

In [ ]:



In [ ]:

$$dX_t=\mu_t dt + \Sigma d W_t$$

Term structure models

The short rate is a linear function of the state variables

$$r_t=\delta_0 + \delta_1 X_t$$

$$\psi_{{\bf x}_t}(T,{\bf c},m) = a_t(T,{\bf c},m) + {\bf b}_t(T,{\bf c},m)\cdot {\bf x}_t$$

$$\chi^0_{{\bf x}_t}(T,{\bf c},m)$$

Exponential

$$\\begin{align*} z_t &= e^{{\bf m}\cdot{\bf x_t} - c_t}\\\\ k_t &= \ln{E\left[e^{{\bf m}\cdot{\bf x_t}}\right]} \\end{align*}$$

where $k_t$ is the exponential compensator of $x_t$.

$$B=\\int_t^T$$

The laplace cumulant of a semimartingale ${\\bf x}_t$ is $\theta$ is given by the following expression.

$$e^{\kappa(\theta)} = {\\tt E}\\left[e^{\theta \cdot x_t}\\right]$$

$$\kappa (u)=u \cdot B + \frac{1}{2} u \cdot C \cdot u + \left(e^{u \cdot x}-1-u\cdot h(x)\right) * \nu$$