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On maximal inequalities via comparison principle
Journal of Inequalities and Applications volume 2015, Article number: 348 (2015)
Abstract
Under certain conditions, we prove a new class of onesided, 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 firstorder differential equations.
Introduction
Let \(B=(B_{t})_{t\geq0}\) be a standard Brownian motion starting at zero, and let τ be any stopping time for B with finite expectation. It is well known that
where \(\sqrt{2}\) is the best possible constant.
The onesided 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
Let \(B=(B_{t})_{t\geq0}\) be a standard Brownian motion starting at \(x\geq0\). Then the weighted maximal inequality
holds for any stopping time τ for B satisfying \(\mathbf{E}_{x}[\tau ]<\infty\). The constants 2 and \(3/2\) in (2) are the best possible.
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
Assume that \(H(\cdot)\) is a positive, continuous and differentiable function defined on \([a,\infty)\) with \(a\geq0\) such that \(H(a)\geq0\). Consider the following optimal stopping problem:
where the supremum is taken over all stopping times τ for B satisfying \(\mathbf{E}_{x,y}[\tau]<\infty\), \(X_{t}=B_{t}\), and \(Y_{t}= (\max_{0\leq s\leq t} X_{s} )\vee y\) with \(0\leq x\leq y\) given and fixed.
Then:

(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 AzémaYor type given by
$$\begin{aligned} \tau_{*}=\inf \bigl\{ t>0: X_{t}\leq g_{*} (Y_{t})\bigr\} , \end{aligned}$$where \(X_{t}=B_{t}\) and \(y\mapsto g_{*}(y)\) is the maximal solution of the firstorder nonlinear differential equation
$$\begin{aligned} g^{\prime}(y)=\frac{1}{2c} \biggl(\frac{yH^{\prime}(y)}{yg(y)} \biggr) \end{aligned}$$(4)under the condition \(0< g(y)< y\).
Remark 3
A characterization of optimal stopping boundaries \(y\mapsto g_{*}(y)\) in terms of the maximal solution of a nonlinear firstorder 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 righthand side of (4) is nondecreasing in \(g(\cdot)\) provided that \(H^{\prime}(\cdot)>0\).
It must be noted that the maximal solution of (4) can only be given explicitly in a few special instances. In the case when \(H(y)=y\) defined on \([0,\infty)\), then
for \(0< y<\infty\) and \(c>2\). This leads to the proof of the weighted maximal inequality (2). In the general case, the maximal solution \(y\mapsto g_{*}(y)\) of (4) cannot be given explicitly. This general case is dealt with in this paper. As a result, we obtain a generalization of the onesided, weighted, maximal inequality (2) and an extension of the wellknown maximal inequality (1).
Throughout the paper, we shall assume the following:

(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 that
$$ \bigl\vert yH^{\prime}(y)\bigr\vert \leq Ky+M $$(6)holds for all (some) \(y>a\).
The condition (A) plays a vital role in this paper. It is useful in establishing a twosided, 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.
Assume that condition (A) holds. Consider the nonlinear equation (4), and an auxiliary comparison nonlinear equation of the form
under the restriction \(0< h(y)< y\).
Notice that (7) admits two positive solutions \(h_{1}(y)\) and \(h_{2}(y)\) such that the solutions satisfy \(h_{1}(y)\leq h_{2}(y)\) for some \(0< y<\infty\), where
is the minimal solution and
is the maximal solution with \(c>2K\) for \(K>0\) and \(M\geq0\).
In the next section, using a comparison principle, a twosided 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.
Main results
The next lemma will play an important role in the proof of our main result, which follows.
Lemma 5
(Comparison principle)
Assume that condition (A) holds. Then the following positive bounds hold for the maximal solution \(y\mapsto g_{*}(y)\) of (4):
for some \(0< y<\infty\), where \(h_{1}(y)\) and \(h_{2}(y)\) are given by (8) and (9), respectively.
Our main result is stated and proved in the next theorem.
Theorem 6
Let \(B=(B_{t})_{t\geq0}\) be a standard Brownian motion starting at \(x\geq0\). Assume that condition (A) holds. Then the weighted maximal inequality
holds for any stopping time τ for B satisfying \(\mathbf{E}_{x}[\tau ]<\infty\).
Proof
Using Lemma 2 and the twosided estimate (10), it follows that
with \(c>2K\) for \(K>0\) and \(M\geq0\).
On the other hand, using the definition of the value function (3), we have
for any stopping time τ for B satisfying \(\mathbf{E}_{x,x}[\tau ]<\infty\).
Now using the upper bound in (12) in the righthand 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. □
Remark 7
Observe that our main result contains Theorem 1 as a special interesting case. This is in the case when \(H(y)=y\), which is clearly a positive, continuous and differentiable function on \([0,\infty)\) with \(H(0)=0\) and satisfying (6) with equality and \(K=1\) and \(M=0\).
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
For a standard Brownian motion \(B=(B_{t})_{t\geq0}\) starting at zero, in this case we have
as a consequence of our main result. Now minimizing the righthand side of the inequality (14) with respect to \(K>0\) and assuming that \(M>0\), it follows that
for any stopping time τ for B satisfying \(\mathbf{E}[\tau]<\infty\).
References
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Makasu, C. On maximal inequalities via comparison principle. J Inequal Appl 2015, 348 (2015). https://doi.org/10.1186/s1366001508733
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DOI: https://doi.org/10.1186/s1366001508733
Keywords
 Brownian motion
 comparison principle
 nonlinear differential equations