Probabilities of Absorption in Asymmetrical Random Walks
Consider a discrete-time random walk on the real numbers ℝ. Let:
- s ∈ ℝ be the initial position of the walker.
- L < s < U be the lower and upper absorbing barriers, respectively.
- p ∈ (0, 1) be the probability of an upward increment.
- u ∈ ℝ, u > 0 be the magnitude of the upward increment.
- l ∈ ℝ, l > 0 be the magnitude of the downward decrement.
Determine the probability, P(s), of the random walk being absorbed at the upper barrier (U) before being absorbed at the lower barrier (L), assuming a potentially infinite number of steps are allowed.
Questions for Investigation
- Existence of a Closed-Form Solution: Does there exist a general closed-form solution for P(s) that depends only on the parameters p, L, U, s, u, and l with real number values? If not, provide a proof or a counterexample to justify the non-existence.
- Methods for Specific Cases: Under what specific conditions can closed-form expressions for P(s) be obtained? Are the same mathematical techniques still applicable (difference equations, generating functions, martingales), or do new approaches become necessary when dealing with real-valued walks?
- Computational Considerations: Since a closed-form solution might not exist, discuss the challenges of numerically finding P(s). Discuss the limitations of numerical and simulation-based approaches, and potential strategies to address the complexities arising from the real-valued random walk.
Specifically, I'd like your guidance on:
- Mathematical Field: What area of mathematics is most relevant to analyzing this class of random walks?
- Book Recommendations: My background is in calculus, linear algebra, and basic number theory. Which books or resources would bridge the gap to the necessary concepts for understanding and potentially solving this problem?
- Role of Measure Theory: Would a deeper understanding of measure theory significantly enhance my ability to analyze this random walk, or are other mathematical tools more central to the solution approaches?"
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This is a high-level research question that will take multiple days to investigate, write proofs for, and test simulations. I would suggest raising the bounty to at least $300.
I wonder if this kind of thing still needs to be researched, and maybe somewhere in the advanced textbook, even more, general cases are described and analyzed. So, the only thing to do is to find and interpret them from a more specific problem, in other words, to deduce from them.
Hi, I believe I have some partial answers to this problem - I can obtain the value of the probability to within an error amount proportional to the max step size over the interval length, thus for small step sizes or large intervals the answer will be very accurate. Further, there is a reformulation in terms of an asymmetric 2d random walk on the integer lattice which may be helpful. Unfortunately my current credit is not enough to accept the problem. (1/2)
If you are happy with a partial answer, I would love to take the problem. Could you maybe lower the bounty slightly? My credit is $100, so I think I can take up to $115.
There is a whole body of literature for this problem. There are exact formulas for multiple particular cases. I don't know how thorough an answer to this problem should be.