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6

Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic function of the realised interest rate $R_t=\int_0^t r_s\text{d}s$ is \begin{align*} \varphi_{R_t}(u)=\mathbb{E}\left[e^{iuR_t}\right] = A_t(u)e^{B_t(u)r_0}, \end{...

4

A partial but general answer: Let $\mathcal{F}_t^W$ be the filtration generated by $W$. Since $X_T = \int_t^T v_u du$ is $\mathcal{F}_T^W$ measurable, the Clark-Ocone-Haussman formula states $$X_T = E_t[X_T] + \int_t^T E_u \left[ D_u^W X_T \right] dW_u$$ with $D_u^W X_T$ denoting the Malliavin derivative of $X_T$ with respect to $W_u$. Hence,  Var(X_T) = ...

5

The probablility of a jump of $J = \phi$. ( in either direction so I'll assume $\frac{\phi}{2} =$ probability of J and $\frac{\phi}{2} =$ probability of -J ). The probability of a jump of $0 = (1-\phi)$. So, the expectation of the of jump amount, MM, $= E(MM) = \frac{\phi}{2} \times J + \frac{\phi}{2} \times -J + (1-\phi) \times 0 = 0$ The variance, \$\...

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