$$
\mathbb{E}[d S]=r S d t=\mu S d t+\mathbb{E}[J-1] S \lambda(t) d t
$$
It follows that the risk-neutral drift is given by $\mu=r+\mu_{J}$ with
$$
\mu_{J}=-\lambda(t) \mathbb{E}[J-1]
$$
$k:=\log (K / F)$ defined by
$$
\hat{G}(u, \tau)=\int_{-\infty}^{\infty} e^{\mathrm{i} u k} G(k, \tau) d x
$$
MATH0095 COURSE NOTES ：
Because the Black-Scholes formula $C$ for a call option is linearly homogenous in the stock price $S$ and the strike price $K$, we have the relation
$$
C=S \frac{\partial C}{\partial S}+K \frac{\partial C}{\partial K}
$$
It follows that
$$
K^{2} \frac{\partial^{2} C}{\partial K^{2}}=S^{2} \frac{\partial^{2} C}{\partial S^{2}}
$$
Also, in the jump-to-ruin case with zero interest rates and dividends, we have
$$
\frac{\partial C}{\partial T}=\frac{1}{2} \sigma^{2} S^{2} \frac{\partial^{2} C}{\partial S^{2}}+\lambda S \frac{\partial C}{\partial S}-\lambda C
$$