The arctangent law for a certain random time related to a one-dimensional diffusion

TL;DR

This paper generalizes a known result for Brownian motion to a broader class of one-dimensional diffusion processes, analyzing the distribution of the first time after a given point when the process exceeds its previous maximum.

Contribution

It extends the arctangent law for the first exceeding time from Brownian motion to general one-dimensional diffusions.

Findings

Derived the distribution of the first exceeding time for general diffusions.

Established the arctangent law in a broader diffusion context.

Generalized Papanicolaou's result beyond Brownian motion.

Abstract

For a time-homogeneous, one-dimensional diffusion process $X(t),$ we investigate the distribution of the first instant, after a given time $r,$ at which $X(t)$ exceeds its maximum on the interval $[0,r],$ generalizing a result of Papanicolaou, which is valid for Brownian motion.

Equations

$$dX(t)=\mu (X(t))dt+\sigma (X(t))d{B}_t,X(0)=\eta \in I,$$

$$S=S(r):=\inf \{t\ge r:X(t)\ge M_r\}-r.$$

$$P\{S_B(r)\le t\}=\frac{2}{\pi }\arctan \left(\sqrt{\frac{t}{r}}\right),t\ge 0,$$

$$f_{S_B(r)}(t)=\frac{\sqrt{r}}{\pi (r+t)\sqrt{t}},t\ge 0.$$

$$\{\begin{array}{l}Lw(x)=0,x\in I\\ w(0)=0,w^{\prime }(0)=1,\end{array}$$

$$Lh=\frac{1}{2}{\sigma }^2(x)\frac{d^{2}h}{d{x}^2}+\mu (x)\frac{dh}{dx},h\in C^2(I).$$

$$w(x)={\int }_0^x\exp \left(-{\int }_0^t\frac{2\mu (z)}{{\sigma }^2(z)}dz\right)dt.$$

$$\zeta (t)=w(\eta )+{\int }_0^t{w}^{\prime }(w^{-1}(\zeta (s)))\sigma (w^{-1}(\zeta (s)))d{B}_s,$$

$$\rho (t)\doteq ⟨\zeta {⟩}_t={\int }_0^t[w^{\prime }(X(s))\sigma (X(s)){]}^{2}ds,t\ge 0.$$

$$X(t)=w^{-1}(\hat{B}(\rho (t))+w(\eta )).$$

$$dX(t)=\frac{1}{2}\sigma (X(t)){\sigma }^{\prime }(X(t))dt+\sigma (X(t))d{B}_t,X(0)=x,$$

$$P\{S(r)\le t\}=\frac{2}{\pi }\arctan \left(\sqrt{\frac{\rho (t+r)-\rho (r)}{\rho (r)}}\right),t\ge 0,$$

$$f_{S(r)}(t)=\frac{{\rho }^{\prime }(t+r)\sqrt{\rho (r)}}{\rho (t+r)\sqrt{\rho (t+r)-\rho (r)}}.$$

$$\alpha (t)\le \rho (t)\le \beta (t),\forall t\ge 0.$$

$$P\{S(r)\le t\}\le \frac{2}{\pi }\arctan \left(\sqrt{\frac{\beta (t+r)-\alpha (r)}{\alpha (r)}}\right),t\ge 0.$$

$$P\{S(r)\le t\}\ge \frac{2}{\pi }\arctan \left(\sqrt{\frac{\alpha (t+r)-\beta (r)}{\beta (r)}}\right),t>\bar{t}.$$

$$M_r=\underset{t\in [0,r]}{\max }X(t)=\underset{t\in [0,r]}{\max }{w}^{-1}\left(\hat{B}(\rho (t))+w(\eta )\right)=w^{-1}\left(\underset{t\in [0,\rho (r)]}{\max }{\hat{B}}_t+w(\eta )\right),$$

$$\tau_{r}:=\inf\Big{\{}t\geq r:X(t)\geq M_{r}\Big{\}}=\inf\Big{\{}t\geq r:\widehat{B}(\rho(t))\geq\max_{u\in[0,\rho(r)]}\widehat{B}_{u}\Big{\}}.$$

$$\rho(\tau_{r})=\inf\Big{\{}s\geq\rho(r):\widehat{B}_{s}\geq M^{\widehat{B}}_{\rho(r)}\Big{\}}=S_{\widehat{B}}(\rho(r))+\rho(r),$$

$$P\{S(r)\le t\}=P\left\{{\rho }^{-1}(S_{\hat{B}}(\rho (r))+\rho (r))\le t+r\right\}$$

$$=P\left\{S_{\hat{B}}(\rho (r))+\rho (r)\le \rho (t+r)\right\}=P\left\{S_{\hat{B}}(\rho (r))\le \rho (t+r)-\rho (r)\right\},$$

$${\tau }_{r,\alpha }=\inf \{t\ge r:\hat{B}(\rho (t))\ge \underset{u\in [0,\alpha (r)]}{\max }{\hat{B}}_u\},{\tau }_{r,\beta }=\inf \{t\ge r:\hat{B}(\rho (t))\ge \underset{u\in [0,\beta (r)]}{\max }{\hat{B}}_u\}.$$

$${\tau }_{r,\alpha }\le {\tau }_r\le {\tau }_{r,\beta },$$

$$\rho ({\tau }_{r,\alpha })\le \rho ({\tau }_r)\le \rho ({\tau }_{r,\beta }).$$

$$\inf \{s>\alpha (r):{\hat{B}}_s\ge \underset{u\in [0,\alpha (r)]}{\max }{\hat{B}}_u\}\le \rho ({\tau }_{r,\alpha })$$

$$\rho ({\tau }_{r,\beta })\le \inf \{s>\beta (r):{\hat{B}}_s\ge \underset{u\in [0,\beta (r)]}{\max }{\hat{B}}_u\}.$$

$$S_{\hat{B}}(\alpha (r))+\alpha (r)\le \rho ({\tau }_r)\le S_{\hat{B}}(\beta (r))+\beta (r),$$

$${\rho }^{-1}(S_{\hat{B}}(\alpha (r))+\alpha (r))\le {\tau }_r\le {\rho }^{-1}(S_{\hat{B}}(\beta (r))+\beta (r)),$$

$${\beta }^{-1}(S_{\hat{B}}(\alpha (r))+\alpha (r))\le {\tau }_r\le {\alpha }^{-1}(S_{\hat{B}}(\beta (r))+\beta (r)).$$

$${\beta }^{-1}(S_{\hat{B}}(\alpha (r))+\alpha (r))-r\le S(r)\le {\alpha }^{-1}(S_{\hat{B}}(\beta (r))+\beta (r))-r.$$

Full text

Keywords: One-dimensional diffusion, First-passage time, Maximum value on an interval

Mathematics Subject Classification: 60J60, 60H05, 60H10.

1 Introduction

In this paper, we extend to a one-dimensional diffusion process $X(t)$ the result of (Papanicolaou, 2016) for Brownian motion, concerning the arctangent law for a certain random time.

Indeed, let be $X(t)$ a time-homogeneous, one-dimensional diffusion in the interval $I\subset\mathbb{R}$ which is the solution of the SDE:

[TABLE]

where $B_{t}$ (with $B_{0}=0)$ is standard Brownian motion (BM) and the drift and diffusion coefficients satisfy the usual conditions (see e.g. (Ikeda and Watanabe, 1981)) for existence and uniqueness of the solution of (1.1).

For a fixed time $r>0,$ we consider the maximum $M_{r}:=\max_{0\leq t\leq r}X(t)$ of the diffusion $X$ on the interval $[0,r],$ and we denote by $S$ the following random time:

[TABLE]

Assuming that the initial state $\eta$ is random, our aim is to study the distribution function of $S,$ generalizing the result of (Papanicolaou, 2016), that refers to the case when $X(t)=B^{\eta}_{t}:=\eta+B_{t}$ (i.e. BM starting from the random value $\eta,$ not necessarily zero), and states that:

[TABLE]

where $S_{B}(r)=\inf\{t\geq r:B^{\eta}_{t}\geq\max_{s\in[0,r]}B^{\eta}_{s}\}-r.$ By taking the derivative with respect to $t,$ one obtains the probability density of $S_{B}(r):$

[TABLE]

Notice that the expectation, $E(S_{B}(r)),$ turns out to be infinite.

The knowledge of the distribution of $S$ is relevant in various diffusion models used in applied sciences, such as Mathematical Finance, Biology, Physics, Hydraulics, etc., whenever the time evolution of the phenomenon under study is described by a diffusion $X(t);$ in fact, one is often interested to find the first instant, after a given time $r,$ at which $X(t)$ exceeds the maximum value attained in the time interval $[0,r],$ namely in times prior to $r.$ For instance, in the Economy framework, if we let $r$ vary in $(0,+\infty),$ the process $S(r),$ so obtained, is related to the drawdown process, which measures the fall in value of $X(t)$ from its running maxima, and is frequently used as performance indicator in the fund management industry (see e.g. (Dassios and Lim, 2017) and references therein). Indeed, $S(r)$ can be expressed in terms of the time elapsed since the last time the maximum is achieved, that was studied in (Dassios and Lim, 2017).

2 The result

Let $w(x)\in C^{2}(I)$ be the scale function associated to the diffusion $X(t)$ driven by the SDE (1.1), that is, the solution of:

[TABLE]

where $L$ is the infinitesimal generator of $X$ defined by:

[TABLE]

Actually, the scale function can be taken as any function $\widetilde{w}=aw+b,$ with $a>0$ and $b\in\mathbb{R}$ (see e.g. (Karlin and Taylor, 1975)); we chose the initial conditions of (2.1), for the sake of simplicity.

As easily seen, if the integral $\int_{0}^{t}\frac{2\mu(z)}{\sigma^{2}(z)}\ dz$ converges, the problem (2.1) has solution:

[TABLE]

If $\zeta(t):=w(X(t)),$ by It${\rm\hat{o}}$’s formula one obtains

[TABLE]

that is, the process $\zeta(t)$ is a local martingale, whose quadratic variation is

[TABLE]

The (random) function $\rho(t)$ is differentiable, increasing, and $\rho(0)=0.$ If $\rho(+\infty)=+\infty,$ it can be shown (see e.g. (Revuz and Yor, 1991)) that there exists a BM $\widehat{B}$ such that $\zeta(t)=\widehat{B}(\rho(t))+w(\eta);$ thus, since $w$ is invertible, the solution $X(t)$ to (1.1) can be written in the form

[TABLE]

In this way, $X$ is obtained from BM by a space transformation and a random time-change (see e.g. the discussion in (Abundo, 2012)).

Definition 2.1

We say that the diffusion $X(t)$ (with $X(0)=x)$ is *conjugated * to BM (see also (Abundo, 2012)), if there exists an increasing differentiable function $v(x)$ with $v(0)=0,$ such that $X(t)=v^{-1}\left(B_{t}+v(x)\right),$ for any $t\geq 0.$ **

Remark 2.2

Diffusions conjugated to BM are special cases of (2.6), for $\rho(t)=t,\ \widehat{B}_{t}=B_{t},$ and $w=v$ (however, it is not required that $v^{\prime}(0)=1)$. **

A class of diffusions conjugated to BM is given by processes $X(t)$ which are solutions of SDEs such as:

[TABLE]

with $\sigma(\cdot)\geq 0.$ Indeed, if the integral $v(x):=\int^{x}\frac{1}{\sigma(r)}dr$ is convergent, by It${\rm\hat{o}}$’s formula, one obtains $X(t)=v^{-1}(B_{t}+v(x)).$

Explicit examples of diffusions conjugated to BM (see also (Abundo, 2012)) are:

$\bullet$ the diffusion in $I=\rm I\!R$ driven by the SDE $dX(t)=\frac{1}{3}X(t)^{1/3}dt+X(t)^{2/3}dB_{t},\ X(0)=x,$ which is conjugated to BM via the the function $v(x)=3x^{1/3},$ that is, $X(t)=(x^{1/3}+\frac{1}{3}B_{t})^{3};$

$\bullet$ the diffusion in $I=[0,+\infty)$ driven by the SDE $dX(t)=\frac{1}{4}dt+\sqrt{X(t)\vee 0}\ dB_{t},\ X(0)=x\geq 0$ (Feller process), which is conjugated to BM via the the function $v(x)=2\sqrt{x},$ that is, $X(t)=\frac{1}{4}(B_{t}+2\sqrt{x})^{2};$

$\bullet$ the diffusion in $I=[0,1]$ driven by the SDE

$dX(t)=(\frac{1}{4}-\frac{1}{2}X(t))dt+\sqrt{X(t)(1-X(t))\vee 0}\ dB_{t},\ X(0)=x\in[0,1]$ (Wright-Fisher like process), which is conjugated to BM via the the function $v(x)=2\arcsin\sqrt{x},$ that is, $X(t)=\sin^{2}(B_{t}/2+\arcsin\sqrt{x}).$

If we drop the requirement that $v(0)=0$ in Definition 2.1, then, for $\sigma>0,$ the diffusion in $I=(0,+\infty)$ driven by the SDE $dX(t)=\frac{\sigma^{2}}{2}X(t)dt+\sigma X(t)dB_{t},\ X(0)=x>0$ (a special case of geometric BM) is conjugated to BM via the function $v(x)=\frac{\ln x}{\sigma},$ that is, $X(t)=\exp(\sigma B_{t}+\ln x).$

The class of processes $X$ given by (2.6) with $\rho(t)$ deterministic, includes, besides diffusions conjugated to BM, the integral of Gauss-Markov processes (see (Abundo, 2015), (Abundo, 2013)), e.g. integrated BM  $X(t)=\int_{0}^{t}B_{s}ds,$ represented by $X(t)=\widehat{B}(\rho(t))$ with $\rho(t)=t^{3}/3,$ and integrated Ornstein-Uhlenbeck (OU) process $X(t)=\int_{0}^{t}Y(s)ds,$ where $Y(t)$ is OU process (see (Abundo, 2013) for the explicit representation of $X$ in the form (2.6)).

The announced result is:

Theorem 2.3

Let $X(t)$ be the solution of the SDE (1.1) and suppose that the scale function $w$ of $X,$ given by (2.1) exists; for fixed $r>0,$ let $S(r)$ be the random time defined by (1.2). With the previous notations, suppose that the function $\rho$ satisfies the condition $\rho(+\infty)=+\infty;$

(i) if $\rho(t)$ is deterministic, then the probability distribution of $S(r)$ is:

[TABLE]

and its density is:

[TABLE]

(ii) If $\rho$ is not deterministic, let us suppose that there exist two deterministic, continuous increasing functions $\alpha(t)$ and $\beta(t),$ with $\alpha(0)=\beta(0)=0,$ such that

[TABLE]

Then:

[TABLE]

Moreover, if there exists $\bar{t}>0$ such that $\alpha(r+\bar{t})\geq\beta(r),$ then:

[TABLE]

Proof.  Under the hypothesis, the representation (2.6) of $X$ holds.

(i) Suppose that $\rho(t)$ is deterministic. Then:

[TABLE]

and so:

[TABLE]

Thus, by recalling the definition of $S_{B}(r),$ and taking $\widehat{B}$ in place of $B$ and $\rho(r)$ in place of $r,$ we get:

[TABLE]

where $M^{\widehat{B}}_{r}=\max_{u\in[0,r]}\widehat{B}_{u}\ .$ Therefore, $\tau_{r}=\rho^{-1}\left(S_{\widehat{B}}(\rho(r))+\rho(r)\right),$ and $S(r)=\tau_{r}-r=\rho^{-1}\left(S_{\widehat{B}}(\rho(r))+\rho(r)\right)-r.$ Finally:

[TABLE]

[TABLE]

from which (2.8) follows, by using (1.3); formula (2.9) is obtained by taking the derivative with respect to $t.$

(ii) Suppose that $\rho(t)$ is not deterministic, and the bounds (2.10) hold; set:

[TABLE]

As easily seen, one has:

[TABLE]

that implies:

[TABLE]

Moreover, since $\rho(\tau_{r,\alpha})=\inf\{s>\rho(r):\widehat{B}_{s}\geq\max_{u\in[0,\alpha(r)]}\widehat{B}_{u}\},$ and $\rho(\tau_{r,\beta})=\inf\{s>\rho(r):\widehat{B}_{s}\geq\max_{u\in[0,\beta(r)]}\widehat{B}_{u}\},$ we have :

[TABLE]

and

[TABLE]

Thus, recalling the definition of $S_{\widehat{B}}(r),$ from (2.17) we get:

[TABLE]

and so:

[TABLE]

where $\rho^{-1}$ is the “inverse” of the random function $\rho,$ which is defined by $\rho^{-1}(s):=\inf\{t>0:\rho(t)>s\}.$ Since (2.10) implies that $\beta^{-1}(s)\leq\rho^{-1}(s)\leq\alpha^{-1}(s),$ we obtain:

[TABLE]

Therefore:

[TABLE]

From the first inequality in (2.23), it follows that:

[TABLE]

that proves (2.11). Moreover, if there exists $\bar{t}>0$ such that $\alpha(r+\bar{t})\geq\beta(r),$ we have $\alpha(r+t)\geq\beta(r)$ for $t>\bar{t},$ because $\alpha(t)$ is increasing; then, for $t>\bar{t}$ the second inequality in (2.23) implies:

[TABLE]

which proves (2.12). The condition $\alpha(t+r)\geq\beta(r)$ is necessary so that $S_{\widehat{B}}(\beta(r))\geq 0;$ of course, if a value $\bar{t}>0,$ such that $\alpha(r+\bar{t})\geq\beta(r),$ does not exist, the inequality (2.12) loses meaning, because the square root is not defined. $\Box$

Remark 2.4

Notice that (2.8) is independent of the scale function $w;$ in particular, if $X$ is conjugated to BM via the function $v,$ being $\rho(t)=t,$ one obtains that the distribution of $S(r)$ is the same as that of $S_{B}(r),$ given by (1.3).

For an example of diffusion $X$ for which $\rho$ is not deterministic, but satisfies the bounds (2.10) with $\alpha(t)$ close to $\beta(t),$ see Example 4 of (Abundo, 2012). **

Remark 2.5

Let us suppose that $\rho(t)$ is deterministic and there exists $\gamma>0$ such that $\rho(t)\sim const\cdot t^{\gamma},$ as $t\rightarrow+\infty;$ then, from (2.9) it easily follows that $E(S(r))<+\infty,$ provided that $\gamma>2.$ This is not the case of BM, because $\rho(t)=t;$ instead, it holds e.g. for integrated BM  $X(t)=\int_{0}^{t}B_{s}ds,$ being $X(t)=\widehat{B}(\rho(t))$ with $\rho(t)=t^{3}/3$ (see (Abundo, 2015), (Abundo, 2013)). **

Remark 2.6

Let us suppose that $\rho(t)$ is deterministic; for given $r_{1},r_{2}$ with $0\leq r_{1}<r_{2},$ set $M_{[r_{1},r_{2}]}:=\max_{t\in[r_{1},r_{2}]}X(t),$ and $S(r_{1},r_{2})=\inf\{t\geq r_{2}:X(t)=M_{[r_{1},r_{2}]}\}-r_{2}.$ Then, by using the arguments of the proof of Theorem 2.3 and the Remark of (Papanicolaou, 2016) which refers to BM, one obtains:

[TABLE]

Remark 2.7

Let us suppose that $\rho(t)$ is deterministic, and for $r>0,$ define $L(r)=\min_{t\in[0,r]}X(t),$ and $U(r)=\inf\{t\geq r:X(t)\geq L(r)\}-r.$ Then, by using the fact, proved in (Papanicolaou, 2016), that $S_{B}(r)$ and $U_{B}(r):=\inf\{t\geq r:B^{\eta}_{t}\geq L_{B}(r)\}-r$ have the same distribution (here $L_{B}(r)=\min_{t\in[0,r]}B^{\eta}_{t}),$ and by arguments analogous to those in the proof of Theorem 2.3, we conclude that also $U(r)$ and $S(r)$ have a common distribution.

3 Conclusions and final remarks

We have considered a time-homogeneous one-dimensional diffusion process $X$ in an interval $I\subset\mathbb{R},$ driven by the SDE (1.1); for a given time $r,$ under suitable conditions, we have found the distribution of the time $S(r)$ required (after $r)$ for $X(t)$ to exceed its maximum $M_{r}=\max_{t\in[0,r]}X(t)$ on the interval $[0,r],$ generalizing the result in (Papanicolaou, 2016), which refers to BM. Indeed, we have reduced $X$ to BM (see (2.6)) by a space transformation, given by the scale function $w(x),$ and a random time-change $\rho(t),$ under the assumption that $\rho(+\infty)=+\infty$ ($\rho(t)$ is the quadratic variation of the space-transformed process). Thus, we have shown that, when $\rho(t)$ is deterministic, $S(r)$ follows a compound arctangent law; note that, in this case $X(t)$ solves also the SDE:

[TABLE]

where $w^{\prime}(x)$ and $w^{\prime\prime}(x)$ denote first and second derivative of $w(x),$ and $\widetilde{B}$ is BM (see also (Abundo, 2017)). The class of processes $X,$ for which the distribution of $S(r)$ has been found, includes, besides diffusions conjugated to BM (see e.g. (Abundo, 2012)), integrated BM and integrated Ornstein-Uhlenbeck process (see (Abundo, 2015), (Abundo, 2013)).

As a curiosity, we note that a number of results are known, which regard inverse trigonometric laws for some random times associated to BM; for instance, the density (1.4) appears as the conditional density of the second inter-passage time of BM through a level, with the condition that the first-passage time is $r$ (see eq. (2.15) of (Abundo, 2016)).

The arc-sine law is valid for the time $\tau$ spent by BM on the positive half-line during the time interval $[0,r],$ that is, $P(\tau\leq t)=\frac{2}{\pi}\arcsin\left(\sqrt{\frac{t}{r}}\ \right),\ t\in[0,r]$ (see (Levy, 1965)); moreover, a compound arc-sine law holds for the first instant $\theta$ at which a diffusion $X$ of the form (2.6), with $\rho(t)$ deterministic, attains the maximum in the interval $[0,r],$ namely $P(\theta\leq t)=\frac{2}{\pi}\arcsin\left(\sqrt{\frac{\rho(t)}{\rho(r)}}\ \right),\ t\in[0,r]$ (see (Abundo, 2006), and (Levy, 1965) in the case of BM, i.e. $\rho(t)=t).$

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