Figure out the boundary conditions for a pure-jump and a jump-diffusion

[jlperla]

Put in a small document on boundary conditions in the main folder.

To simplify:

• Assume that the jump is always forwards, and it is a deterministic number.
• define it on $z \in [0,1]$
• Have a reflecting barrier at the bottom and consider 2 cases: a reflecting barrier at the top, and an absorbing barrier.

First task:

• [ ] Is it possible to have a different payoff depending on how far over the absorbing barrier they can go? This would be a property of jump processes. Think you would see about this stuff if you look up optimal stopping with a jump-diffusion.

Main task: figure out what the boundary conditions look like in these two cases:

• [ ] First, a pure jump process, reflecting barrier
• [ ] a pure jump process with an absorbing barrier (and a payoff of 0). Still linear
• [ ] a pure jump process with an absorbing barrier, and a payoff as a function of the terminal state
• [ ] jump-diffusion, reflecting
• [ ] jump-diffusion absorbing linear
• [ ] jump-diffusion absorbing affine payoff.
• [ ] forward jump-diffusion with an absorbing barrier on the bottom. That is, it cannot jump across the boundary. This is probably the same as the pure diffusion one (i.e. the jump doesn't effect anything), but we need to make sure.
• [ ] forward jump-diffusion with a reflecting barrier on the bottom. Again, the guess is that the jumps won't effect anything, but we need to be sure.

[jlperla]

I think that a good way to look into this would be to find the https://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula style formula for a jump-diffusion, given a stopping time associate with crossing the boundary.

The exact definition of the stopping time here might matter for the formulas people use for this...

[jlperla]

@mszkup I wanted to point you towards this issue, as I think Fernando will need someone far more competent than myself to check the results.

To summarize: we need to make sure that we have the clear general case for the absorbing and reflecting boundary values of a jump-diffusion process. Of course, we may have a simplified version of them for our particular problem, but with finite-difference methods we need to be extremely careful with the number of degrees of freedom/etc.

[FernandoKuwer]

Jesse and Michal, this is an update to keep you informed about the task: So far, I am still working on the value function problem. The reason it is taking so long to answer it is because, in order to have a proper solution, I have to impose continuity for the value function around the absorbing boundary (here, my definition of a "proper solution" is to satisfy theorem 2.2 on Øksendal and Sulem). This ends up implying one of two: i) the solution ignores the boundary condition V(x_low,t) = g(t), forcing g to be a function of X or ii) the solution implies a constant value function for all continuity area (the space in which the agent prefers to not stop immediately), which makes V not to change after crossing the boundary condition. The books (and papers/notes that I could find) never talk directly about this problem, but after reading a bit my intuition was that V could change after crossing the boundary. Due to this continuity requirement on the boundaries of the closure of the "solvency region", I’m not so sure anymore. Therefore, in order to solve this as soon as possible, I thought that it might be a good idea if I ask for references and maybe some insights to some faculty in the Math department, what do you think about that? I have a friend who is a student there and could help me pointing out which professor is more likely to help. Also, I would, off course, just drop by his/her office and discuss this specific case, not involving this faculty any further in the research. (Another possibility is to continue looking in the literature for different theorems on how to solve this class of problems, but this might take longer)

[jlperla]

I have to impose continuity for the value function around the absorbing boundary (here, my definition of a "proper solution" is to satisfy theorem 2.2 on Øksendal and Sulem).

That shouldn't be a problem, since value functions always end up continuous. Similarly, it means that we probably should assume that the payoff is continuous as well.

This ends up implying one of two: i) the solution ignores the boundary condition V(x_low,t) = g(t), forcing g to be a function of X

I am not sure what the x_low there is? Lets stick with the notation I put in the issue where we will only look at absorbing/reflecting boundaries for jumps above the x=1.

Given what we have above, lets assume that the region is x \in [0,1] for the "interior" of the operator. Now, of course it would be possible for jumps to occur more than an infinitesimal above the 1 threshold. Lets assume that the payoff is b(x,t) for the payoff at time t if the agent ends up in state x. Note that b(x,t) should be defined on x in[1, ???]where the???is the maximum jump size that can occur if the point is arbitrarily close tox=1`.

So, with this I am not sure I understand your statement about V(x_low,t) = b(t). Are you saying that the payoff can be a function of the X? If so, that would be preferable. Otherwise, we the payoff would just have to be b(t) instead of b(x,t).

Now, all of this is to say that: I am not sure if having the x dependent payoff at stopping is possible, but it seems to be the natural way to do it.... basically, the random stopping time is when x >= 1 and the payoff at that point is g(t, x) which could depend on exactly where the x "lands".

[mschauer]

There are two ways to deal with a space homogenous jump distribution when an absorbing boundary comes into play: jumps which formerly would go beyond the boundary do not happen and the jump size distribution is truncated, or jumps which formerly would go beyond the boundary end at the boundary and the jump size distribution has a point mass at the distance to the boundary.

[jlperla]

@mschauer Thanks you so much!

I think we want to go with the latter, which might be easier to implement more generally and to nest the reflecting boundary. It also seems like it might nest the first case (i.e. truncation to a point) for typical problems?

Basically, we have the "extension" of the domain for the maximum size of the jump, and then have a "payoff" for the absorbing boundary as a function of the state where it formally jumps over. Does that work?