Scaled Penalization of Brownian Motion with Drift and the Brownian Ascent

Panzo, Hugo

doi:10.1007/978-3-030-28535-7_12

Hugo Panzo¹⁹

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2252))

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Abstract

We study a scaled version of a two-parameter Brownian penalization model introduced by Roynette-Vallois-Yor (Period Math Hungar 50:247–280, 2005). The original model penalizes Brownian motion with drift $h\in \mathbb {R}$ by the weight process ${\big (\exp (\nu S_t):t\geq 0\big )}$ where $\nu \in \mathbb {R}$ and (S_t : t ≥ 0) is the running maximum of the Brownian motion. It was shown there that the resulting penalized process exhibits three distinct phases corresponding to different regions of the (ν, h)-plane. In this paper, we investigate the effect of penalizing the Brownian motion concurrently with scaling and identify the limit process. This extends an existing result for the ν < 0, h = 0 case to the whole parameter plane and reveals two additional “critical” phases occurring at the boundaries between the parameter regions. One of these novel phases is Brownian motion conditioned to end at its maximum, a process we call the Brownian ascent. We then relate the Brownian ascent to some well-known Brownian path fragments and to a random scaling transformation of Brownian motion that has attracted recent interest.

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Acknowledgements

The author would like to thank Iddo Ben-Ari for his helpful suggestions and encouragement and also Jim Pitman and Ju-Yi Yen for their tips on the history of the Brownian meander and co-meander as well as pointers to the literature.

Appendix 1: Normalized Brownian Excursion, Meander and Co-meander

The normalized Brownian excursion, meander and co-meander can be constructed from the excursion of Brownian motion which straddles time 1. In fact, this is usually how these processes are defined, see Chapter 7 in [35]. Define $g_1=\sup \{t<1:X_t=0\}$ as the last zero before time 1 and $d_1=\inf \{t>1:X_t=0\}$ as the first zero after time 1. Then the normalized excursion, meander and co-meander have representation

$$\displaystyle \begin{aligned} (\mathfrak{e}_s:0\leq s\leq 1)\stackrel{\mathcal{L}}{=}\left(\frac{|X_{g_1+s(d_1-g_1)}|}{\sqrt{d_1-g_1}}:0\leq s\leq 1\right), \end{aligned}$$

$$\displaystyle \begin{aligned}(\mathfrak{m}_s:0\leq s\leq 1)\stackrel{\mathcal{L}}{=}\left(\frac{|X_{g_1+s(1-g_1)}|}{\sqrt{1-g_1}}:0\leq s\leq 1\right)\end{aligned}$$

and

$$\displaystyle \begin{aligned}\left(\tilde{\mathfrak{m}}_s:0\leq s\leq 1\right)\stackrel{\mathcal{L}}{=}\left(\frac{|X_{d_1+s(1-d_1)}|}{\sqrt{d_1-1}}:0\leq s\leq 1\right),\end{aligned}$$

respectively, where the right-hand sides are under P₀.

The laws of the meander and co-meander are absolutely continuous with respect to each other and to the law of the Bessel(3) process starting at 0. More specifically, for any measurable $F:\mathcal {C}\big ([0,1];\mathbb {R}\big )\to \mathbb {R}_+$ we have

$$\displaystyle \begin{aligned} E[F(\mathfrak{m}_\bullet)]&=\sqrt{\frac{\pi}{2}}E_0\left[F(R_\bullet)\frac{1}{R_1}\right]{} \end{aligned} $$

(12.48)

$$\displaystyle \begin{aligned} E[F(\tilde{\mathfrak{m}}_\bullet)]&=E_0\left[F(R_\bullet)\frac{1}{R_1^2}\right]{} \end{aligned} $$

(12.49)

$$\displaystyle \begin{aligned} E[F(\mathfrak{m}_\bullet)]&=\sqrt{\frac{\pi}{2}}E\left[F(\tilde{\mathfrak{m}}_\bullet)\tilde{\mathfrak{m}}_1\right].{}\end{aligned} $$

(12.50)

The first of these relations (12.48) is known as Imhof’s relation [12, 28], while (12.49) appears as Theorem 7.4.1. in [35] and (12.50) follows from a combination of the previous two.

Appendix 2: Absolute Continuity Relations

Here we collect some useful absolute continuity relations between the laws of various processes. While the statements involve bounded measurable path functionals F, they are also valid for non-negative measurable F. The results given without proof can be found in the literature as indicated. The first two relations give us absolute continuity for Brownian motion and Bessel(3) processes starting at different points, as long as we are willing to ignore the initial [0, δ] segment of the path. See Definition 12.3 for notation that makes this precise.

Lemma 12.3

Let $x\in \mathbb {R}$, y > 0, and 0 < δ ≤ 1. Then for any bounded measurable $F:\mathcal {C}\big ([0,1];\mathbb {R}\big )\to \mathbb {R}$ we have

$$\displaystyle \begin{aligned}E_0\left[F_\delta(X_\bullet)\right]=E_x\left[F_\delta(X_\bullet)\exp\left(\frac{x^2-2xX_\delta}{2\delta}\right)\right]\end{aligned} $$

and

$$\displaystyle \begin{aligned}E_0\left[F_\delta(R_\bullet)\right]=E_y\left[F_\delta(R_\bullet)\frac{yR_\delta\exp\left(\frac{y^2}{2\delta}\right)}{\delta\sinh\left(\frac{yR_\delta}{\delta}\right)}\right].\end{aligned} $$

Proof

We only prove the first statement as the same argument applies to the second; see (12.51) and (12.52) for the Bessel(3) transition densities. First note that by the definition of F_δ and the Markov property we have for any $z\in \mathbb {R}$

$$\displaystyle \begin{aligned}E_0\left[F_\delta(X_\bullet)\middle|X_\delta=z\right]=E_x\left[F_\delta(X_\bullet)\middle|X_\delta=z\right].\end{aligned}$$

Now by conditioning on X_δ with p_δ(⋅, ⋅) denoting the transition density of Brownian motion at time δ, we can write

$$\displaystyle \begin{aligned} \begin{aligned} E_0\left[F_\delta(X_\bullet)\right]&=\int_{-\infty}^\infty E_0\left[F_\delta(X_\bullet)\middle|X_\delta =z\right]p_\delta(0,z)dz\\ &=\int_{-\infty}^\infty E_x\left[F_\delta(X_\bullet)\middle|X_\delta =z\right]\frac{p_\delta(0,z)}{p_\delta(x,z)}p_\delta(x,z)dz\\ &=\int_{-\infty}^\infty E_x\left[F_\delta(X_\bullet)\frac{p_\delta(0,X_\delta)}{p_\delta(x,X_\delta)}\middle|X_\delta =z\right]p_\delta(x,z)dz\\ &=E_x\left[F_\delta(X_\bullet)\frac{\exp\left(-\frac{X_\delta^2}{2\delta}\right)}{\exp\left(-\frac{(X_\delta-x)^2}{2\delta}\right)}\right]\\ &=E_x\left[F_\delta(X_\bullet)\exp\left(\frac{x^2-2xX_\delta}{2\delta}\right)\right]. \end{aligned} \end{aligned}$$

□

The next relation results from an h-transform of Brownian motion by the harmonic function h(x) = x. See Section 1.6 of [35].

Proposition 12.8

Let x > 0. Then for any bounded measurable $F:\mathcal {C}\big ([0,1];\mathbb {R}\big )\to \mathbb {R}$ we have

$$\displaystyle \begin{aligned}E_x[F(R_\bullet)]=E_x\left[F(X_\bullet)\frac{X_1}{x};I_1>0\right].\end{aligned}$$

The law of a Bessel(3) process run up to the last hitting time of x > 0 is, after rescaling, absolutely continuous with respect to the law of a Bessel(3) process run up to a fixed time. This is a special case of Théorème 3 in [4]; see also Theorem 8.1.1. in [35].

Theorem 12.8

Let γ_x be the last hitting time of x > 0 by the Bessel(3) process R. Then for any bounded measurable $F:\mathcal {C}\big ([0,1];\mathbb {R}\big )\to \mathbb {R}$ we have

$$\displaystyle \begin{aligned}E_0\left[F\left(\frac{R_{\bullet \gamma_x}}{\sqrt{\gamma_x}}\right)\right]=E_0\left[F(R_\bullet)\frac{1}{R_1^2}\right].\end{aligned}$$

Appendix 3: Path Decompositions

Denisov’s path decomposition [8] asserts that the pre and post-maximum parts of a Brownian path are rescaled independent Brownian meanders. See Corollary 17 in Chapter VIII of [1] for an extension to strictly stable Lévy processes.

Theorem 12.9 (Denisov)

Let Θ denote the almost surely unique time at which the Brownian motion W attains its maximum over the time interval [0, 1]. Then the transformed pre-maximum path

$$\displaystyle \begin{aligned}\left(\frac{W_\Theta-W_{\Theta-s\Theta}}{\sqrt{\Theta}}:0\leq s\leq 1\right)\end{aligned}$$

and the transformed post-maximum path

$$\displaystyle \begin{aligned}\left(\frac{W_\Theta-W_{\Theta+s(1-\Theta)}}{\sqrt{1-\Theta}}:0\leq s\leq 1\right)\end{aligned}$$

are independent Brownian meanders which are independent of Θ.

Williams’ path decomposition for Brownian motion with drift h < 0 splits the path at the time of the global maximum Θ_∞ by first picking an Exponential(−2h) distributed S_∞ and then running a Brownian motion with drift − h until it hits the level S_∞ for the pre-maximum path and then running Brownian motion with drift h conditioned remain below S_∞ for the post-maximum path. See Theorem 55.9 in Chapter VI of [24] for the following more precise statement.

Theorem 12.10 (Williams)

Suppose h < 0 and consider the following independent random elements:

1.
(X_t : t ≥ 0), a Brownian motion with drift − h starting at 0;
2.
(R_t : t ≥ 0), a Brownian motion with drift h starting at 0 conditioned to be non-positive for all time;
3.
and l, an Exponential(−2h) random variable.

Let $\tau _l=\inf \{t:X_t=l\}$ be the first hitting time of the level l by X. Then the process

$$\displaystyle \begin{aligned}\widetilde{X}_t = \left\{ \begin{array}{ll} X_t & : 0\leq t\leq \tau_l \\ \\ l+R_{t-\tau_l} & : \tau_l< t. \end{array} \right. \end{aligned}$$

is Brownian motion with drift h starting at 0.

Williams’ time reversal connects the laws of Brownian motion run until a first hitting time and a Bessel(3) process run until a last hitting time, see Theorem 49.1 in Chapter III of [23].

Theorem 12.11 (Williams)

Let $\tau _1=\inf \{t:X_t=0\}$ be the first hitting time of 1 by the Brownian motion X started at 0. Let $\gamma _1=\sup \{t:R_t=1\}$ be the last hitting time of 1 by the Bessel(3) process R started at 0. Then the following equality in law holds:

$$\displaystyle \begin{aligned}(1-X_{\tau_1-t}:0\leq t\leq\tau_1)\stackrel{\mathcal{L}}{=}(R_t:0\leq t\leq\gamma_1).\end{aligned}$$

Appendix 4: Density and Distribution Formulas

The Bessel(3) transition density formulas

$$\displaystyle \begin{aligned} p_t(0,y)=\sqrt{\frac{2}{\pi t^3}}y^2\exp\left(-\frac{y^2}{2t}\right)dy \end{aligned} $$

(12.51)

and

$$\displaystyle \begin{aligned} p_t(x,y)=\sqrt{\frac{2}{\pi t}}\frac{y}{x}\sinh\left(\frac{xy}{t}\right)\exp\left(-\frac{x^2+y^2}{2t}\right)dy, \end{aligned} $$

(12.52)

valid for y ≥ 0 and x, t > 0, can be found in Chapter XI of [22]. The density formula for the endpoint of a Brownian meander

$$\displaystyle \begin{aligned} P(\mathfrak{m}_1\in dy)=y\exp\left(-\frac{y^2}{2}\right)dy,~y\geq 0 \end{aligned} $$

(12.53)

follows from (12.51) together with the Imhof relation (12.48). The well-known arcsine law for the time of the maximum of Brownian motion states that

$$\displaystyle \begin{aligned} P_0(\Theta_1\in du)=\frac{1}{\pi\sqrt{u(1-u)}}du,~0<u<1. \end{aligned} $$

(12.54)

Using Denisov’s path decomposition Theorem 12.9, the densities (12.53) and (12.54) can be combined to yield the joint density

$$\displaystyle \begin{aligned} \begin{aligned} P_0&\left(S_1\in dx, S_1-X_1\in dy, \Theta_1\in du\right)\\ &=\frac{xy}{\pi\sqrt{u^3(1-u)^3}}\exp\left(-\frac{x^2}{2u}-\frac{y^2}{2(1-u)}\right)dxdydu \end{aligned} \end{aligned} $$

(12.55)

which holds for x, y ≥ 0 and 0 < u < 1.

The maximum of a Brownian bridge from 0 to a of length T > 0 has distribution

$$\displaystyle \begin{aligned} P_0\left(\sup_{0\leq s\leq T}X_s\geq b\middle|X_T=a\right)=\exp\left(-\frac{2b(b-a)}{T}\right) \end{aligned} $$

(12.56)

where $b\geq \max \{0,a\}$, see (4.3.40) in [13].

Appendix 5: Asymptotic Analysis Tools

The following versions of these standard results in asymptotic analysis can be found in [17] and [32], respectively.

Lemma 12.4 (Watson’s Lemma)

Let q(x) be a function of the positive real variable x, such that

$$\displaystyle \begin{aligned}q(x)\sim\sum_{n=0}^\infty a_n x^{\frac{n+\lambda-\mu}{\mu}}~\mathit{\text{as}}~x\to 0,\end{aligned}$$

where λ and μ are positive constants. Then

$$\displaystyle \begin{aligned}\int_0^\infty q(x)e^{-t x}dx\sim\sum_{n=0}^\infty \Gamma\left(\frac{n+\lambda}{\mu}\right)\frac{a_n}{t^{\frac{n+\lambda}{\mu}}}~\mathit{\text{as}}~t\to\infty.\end{aligned}$$

Theorem 12.12 (Laplace’s Method)

Define

$$\displaystyle \begin{aligned}I(t)=\int_a^b f(x)e^{-t h(x)}dx\end{aligned}$$

where −∞≤ a < b ≤∞ and t > 0. Assume that:

1.
h(x) has a unique minimum on [a, b] at point x = x₀ ∈ (a, b),
2.
h(x) and f(x) are continuously differentiable in a neighborhood of x₀ with f(x₀) ≠ 0 and
$$\displaystyle \begin{aligned}h(x)=h(x_0)+\frac{1}{2}h^{\prime\prime}(x_0)(x-x_0)^2+O\big((x-x_0)^3\big)~\mathit{\text{as}}~x\to x_0,\end{aligned}$$
3.
the integral I(t) exists for sufficiently large t.

Then

$$\displaystyle \begin{aligned}I(t)=f(x_0)\sqrt{\frac{2\pi}{t h^{\prime\prime}(x_0)}}e^{-t h(x_0)}\left(1+O\left(\frac{1}{\sqrt{t}}\right)\right)~\mathit{\text{as}}~t\to\infty.\end{aligned}$$

Appendix 6: Convergence Lemma

Here we give a Fatou-type lemma that helps streamline the proofs of the main theorems.

Lemma 12.5

Suppose {F_t}_t≥0, F, {X_t}_t≥0, and X are all integrable functions defined on the same measure space ( Ω, Σ, μ) such that {|F_t|}_t≥0 are bounded by M > 0, {X_t}_t≥0 are non-negative, $\int X_t d\mu \to \int X d\mu $, and both F_t → F and X_t → X μ-almost surely. Then we have

$$\displaystyle \begin{aligned}\lim_{t\to\infty}\int F_t X_t d\mu=\int FX d\mu.\end{aligned} $$

Proof

Notice that (M + F_t)X_t is non-negative for all t ≥ 0. So by Fatou’s lemma we have

$$\displaystyle \begin{aligned}M\int X d\mu+\liminf_{t\to\infty}\int F_t X_t d\mu=\liminf_{t\to\infty}\int (M+F_t)X_t d\mu\geq\int(M+F)X d\mu\end{aligned} $$

which implies

$$\displaystyle \begin{aligned}\liminf_{t\to\infty}\int F_t X_t d\mu\geq\int FX d\mu.\end{aligned} $$

Similarly, (M − F_t)X_t is non-negative for all t ≥ 0, hence

$$\displaystyle \begin{aligned}M\int X d\mu-\limsup_{t\to\infty}\int F_t X_t d\mu=\liminf_{t\to\infty}\int (M-F_t)X_t d\mu\geq\int (M-F)X d\mu\end{aligned} $$

which implies

$$\displaystyle \begin{aligned}\limsup_{t\to\infty}\int F_t X_t d\mu\leq\int FX d\mu.\end{aligned} $$

Together these inequalities imply that

$$\displaystyle \begin{aligned}\lim_{t\to\infty}\int F_t X_t d\mu=\int FX d\mu.\end{aligned}$$

□

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Department of Mathematics, University of Connecticut, Storrs, CT, USA
Hugo Panzo

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Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France
Catherine Donati-Martin
Institut Elie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France
Antoine Lejay
Laboratoire de Mathématiques de Versailles, Université de Versailles-St-Quentin, Versailles, France
Alain Rouault

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Panzo, H. (2019). Scaled Penalization of Brownian Motion with Drift and the Brownian Ascent. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités L. Lecture Notes in Mathematics(), vol 2252. Springer, Cham. https://doi.org/10.1007/978-3-030-28535-7_12

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