August 24th, 2016, 3:35 pm

For just about any n-dimensional continuous-time Markov process [$]\{X_t\}[$], you can define a *local *Levy measure [$]\nu_x(dy)[$] that characterizes the jump possibilities from [$]X_{t-} = x[$] to [$]X_t = x + y[$]. Note that [$](x,y)[$] are both [$]n[$]-vectors, I am assuming a time-homogeneous process (which includes all of your examples), and my notation may be somewhat idiosyncratic (but I think it will be clear by the end of the post).

Since the 2-dimensional Heston model no jumps, its local Levy measure is zero.

For the Bates ('96) model, the 2-dimensional process can jump in only one coordinate direction [$]x_1[$], where [$]x_1[$] is either the stock price or the log-stock price, whatever you like. So, I would say [$]\nu_{x_1,x_2}(dy_1,dy_2) = \nu_{x_1}(dy_1) \delta(dy_2)[$], where the dependence on [$](x_1,y_1)[$] is the same as the Merton jump-diffusion model and [$]\delta[$] is the Dirac measure. For example, with log-stock price as the first coordinate, [$]\nu_{x_1,x_2}(dy_1,dy_2) = \lambda \frac{e^{-(y_1-\mu_J)^2/2 \sigma_J^2}}{\sqrt{2 \pi \sigma_J^2}} dy_1 \delta(dy_2)[$], where [$]\lambda[$] is the Poisson intensity for the jumps.

Why? Because, basically, for the Bates model, you want the generator for the jump part to come out as follows:

[$]\mathcal{A} f(t,x,v) = \int \int [f(t,x + \xi_1, v + \xi_2] - f(t,x,v)] \nu_{x,v}(d \xi_1, d \xi_2) = \lambda \int [f(t,x + \xi_1,v] - f(t,x,v)] p_J(\xi_1) d \xi_1[$].

Now all the coordinates are scalars and [$]p_J(\xi_1) = \frac{e^{-(\xi_1-\mu_J)^2/2 \sigma_J^2}}{\sqrt{2 \pi \sigma_J^2}}[$] is exactly the same normal density for the jump sizes as in Merton's jump-diffusion.

There is a general discussion of generators with local Levy measures on pgs 38-39 of "Option Valuation under Stochastic Volatility II", and an example of a 2d jump-diffusion that can only jump in one direction on pg. 84. If there is any interest, I could post pgs 38-39 as an excerpt.