Has this problem been solved?

I release a particle. Its vertical location is a random function of time. There is a barrier above it, and a barrier below it. The heights of both barriers are non-random (known) functions of time.


What is the probability the particle will hit the top barrier first, the bottom barrier first, or neither, within time t?


eLROY

No offense eLROY..who cares?

Then this may be of interest to you.

Buy two June Eurocurreny

Sell three June Swiss Franc..do it right now on Globex

If the upper and lower bound functions are known you can simulate it and get very acurate answers. If they have some additional properties then there are analytical answers to your questions. The fact you think this is important for a trade, and don't know stochstic calc means you should not do the trade - (-:

No, I was wondering if it had been solved, and I guess you're saying it hasn't been. So I have a simpler question. What is the term for the 3D volume that is all possible paths between two points in a 2D surface stacked on top of each other?


In other words, at each of the two points it is ininitely high, and at the saddle between them, it is at its lowest and most dispersed, right? It's like, hold 100 pieces of string, one end in each hand, and let them go slack by moving your hands together. The string will be the most spread out in the middle.


It's like a normal distribution that runs forward then runs in reverse, or something. What's the name for that, so I can hunt down some earlier mathematical work on adding and removing paths and barriers - on cutting strings in effect?


Thanks,


eLROY


P.S. I am not sure you need to do any "stochastic calculus" to run a monte carlo, or do analytic estimates of this - much less trade it. I have already developed a number of analytic approaches in my head, on the areas of certain bisections and slices, relative volumes if you cut out different regions, the commonality of strings to various sequential groupings involving barriers, and so forth. But I would still much appreciate any further "analytic" tips you can give me, SC!

Very common in the FX options world. FX is modelled as more or less random walk, and barrier options are quite common.


I don't think there are any closed form solutions to this problem... people either use monte carlo or trinomial models

It has been solved for simple random walks (simple random walk with absorbing barriers) and Brownian motion....ans yes you don't need any stochastic calc. for this.