Exercising control when confronted by a (Brownian) spider: part II

Philip A. Ernst ()
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Philip A. Ernst: Imperial College London, South Kensington Campus

Queueing Systems: Theory and Applications, 2025, vol. 109, issue 2, No 4, 20 pages

Abstract: Abstract This paper is concerned with the Brownian “spider process,” also known as Walsh Brownian motion, as first introduced in the epilogue of Walsh (Astérisque 52:37–45, 1978). We revisit a longstanding problem due to L.E. Dubins, stated as follows: find $$C_n$$ C n for all integer n in the inequality $$\begin{aligned} \mathbb {E}\left[ S_1(\tau )+S_2(\tau )+\cdots +S_n(\tau )\right] \le C_n \sqrt{\mathbb {E}\left[ \tau \right] }, \end{aligned}$$ E S 1 ( τ ) + S 2 ( τ ) + ⋯ + S n ( τ ) ≤ C n E τ , where there are n spider arms and $$S_i(\tau )$$ S i ( τ ) is the supremum of reflected Brownian motion on rib i up to the stopping time $$\tau $$ τ . No generalization beyond two arms has been solved for Dubins’ problem. This article considers the case $$n=3$$ n = 3 , revealing its considerable complexity. Letting $$s_1$$ s 1 , $$s_2$$ s 2 , and $$s_3$$ s 3 denote the distances that have already been covered on each of the respective ribs at time 0, we provide optimal reward functions for three different regions $$(s_1,s_2,s_3)$$ ( s 1 , s 2 , s 3 ) of the state space.

Keywords: Spider process; Walsh Brownian motion; Optimal stopping; Dynamic programming; Martingale inequalities; Primary 60G40; 90C39; Secondary 60G44 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s11134-025-09942-5

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