Hy everyone,
My first question so please be gentle.
I need to find the following PIDE for a Lévy market
$$ -rf(x,t) +\partial_2 f(x,t)+(r-\frac{c}{2}) \partial_1 f(x,t)+\frac{c}{2} \partial_1^2 f(x,t) \\ +\int_R \Big(f(x+z,t)-f(x,t)-\partial_1f(x,t)(e^z-1)\Big)v(dz) $$
Where $f(x,t)$ is the price of the option at time $t$ and the driving Lévy process(not the underlying) is at $x$. $ \partial_1 f(x,t)$ is the derivative with respect to the first argument and $\partial_2 f(x,t)$ is the derivative with respect to the second argument. $N(x,t)$ is the nondeterministic jump measure (basically counting jumps of the height $x$ in the $ [0,t]$), $v(x)$ is the Lévy measure(basically the expectation of $N$). This PIDE solves the price for a European option in an exponential Lévy market, this PIDE is correct. My issue is the term
$$\partial_1f(x,t)(e^z-1)$$
In every proof I see I cannot understand where it comes from, I am breaking.
The main part of deriving the formula is just a basic application of Itô's formula for Lévy processes.
$$\begin{eqnarray}d(e^{-rt}f(L_{t-},t)) = & \\ & \hspace{-3cm} -re^{-rt}f(L_{t-},t)dt+e^{-rt}\partial_2 f(L_{t-},t)dt+e^{-rt}\partial_1 f(L_{t-},t) dL_t +\frac{1}{2} e^{-rt}\partial_1^2 f(L_{t-},t)dL^c_t \\ & \hspace{-7cm} + e^{-rt} \int_R \Big(f(L_{t-}+x,t)-f(L_{t-},t)-\partial_1f(L_{t-},t)x\Big)N(dx,dt)\\ & \hspace{-5cm} = e^{-rt} \{ -rf(L_{t-},t)dt +\partial_2 f(L_{t-},t)dt+\partial_1 f(L_{t-},t)\sqrt{c}dW_t+\frac{1}{2} \partial_1^2 f(L_{t-},t)cdt \\ & \hspace{-4cm} +\int_R \Big(\partial_1f(L_{t-},t)x\Big)(N-v)(dx,dt) + \int_R \Big(f(L_{t-}+x,t)-f(L_{t-},t)-\partial_1f(L_{t-},t)x\Big)(N-v)(dx,dt) \\ & \hspace{-9cm} +\int_R \Big(f(L_{t-}+x,t)-f(L_{t-},t)-\partial_1f(L_{t-},t)x\Big)v(dx) dt \} \end{eqnarray}$$
And this is where my issue lies. As far as I see it (the first) $\int_R \Big(\partial_1f(L_{t-},t)x\Big)(N-v)(dx,dt)$ in the second formula comes only from the term $e^{-rt}\partial_1 f(L_{t-},t) dL_t$ in the first formula. But as far as I see it $\partial_1 f(L_{t-},t) dL_t=\partial_1 f(L_{t-},t) dL_t^c +\int_R (\partial_1f(L_{t-},t)x)N(dx,dt)$ so we end up missing a $-\int_R \Big(\partial_1f(L_{t-},t)x\Big) v(dx)dt$.