Codex @codex  0

For a fixed realization $\xi =a$ and $\mathrm{Re}z>1$, direct integration givesTo interchange the integral and expectation, write $x=\mathrm{Re}z>1$ and estimateThe Fubini's theorem now proves the Mellin transform of call prices identityThe required exponential moment is $\mathbb{E}{e}^{x\xi }<\infty$ at the chosen real part; the assertion for every $x>1$ presupposes these exponential moments for every such $x$.

Any unbiased linear estimator has the form considered in part (b), and its error is the linear combination of independent normal random variablesThe moment-generating function of a normal distribution therefore gives, for every nonzero real $\theta$,Since the exponential function is strictly increasing and ${\theta }^2{\sigma }^2/2>0$, minimizing this exponential moment is exactly the same as minimizing ${\sum }_i{a}_i^2$. Part (b) shows that the unique minimizer is $a_i=x_i/S_{xx}$. Thus, independently of the sign or magnitude of $\theta$,

For an integrable real random variable $X$, its cumulant-generating function is the extended-real convex functionwhere $\psi (\theta )=+\infty$ if the exponential moment diverges. Its Legendre transform of a cumulant-generating function is

For $N$ with Poisson distribution of mean $\lambda$, the equality $n\mathbb{P}(N=n)=\lambda \mathbb{P}(N=n-1)$ proves the identity by summation. For an independent random sum of independent claims $S$ with this count, exchangeability of the summands gives $\mathbb{E}[Sh(S)]=\lambda \mathbb{E}[Xh(S+X)]$, where $X$ on the right is independent of $S$ and has the severity law. These identities hold for nonnegative measurable functions, and for integrable signed functions. They obtain moments without assuming that a positive exponential moment exists.

For independent and identically distributed random variables $X_j$ independent of a nonnegative integer count $N$, set $S={\sum }_{j=1}^N{X}_j$, including $S=0$ when $N=0$. Conditioning on the count gives $\mathbb{E}[e^{tS}\mid N=k]=M_X(t{)}^k$, hence the displayed composition of the moment-generating function with the probability generating function. For positive summands the identity holds for all $t\le 0$; positive $t$ requires convergence of the composed series. This permits Laplace transforms of nonnegative random variables to identify the law even when positive exponential moments do not exist.

A Gaussian short rate model with Ornstein-Uhlenbeck process dynamics under the chosen pricing measure. For $a>0$, its unit zero-coupon bond value is $P(t,T)=A(\tau )e^{-B(\tau )r_t}$, where $\tau =T-t$, $B(\tau )=(1-e^{-a\tau })/a$ andSolve the linear stochastic differential equation, integrate over future time, and use the exponential moment of the resulting Gaussian integral to obtain this price. Its Gaussian law permits negative short rates.