Monotonicity of the optimal stopping region for the duration problem

A Bernstein-basis proof of the stopping-region monotonicity

Source correction

There are two typographical indexing errors in the source formulas. They can be resolved from the probability model itself.

If there are n stages to go, including the present record of value x, its rank remains at most two at stage r+1 precisely when at most one of the first r future observations exceeds x. Consequently

$$\begin{aligned} U_n(x) &=\sum_{r=0}^{n-1}\bigl(x^r+r(1-x)x^{r-1}\bigr)\\ &=2\sum_{r=0}^{n-2}x^r-(n-2)x^{n-1}. \tag{1} \end{aligned}$$

The summand with r=0 in the first line is interpreted as 1. In particular, U_1=1 and U_2=2. Kurushima--Ano's preceding probability sum is exactly the first line of (1), but both that paper and the later survey print -n x^{n-1} in the simplified formula. That cannot be correct: it would give U_1=-1, and direct simplification of the preceding line gives -(n-2)x^{n-1}.

We use the definition

$$G_n(x)=U_n(x)-\sum_{k=1}^{n-1}x^{k-1} \int_x^1U_{n-k}(y)\,dy. \tag{2}$$

This is the expected advantage of stopping over one-stage look-ahead.

Power coefficients

Let H_m=1+1/2+...+1/m, with H_0=0. Since

$$\int_0^1U_m(y)\,dy=2H_m-1,$$

expanding (2) gives

$$\boxed{ G_n(x)=\sum_{j=0}^{n-1} \bigl(3-2H_{n-j-1}+2H_j\bigr)x^j-2n x^{n-1}.} \tag{3}$$

This is also the coefficient representation in Kurushima--Ano. Equivalently, the fully expanded formula is

$$\begin{aligned} G_n(x)={}&3\sum_{k=1}^{n}x^{k-1}-2nx^{n-1} -2\sum_{k=1}^{n-1}H_{n-k}x^{k-1} +2\sum_{k=1}^{n-1}H_kx^k. \tag{4} \end{aligned}$$

The displayed threshold equation in both sources has H_{n-k-1} where (4) requires H_{n-k}. Formula (3), the probability calculation, and the defining integral all agree with (4).

Bernstein coefficients

Put d=n-1 and write G_n in the degree-d Bernstein basis:

$$G_n(x)=\sum_{k=0}^{d}\beta_k {d\choose k}x^k(1-x)^{d-k}. \tag{5}$$

For k<d, put r=d-k. Then

$$\boxed{\displaystyle \beta_k=\frac{n}{r+1}\bigl(3-2H_r\bigr),\qquad \beta_d=n.} \tag{6}$$

To verify (6), if c_j=3-2H_{d-j}+2H_j denotes the power coefficient before the final correction, the power-to-Bernstein conversion is

$$\beta_k=\sum_{j=0}^{k}c_j \frac{{k\choose j}}{{d\choose j}}.$$

The two elementary identities

$$\sum_{j=0}^{k}\frac{{k\choose j}}{{d\choose j}} =\frac{d+1}{r+1},$$

and

$$\sum_{j=0}^{k}(H_{d-j}-H_j) \frac{{k\choose j}}{{d\choose j}} =H_r\frac{d+1}{r+1}$$

give the first part of (6). They follow on putting j=k-l from the hockey-stick identity and its harmonic convolution. Finally, beta_d=G_n(1)=U_n(1)=n.

One sign change

For r>=3, H_r>3/2, while H_2=3/2 and H_1=1. Thus, in increasing order of k, the nonzero Bernstein coefficients consist of

• negative coefficients through k=d-3;
• beta_{d-2}=0;
• beta_{d-1}=n/2>0 and beta_d=n>0.

There is exactly one sign variation when n>=4. Set t=x/(1-x). For 0<x<1, equation (5) becomes

$$G_n(x)=(1-x)^d \sum_{k=0}^{d}\beta_k {d\choose k}t^k. \tag{7}$$

The polynomial in t has one coefficient sign change, so Descartes' rule of signs gives at most one root in (0,infinity), hence G_n has at most one root in (0,1).

For n>=4,

$$G_n(0)=3-2H_{n-1}<0, \qquad G_n(1)=n>0.$$

Therefore G_n has exactly one root s_n in (0,1), is negative before s_n, and positive after it. The small cases are

$$G_1(x)=1,\qquad G_2(x)=1+x,\qquad G_3(x)=3x.$$

It follows for every n and every 0<=x<=y<=1 that

$$G_n(x)\geq0\quad\Longrightarrow\quad G_n(y)\geq0.$$

This proves the monotonicity assertion in MathDB #333222 and establishes the threshold form of the one-stage-look-ahead stopping region for the intended expected-duration model.

Verification record

• Kurushima--Ano primary PDF: https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1682-07.pdf
• 2016 survey: https://arxiv.org/html/1605.08364
• MathDB: https://mathdb.com/p/333222
• Exact coefficient checks: verify_bernstein.py
• Dependency and source audit: proof-skeleton.md

No indexed published resolution of this exact monotonicity assertion was located through 2026-08-18. This is an internally verified proof, not yet a claim of external peer review or publication.