Makasu Journal of Inequalities and Applications (2015) 2015:348 DOI 10.1186/s13660-015-0873-3
RESEARCH
Open Access
On maximal inequalities via comparison principle Cloud Makasu* *
Correspondence:
[email protected] Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, Cape Town, South Africa
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: Brownian motion; comparison principle; nonlinear differential equations
1 Introduction Let B = (Bt )t≥ 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 E max |Bt | ≤ E[τ ], ≤t≤τ
()
√ where is the best possible constant. The one-sided maximal inequality () is proved in Dubins et al. []; see also []. Recently, using a similar method of proof to Dubins et al. [], the present author has proved the following weighted maximal inequality for a standard Brownian motion. Theorem Let B = (Bt )t≥ be a standard Brownian motion starting at x ≥ . Then the weighted maximal inequality Ex |Bτ | max |Bt | ≤ Ex [τ ] + x ≤t≤τ
()
holds for any stopping time τ for B satisfying Ex [τ ] < ∞. The constants and / in () 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 []. Lemma Assume that H(·) is a positive, continuous and differentiable function defined on [a, ∞) with a ≥ such that H(a) ≥ . Consider the following optimal stopping problem: u(x, y) := sup Ex,y Xτ H(Yτ ) – cτ (c > ),
()
τ
© 2015 Makasu. 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.
Makasu Journal of Inequalities and Applications (2015) 2015:348
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where the supremum is taken over all stopping times τ for B satisfying Ex,y [τ ] < ∞, Xt = |Bt |, and Yt = (max≤s≤t Xs ) ∨ y with ≤ x ≤ y given and fixed. Then: (a) the value function u(x, y) is of the form u(x, y) =
cx + xH(y) – cxg∗ (y) + c(g∗ (y)) , g∗ (y) < x ≤ y, xH(y), ≤ x ≤ g∗ (y),
(b) the optimal stopping time τ∗ is of the Azéma-Yor type given by
τ∗ = inf t > : Xt ≤ g∗ (Yt ) , where Xt = |Bt | and y → g∗ (y) is the maximal solution of the first-order nonlinear differential equation g (y) =
yH (y) c y – g(y)
()
under the condition < g(y) < y. Remark A characterization of optimal stopping boundaries y → 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 []. Remark For the existence of the maximal solution y → g∗ (y) of (), note that the righthand side of () is nondecreasing in g(·) provided that H (·) > . It must be noted that the maximal solution of () can only be given explicitly in a few special instances. In the case when H(y) = y defined on [, ∞), then
+ – y g∗ (y) = c
()
for < y < ∞ and c > . This leads to the proof of the weighted maximal inequality (). In the general case, the maximal solution y → g∗ (y) of () cannot be given explicitly. This general case is dealt with in this paper. As a result, we obtain a generalization of the one-sided, weighted, maximal inequality () and an extension of the well-known maximal inequality (). Throughout the paper, we shall assume the following: (A) H is a positive, continuous, and differentiable function defined on [a, ∞) with a ≥ such that H(a) ≥ and there exist constants K > and M ≥ such that yH (y) ≤ Ky + M holds for all (some) y > a.
()
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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 → g∗ (y) of (). This estimate is obtained via a comparison principle (see for instance [–] etc.), which is an important tool in the theory of ordinary differential equations. Assume that condition (A) holds. Consider the nonlinear equation (), and an auxiliary comparison nonlinear equation of the form h (y) =
Ky + M c y – h(y)
()
under the restriction < h(y) < y. Notice that () admits two positive solutions h (y) and h (y) such that the solutions satisfy h (y) ≤ h (y) for some < y < ∞, where
M K – – y– h (y) = c c( – –
() K ) c
is the minimal solution and
M K + – y– h (y) = c c( + –
() K ) c
is the maximal solution with c > K for K > and M ≥ . In the next section, using a comparison principle, a two-sided explicit estimate of the maximal solution y → g∗ (y) of () is given based on the maximal and minimal solutions of (). Our main emphasis is on the explicit estimates obtained. This permits the generalization of the weighted maximal inequality () 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 (Comparison principle) Assume that condition (A) holds. Then the following positive bounds hold for the maximal solution y → g∗ (y) of (): h (y) ≤ g∗ (y) ≤ h (y)
()
for some < y < ∞, where h (y) and h (y) are given by () and (), respectively. Our main result is stated and proved in the next theorem. Theorem Let B = (Bt )t≥ be a standard Brownian motion starting at x ≥ . Assume that condition (A) holds. Then the weighted maximal inequality K M Ex |Bτ |H max |Bt | ≤ KEx [τ ] + xH(x) + x + Mx + ≤t≤τ K holds for any stopping time τ for B satisfying Ex [τ ] < ∞.
()
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Proof Using Lemma and the two-sided estimate (), it follows that u(x, x) = cx + xH(x) – cxg∗ (x) + c g∗ (x)
K Mx ≤ cx + xH(x) – c – – x + c – –
M K + – x– +c c c( + – K )
K c
()
c
with c > K for K > and M ≥ . On the other hand, using the definition of the value function (), we have Ex,x |Bτ |H max |Bt | ≤ cEx,x [τ ] + u(x, x) ≤t≤τ
()
for any stopping time τ for B satisfying Ex,x [τ ] < ∞. Now using the upper bound in () in the right-hand side of the inequality () and passing to the limit as c ↓ K , we obtain the weighted maximal inequality (). This completes the proof of the assertion. Remark Observe that our main result contains Theorem as a special interesting case. This is in the case when H(y) = y, which is clearly a positive, continuous and differentiable function on [, ∞) with H() = and satisfying () with equality and K = and M = .
3 Illustrative example In this section, we shall now consider one example satisfying condition (A). Let H(y) = sin y for < y < π . Clearly, H is a positive, continuous and differentiable function for < y < π . In this case, we have |yH (y)| ≤ y satisfying () with K = and M = . Hence, the weighted maximal inequality () holds in the present example. Note added in proof For a standard Brownian motion B = (Bt )t≥ starting at zero, in this case we have M E |Bτ |H max |Bt | ≤ KE[τ ] + ≤t≤τ K
()
as a consequence of our main result. Now minimizing the right-hand side of the inequality () with respect to K > and assuming that M > , it follows that E |Bτ |H max |Bt | ≤ M E[τ ] ≤t≤τ
for any stopping time τ for B satisfying E[τ ] < ∞. Competing interests The author declares that he has no competing interests. Received: 14 May 2015 Accepted: 12 October 2015
()
Makasu Journal of Inequalities and Applications (2015) 2015:348
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