<t>you can reduce the problem to solving it for a regular compound poisson process.first just consider the jump times of the two processes, which will be independent standard poissons. then you can calculate that the sum, say , will be a poisson process with intensity a_1 + a_2. you want to make the...

i know there has been a lot of abstract theory developed, usually with proportional transaction costs. do people just sort of manipulate the frictionless models in an ad hoc way or are there any systematic models used?

what about using a levy copula to construct a 2-dimensional levy process

If , then .

<t>williams is a great book, but its not on stochastic calculus. its like a first course in graduate probability. i think oksendal is kind of bad at writing text books, but "SDE" is better than his other ones. karatzas and shreve is the most famous but it really kind of sucks. for pure math approach...

<t>This is just what I thought, I'm not sure if it's right. When you have a weak solution to a diffusion, you have a measure on the space of continuous functions, which is equivalent to specifying all finite dimensional distributions of the process. But if a stopping time takes on uncountably many v...

<t>I'm not sure about the last part of this argument. Here is the solution I came up with. It might be similar, it definitely starts the same way. The game is to reduce things to the case of just one random variable which was solved above. Without loss of generality, assume that all of the random va...

For the inductive step jurowilmott11 mentions, you can also use the fact that the supremum of a finite collection of real valued random variables is again a real valued random variable. But you can't use this reasoning for infinitely many.

<t>Each of the random variables has a trouble spot where it blows up. But outside of an arbitrarily small interval around that one bad point, the random variables are already integrable. By choosing the "right" intervals around the bad points you can make sure that you don't land in them too much. <...

If the sequence is finite, the problem is easier to solve, but it is still true in the infinite case.

You are given a sequence of random variables defined on some probability space, say with measure P. Find an equivalent (probability) measure Q such that under Q, all of the random variables are integrable.

<t>I think you can additionally do a variance reduction thing on your monte carlo simulations. For example I think you can compute an appropriate weighted sum of two call options, each on just one of the stocks, which hopefully can be done analytically. Then when you do simulations, you tweak your r...

Yeah i do usually think about conditioning as partitioning the sample space and then averaging over those partitions, as you mention. The only drawback is that this way typically only makes sense when the sample space is finite.

<t> refers to the space of random variables such that is finite. For finite measure spaces, when . In particular . To define the notion of an orthogonal projection, your space needs to have some geometry, which means it needs to have an inner product, ie it needs to be a Hilbert space. has such an i...

<t>When your random variables are square-integrable, you can think of conditioning as an orthogonal projection onto a smaller subspace, and so you can project all at once or do it in a couple steps. For just L1 random variables you need a couple more lines of math, but the intuition is still the sam...