Hi, 

I'm looking for an analogy of Feynman-Kac (but ideally a fairly general) that would be able to incorporate the jump processes. I.e. Feynman-Kac analogy for 
[$]dx(t) = \mu (t,x(t))dt + \sigma (t,x(t))dW(t) + dJ(t,x(t))[$], where [$]J[$] is some sort of a pure jump process.
Also, I'm interested in the role of a 'compensator' which is deducted from drift in connection with the possible Feynman-Kac theorem. For example in [Merton 1976] we have ([$]x[$] is a stock process here):
[$]\frac{{dx(t)}}{{x(t)}} = (r - \lambda \alpha )dt + \sigma dW(t) + (\eta  - 1)dN(t)[$], where [$](\eta  - 1)[$] is lognormal, [$]\alpha  = {\mathbb{E}^\mathbb{Q}}[\eta  - 1][$], [$]N[$] is a standard Poisson process with intensity [$]\lambda[$]. Here, it is clear to me that the compensator term [$](\lambda \alpha)[$] ensures the jump process being a [$]\mathbb{Q}[$]-martingale. I guess the compensator needs to be there to ensure that the local rate of return is [$]r[$].
Then, the PIDE originating from this jump diffusion is
[$]\frac{{\partial V}}{{\partial t}} + x(r - \lambda \alpha )\frac{{\partial V}}{{\partial x}} + \frac{1}{2}{\sigma ^2}{x^2}\frac{{{\partial ^2}V}}{{\partial {x^2}}} + \lambda \left( {\int\limits_\eta  {\left( {V(t,x\eta ) - V(t,x)} \right){f_\eta }(\eta )d\eta } } \right) - rV = 0[$].

So my 'non-technical' approach (based on 'if-then' by observing the dynamics of [$]x[$] and the PIDE above) to the Feynman-Kac for jump diffusion is summarized into the following steps:
1) Ensure the drift reflects the compensator of the jump process (such that the jump process together with the compensator is a martingale). That is - express the dynamics in a compensated form such that the compensator is in the drift and the jump component is no longer compensated. (I hope this point is understandable).
2) Apply results from the standard Feynman-Kac for the continuous part of the dynamics of [$]x[$].
3) Add the infinitesimal rate of arrival (here this is the jump intensity [$]\lambda[$]) times integral that captures the expected change of the option price resulting from the jump. 

If I then have two jump diffusion equations with correlated Wiener processes and independent jump components, would the resulting pricing PIDE for [$]V(t,x_1,x_2)[$] be
[$]
\begin{array}{l}
\frac{{\partial V}}{{\partial t}} + ({\mu _1}(t,x) - {c_1})\frac{{\partial V}}{{\partial {x_1}}} + \frac{1}{2}\sigma _1^2(t,x)\frac{{{\partial ^2}V}}{{\partial x_1^2}} + {\lambda _1}\left( {\int\limits_{{\eta _1}} {\left( {V(t,{x_1} + {\eta_1},{x_2}) - V(t,x)} \right){f_{{\eta _1}}}({\eta _1})d{\eta _1}} } \right) + \rho {\sigma _1}(t,x){\sigma _2}(t,x)\frac{{{\partial ^2}V}}{{\partial {x_1}\partial {x_2}}}\\
 + ({\mu _2}(t,x) - {c_2})\frac{{\partial V}}{{\partial {x_2}}} + \frac{1}{2}\sigma _2^2(t,x)\frac{{{\partial ^2}V}}{{\partial x_2^2}} + {\lambda _2}\left( {\int\limits_{{\eta _2}} {\left( {V(t,{x_1},{x_2} + {\eta_2}) - V(t,x)} \right){f_{{\eta _2}}}({\eta _2})d{\eta _2}} } \right) - rV = 0
\end{array}
[$]
? (The changes in option prices resulting from the jumps do not necessarily have to be with respect to [$]{x_i} + \eta_i[$], this depends on the specification of the jump term, in general). Here [$]x = (x_1,x_2)[$], and [$]c_1,c_2[$] being compensators of the jump-diffusing processes [$]x_1,x_2[$]. The drifts are a bit garbled but they have the meaning of being compensated original drifts.
Thanks!
//Edit: Based on Alan's suggestion corrected typo in the "jump" integrals.

Unfortunately, virtually all SDE notations for jump processes are awkward. Your general case notation is somewhat ambiguous, because you haven't connected up the [$]J[$]'s and the [$]\eta[$]'s in your integrals. It can probably be made to generally work (for independent jumps with constant Poisson intensities) if you allow [$]J_i = J_i(x_i,\eta_i)[$]. In other words, you should be able to recover the Merton model from the general case by a suitable specialization, right?   

Another comment is that arranging for the (discounted) process to be a martingale is required only for risk-neutral evolution. But, the Feynman-Kac idea works in general for jump-diffusions or, more generally, for any Levy-type process (which is a class of very general jump-diffusions with local diffusion and jump characteristics).   

Cont & Tankov's book on jump processes is a good reference for this type of stuff. I have some discussion of general Levy-type generators in my own latest book, too.

If you are trying to use jump diffusion underlying then you first should note that there is no risk free hedged portfolio does exist. Thus, using formally BS formula for option price is formally wrong. The statement that BS option is no arbitrage does not accurate. The BS price is derived from the fact of existence of the risk free instantaneous hedged portfolio which presents illustration of no arbitrage portfolio construction. If one uses BS option price formula for jump diffusion then BS hedged portfolio is risky which then goes to BS option price. 

Alan:
1) You are right with the [$]J[$]s in the integral. I corrected several typos in the formula shortly after I posted my question but I forgot to correct this one. In the Merton model it was correct so I hope you don't think it was a mistake but rater a typo. I now corrected this.

2) Yes, I'm aware of what you point out about the Feynman-Kac is a generally valid result and that the exact form of the PDE (PIDE) from SDE + some expectation depends on the measure under which the expectation (and thus also the SDE dynamics) is formed. I guess this is a natural property of conditional expectation which is always a martingale, regardless of the measure under consideration.

I have searched Tavella and Randall so far and it helped me to improve my skills in solving PDEs/PIDEs but it still doesn't provide the reader with a complete "Feynman Kac for jump diffusion" although they give some basic suggestions. I would need a multidimensional version of this theorem anyways. 

My objective is to find a PIDE for a claim which is subject to two interrelated processes (default intensities). Ideally these intensities can experience a common "shock" (jump). This could patch some of the weaknesses of modeling "correlated defaults" where the default correlation over time defined as [$]\rho (t) = {\rm{corr}}[{{\bf{1}}_{\{ {\tau _1} \le t\} }},{{\bf{1}}_{\{ {\tau _2} \le t\} }}][$] is typically very low even if in both intensities (e.g. CIR) the Wiener processes are perfectly correlated.
I guess in the case of a common shock in the resulting PIDE there should be only one "jump" integral such as
[$]\left( {\lambda \int\limits_\eta  {\left[ {V(t,{x_1} + \eta ,{x_2} + \eta ) - V(t,x)} \right]{f_\eta }(\eta )d\eta } } \right)[$] because the shock affects both [$]x_1,x_2[$] simultaneously (and if we further for simplicity assume at the same magnitude). But again, this is my intuition, not supported by a theorem. Of course, I can verify this numerically by solving the 2D-PIDE and compare the result with MC but still that is not bullet-proof for me. I need a theorem and it does not even need to be a general theorem for the whole class of Lévy processes, Poisson jumps will do to catch the main idea. Hopefully Cont and Tankov cover this, at my university (and even in the whole country) no one even knows what I'm talking about.  You have an interesting book, contains much more than I can absorb in my whole life but will consider getting it because some topics might be useful to me.

list1:
Thanks for your comment. As far as I know, there is something like a "minimum variance hedge" applied instead because a perfect hedge is not feasible.

It seems that one interprets the BSE solution as a price of the option based on the fact that BS's hedged portfolio has instant risk free return. If BS portfolio does not riskless the sense of the option price is lost. For jump diffusion underlying you can check the value of its change in thew value and try to verify BS interpretation of the BSE solution as the option price. Subjectively BS price is a price of the option that implies by the market participants who thinking about the option price based on hedge opportunity. If the option buyer(s) buys option to get high return he should ignore BS price and think about the price based on P/L perspective. Hence BS price is rather the price implied by a particular strategy, ie hedging. In order to claim that BS is market price we need to find arguments that justify that all buyers think about the option price based on its hedging possibility. Nevertheless one can use it as an approximation of the market option premium and interpret its hedging property as an approximation of the real unhedged option price defined by its surplus and demand settlement.

I'm looking for an analogy of Feynman-Kac (but ideally a fairly general) that would be able to incorporate the jump processes. I.e. Feynman-Kac analogy for 
[$]dx(t) = \mu (t,x(t))dt + \sigma (t,x(t))dW(t) + dJ(t,x(t))[$], where [$]J[$] is some sort of a pure jump process.

For what it's worth, here's an informal approach for a case that you are interested in: an n-dimensional jump-diffusion where jumps are triggered by a single, constant intensity, Poisson process [$]N_t[$]. Taking a time-homogeneous process for simplicity, the SDE evolution is

[$]dX_t = \mu(X_t) dt + \sigma(X_t) \cdot dW_t + \xi_t \, dN_t[$], where [$]E[dN_t] = \lambda dt[$]. 

Here [$](X_t,\mu, \xi_t)[$] are each n-vectors. (You can take [$]W_t[$] to be a standard n-dimensional BM, in which case [$]\sigma[$] is an n x n matrix here).  Conditional on a jump event, [$]\xi_t[$] is the random n-dimensional jump amplitude, which is independently drawn at each jump time t from a probability density [$]p_J(\xi)[$].

Let [$]E_{t,x}[\cdots][$] denote throughout a time t expectation, conditional on [$]X_t = x[$], and [$]\phi(x)[$] a fixed given function. First, you need to convince yourself that, under any continuous-time Markov process (which is a class including the process we are talking about here), the terminal value function [$]f(t,x) \equiv E_{t,x}[\phi(X_T)][$] always satisfies the evolution equation

[$](*)  \quad (\frac{\partial f}{\partial t} + \mathcal{A} f)(t,x) = 0[$], on [$]t < T[$], where [$]\mathcal{A}[$] is the infinitesimal generator of the process. The generator is generally defined by

[$](**) \quad \mathcal{A} f(x) = \lim_{t \downarrow 0} \frac{E_{t,x}[f(X_t)] - f(x)}{t} = \frac{E_{t,x}[df]}{dt}[$]. (Technically, [$]f(x)[$] must be in the domain of the operator).

 Essentially, (*) holds because the process [$]M_t \equiv f(t,X_t)[$] is a martingale, whether we are talking about real-world or risk-neutral evolutions. The first equality in (**) is the definition of the generator and the second equality is the informal (generalized) Ito's rule version. So you should think of [$]df[$] as the Ito differential, generalized to allow jumps. Note that the second expectation can be taken at an arbitrary time t because the process is time-homogeneous. (There is somewhat of a notational abuse here because the [$]f(\cdot)[$] in the definition of the generator does not depend on t. If that bothers you, use [$]u(x)[$] in the generator definition, keeping [$]f(t,x)[$] for the value function. The key point is that [$]\mathcal{A}[$] only acts on the spatial coordinates, so the presence of t is irrelevant).

When there is a jump possibility of the type we are considering here:

[$]df(t,X_t) = df_c(t,X_t) + \Delta f = df_c + [f(t,X_t + \xi_t) - f(t,X_t)] dN_t[$], where [$]df_c(t,x)[$] is the standard Ito df under a purely continuous process (the diffusion part).   Thus 

[$] \mathcal{A} f(t,x) =  \mathcal{A}_c f(t,x) + \frac{E_{t,x} \left\{ [f(t,X_t + \xi_t) - f(t,X_t)] dN_t \right\}}{dt}[$] .

The second term reduces to   

[$]\frac{\lambda \, dt E_{t,x} \left[ \left( f(t,X_t + \xi_t) - f(t,X_t) \right) | \mbox{a jump occurred} \right]}{dt} =  \lambda \int [f(t,x + \xi) - f(t,x)] p_J(\xi) \, d \xi[$],

where [$]p_J(\xi)[$] is the jump-amplitude probability density, introduced earlier. In the integral, [$]d \xi[$] is the n-dimensional volume element. The last equation, coupled with (*) then provides an informal derivation of the evolution equation you were seeking.

(There have been a zillion small edits, but I think it is stable now).

Alan, 

thanks, this is complete and clear to me. I'm familiar with the infinitesimal generators so it was easy to understand for me. 

So in this case (based on the dynamics of [$]X[$] that you use), [$]N[$] is one-dimensional Poisson process. In turn, the convolution integral captures a jump in [$]f[$] resulting from a simultaneous jump in all the spatial variables. It is nice to see (if I'm not wrong) that there can be some dependence among the jump sizes [$]{\xi _1},{\xi _2},...[$], by the joint density [$]p_J[$], without any further clutter. This is what I needed. I can also see that I wasn't actually wrong with the convolution integral expressing the common shock. This was a special case of your 'General case' jump integral when [$]{\xi _i} \equiv \eta [$] as in such a case the multivariate case reduces to the one-dimensional case where a single jump size affects all the spatial variables.

You should write more books. I took a quick look in Cont and Tankov and I found some errors in this part which (temporarily) demotivated me to continue reading it.

Alan, 

thanks, this is complete and clear to me. I'm familiar with the infinitesimal generators so it was easy to understand for me. 

So in this case (based on the dynamics of [$]X[$] that you use), [$]N[$] is one-dimensional Poisson process. In turn, the convolution integral captures a jump in [$]f[$] resulting from a simultaneous jump in all the spatial variables. It is nice to see (if I'm not wrong) that there can be some dependence among the jump sizes [$]{\xi _1},{\xi _2},...[$], by the joint density [$]p_J[$], without any further clutter. This is what I needed. I can also see that I wasn't actually wrong with the convolution integral expressing the common shock. This was a special case of your 'General case' jump integral when [$]{\xi _i} \equiv \eta [$] as in such a case the multivariate case reduces to the one-dimensional case where a single jump size affects all the spatial variables.

You should write more books. I took a quick look in Cont and Tankov and I found some errors in this part which (temporarily) demotivated me to continue reading it.

It might make sense to look through book I.I.Gikhman , A.V. Skorohod Stochastic Differential Equations, 1972 [$]\S 9[$] p 288. They consider nonhomogeneous case for jump diffusion. There is also revised 2nd ed of the book.

Alan, 

thanks, this is complete and clear to me. I'm familiar with the infinitesimal generators so it was easy to understand for me. 

So in this case (based on the dynamics of [$]X[$] that you use), [$]N[$] is one-dimensional Poisson process. In turn, the convolution integral captures a jump in [$]f[$] resulting from a simultaneous jump in all the spatial variables. It is nice to see (if I'm not wrong) that there can be some dependence among the jump sizes [$]{\xi _1},{\xi _2},...[$], by the joint density [$]p_J[$], without any further clutter. This is what I needed. I can also see that I wasn't actually wrong with the convolution integral expressing the common shock. This was a special case of your 'General case' jump integral when [$]{\xi _i} \equiv \eta [$] as in such a case the multivariate case reduces to the one-dimensional case where a single jump size affects all the spatial variables.

You should write more books. I took a quick look in Cont and Tankov and I found some errors in this part which (temporarily) demotivated me to continue reading it.

Yes, you're absolutely correct on the jump size dependence.

Thanks for the kind remark on the book writing. While I do have a glimmer of an idea for another book, mostly I'm still trying to get the latest one off to a good start. If you happen to buy it and like it, please post an amazon review!