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Mathematical Finance: What are Affine Jump Diffusion (AJD) Models?
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Mathematical Finance: What are Affine Jump Diffusion (AJD) Models?

The state vector $X_t$ follows a Markov process solving the SDE \begin{equation*} dX_t=\mu(X_t)dt +\sigma(X_t)dW_t + dZ_t \end{equation*} where $W$ is an adapted Brownian, and $Z$ is a pure jump process with intensity $\lambda$. The moment generating function of the jump sizes is $\theta(c)=EQ(\exp(cJ))$. Impose an affine structure on $\mu,\sigma\sigma^T,\lambda$ and the discount rate $R$, possibly time dependent: \begin{equation*} \mu(x)=K_0+K_1x \quad (\sigma(x)\sigma(x)^T)_{ij}=(H_0)_{ij}+(H_1)_{ij}x \quad \lambda(x)=L_0+L_1x \quad R(x)=R_0+R_1x \end{equation*} Given $X_0$, the risk neutral coefficients ($K,H,L,\theta,R$) completely determine the discounted risk neutral distribution of $X$. Introduce the transform function \begin{equation*} \psi(u,X_0,T)=EQ\left[\left.\exp\left(-\int_0^{T}R(X_s)ds\right)e^{u^TX_T}\right|F_0\right]=e^{\alpha(0,u)+\beta(0,u)^Tx_0} \end{equation*} where $\alpha$ and $\beta$ solve the Ricatti ODEs subject to $\alpha(T,u)=0,\beta(T,u)=u$: \begin{align*} -\dot{\beta}(t,u)=&K_1^T\beta(t,u)+\tfrac{1}{2}\beta(t,u)^TH_1\beta(t,u)+L_1(\theta(\beta(t,u))-1)-R_1\\ -\dot{\alpha}(t,u)=&K_0^T\beta(t,u)+\tfrac{1}{2}\beta(t,u)^TH_0\beta(t,u)+L_0(\theta(\beta(t,u))-1)-R_0 \end{align*}
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