chi=\left(K,H,\rho\right)\) captures both the distribution of \(Y\) as well as the effects of any discounting.