Everything depends on the model of randomness influencing the differential equation(s). More precisely, on the quantities perturbed by the Levy Jump process(es). Otherwise the subject of simulation is not determined. Let me illustrate this question with the following problem for a single differential equation:

Let the deterministic DE be:

\dot{x}(t) = a x(t) + b x^2(t),

which is equivalent (for positive processes) the following one:

\dot{y}(t) = a + b exp{y(t)}, where y = log{ x }.

Thus some (of many) possible examples of the SDE which can be seen as the counterparts of the above ONE equation are

1. dX(t) = (a X(t) + b X^2(t)) dt + dL(t)

2. dX(t) = (a x(t) + b x^2(t)) dt + g(t,X(t) dL(t)

3. dY(t) = ( a + exp{ Y(t)} ) dt + h(t,Y(t)) dL(t)

. . . .

After deciding about the way the Levy process L={L(t), t >0}, influences the equations, one can start to think about the class of meaningfully admitted processes L . These are some examplary questions to be asked:

- should them be with mean value zero,?

- should their intensities of appearing of signle jumps be finite?

- what type of the distribution of the sizes of jumps would have physical justification?

Answers to all such subproblems depend on the particular problem of further application of the simulation. The way of simulation depends mainly of the intensity of the jumps - is it finite or not. In the first case one can easily manage the problem just b solving the deterministic equation between the instants of the jumps, then generating the next jump which pushes the process into a new value, and then again llowing the system evolve accorting to the determinstic DE.

Possibly the above advices help to strat the research!

Regards, Joachim

Thanks a lot such a wonderful reply, but i am facing the problem in numerical simulation of SDEs with Levy jumps.

Please elaborate the Simulation part also.

First, I do not have access to the paper

Second, about simulation the only advice I can give should be related to the idea of the algorithm, which I have already given above. In some more detail, according to my understandingthe main steps are as follows,

1. determine the deterministic equation

2. determine the way the randomness impacts the evolution

3. assume the model of the random perturbation (properties of the process) with respect to some qualitative features justified by the natural causes of the randomness

4. perform with suitable software step by step the solution, in particular instants adding to the procedures the perturbation generated by an appropriate commands chosen according to the assumptions.

I cannot help much more unless I see the algorithm and/or a model to be generated.

Thanks for thanks, Joachim

Thanks