On maximal inequalities via comparison principle
- Cloud Makasu^{1}Email author
https://doi.org/10.1186/s13660-015-0873-3
© Makasu 2015
Received: 14 May 2015
Accepted: 12 October 2015
Published: 2 November 2015
Abstract
Under certain conditions, we prove a new class of one-sided, weighted, maximal inequalities for a standard Brownian motion. Our method of proof is mainly based on a comparison principle for solutions of a system of nonlinear first-order differential equations.
Keywords
1 Introduction
The one-sided maximal inequality (1) is proved in Dubins et al. [1]; see also [2]. Recently, using a similar method of proof to Dubins et al. [2], the present author has proved the following weighted maximal inequality for a standard Brownian motion.
Theorem 1
The proof of the above theorem is essentially based on the next lemma, which can easily be proved using similar arguments to [2].
Lemma 2
- (a)the value function \(u(x,y)\) is of the form$$\begin{aligned} u(x,y)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} cx^{2}+xH(y)-2cxg_{*}(y)+c (g_{*}(y) )^{2},& g_{*}(y)< x\leq y,\\ xH(y),& 0\leq x\leq g_{*}(y), \end{array}\displaystyle \right . \end{aligned}$$
- (b)the optimal stopping time \(\tau_{*}\) is of the Azéma-Yor type given bywhere \(X_{t}=|B_{t}|\) and \(y\mapsto g_{*}(y)\) is the maximal solution of the first-order nonlinear differential equation$$\begin{aligned} \tau_{*}=\inf \bigl\{ t>0: X_{t}\leq g_{*} (Y_{t})\bigr\} , \end{aligned}$$under the condition \(0< g(y)< y\).$$\begin{aligned} g^{\prime}(y)=\frac{1}{2c} \biggl(\frac{yH^{\prime}(y)}{y-g(y)} \biggr) \end{aligned}$$(4)
Remark 3
A characterization of optimal stopping boundaries \(y\mapsto g_{*}(y)\) in terms of the maximal solution of a nonlinear first-order differential equation for a general class of diffusion processes is given in Peskir [3].
Remark 4
For the existence of the maximal solution \(y\mapsto g_{*}(y)\) of (4), note that the right-hand side of (4) is nondecreasing in \(g(\cdot)\) provided that \(H^{\prime}(\cdot)>0\).
- (A)H is a positive, continuous, and differentiable function defined on \([a,\infty)\) with \(a\geq0\) such that \(H(a)\geq0\) and there exist constants \(K>0\) and \(M\geq0\) such thatholds for all (some) \(y>a\).$$ \bigl\vert yH^{\prime}(y)\bigr\vert \leq Ky+M $$(6)
The condition (A) plays a vital role in this paper. It is useful in establishing a two-sided, explicit estimate of the maximal solution \(y\mapsto g_{*}(y)\) of (4). This estimate is obtained via a comparison principle (see for instance [4–7] etc.), which is an important tool in the theory of ordinary differential equations.
In the next section, using a comparison principle, a two-sided explicit estimate of the maximal solution \(y\mapsto g_{*}(y)\) of (4) is given based on the maximal and minimal solutions of (7). Our main emphasis is on the explicit estimates obtained. This permits the generalization of the weighted maximal inequality (2) to those with explicit constants.
2 Main results
The next lemma will play an important role in the proof of our main result, which follows.
Lemma 5
(Comparison principle)
Our main result is stated and proved in the next theorem.
Theorem 6
Proof
Now using the upper bound in (12) in the right-hand side of the inequality (13) and passing to the limit as \(c\downarrow2K\), we obtain the weighted maximal inequality (11). This completes the proof of the assertion. □
3 Illustrative example
In this section, we shall now consider one example satisfying condition (A). Let \(H(y)=\sin y\) for \(0< y<\pi\). Clearly, H is a positive, continuous and differentiable function for \(0< y<\pi\). In this case, we have \(\vert yH^{\prime}(y)\vert \leq y\) satisfying (6) with \(K=1\) and \(M=0\). Hence, the weighted maximal inequality (11) holds in the present example.
Note added in proof
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
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