Energy saving approximation of Wiener process
under unilateral constraints³³3The work supported by RSF grant 21-11-00047.

Let $AC[0,T]$ denote the space of absolutely continuous functions on the interval $[0,T]$. For $h\in AC[0,T]$ let us call kinetic energy

$$|h|_{T}^{2}:=\int_{0}^{T}h^{\prime}(t)^{2}dt.$$

In the works [1, 3, 7, 8, 9, 10, 11] the approximation of a random process sample path by the function $h$ of smallest energy was considered under various constraints on closeness between $h$ and the sample path. In particular, in [9] the energy saving approximation was studied for a Wiener process $W$ under bilateral uniform constraints.

For $T>0,r>0$ let us define the set of admissible approximations as

$$M^{\pm}_{T,r}:=\left\{h\in AC[0,T]\,\big{|}\,\forall t\in[0,T]:W(t)-r\leq h(t)\leq W(t)+r;h(0)=0\right\}$$

and let

$$I^{\pm}_{W}(T,r):=\inf\left\{|h|_{T}^{2}\>\big{|}\>h\in M^{\pm}_{T,r}\right\}.$$

It was proved in [9] that for every fixed $r>0$ it is true that

$$\frac{I^{\pm}_{W}(T,r)}{T}\stackrel{{\scriptstyle\text{a.s.}}}{{\longrightarrow}}{\mathcal{C}}^{2}\,r^{-2},\qquad\textrm{as }T\to\infty,$$

where ${\mathcal{C}}\approx 0,63$ is some absolute constant (the exact value of ${\mathcal{C}}$ is unknown), i.e. the optimal approximation energy grows linearly in time.

In this work, we are interested in the behavior of a similar quantity under unilateral constraints, i.e. the set of admissible approximations is

$$M_{T,r}:=\left\{h\in AC[0,T]\,\big{|}\,\forall t\in[0,T]:h(t)\geq W(t)-r;h(0)=0\right\},$$

and we are now interested in the behavior of

$$I_{W}(T,r):=\inf\left\{|h|_{T}^{2}\>\big{|}\>h\in M_{T,r}\right\}.$$

It is technically more convenient to translate the initial value of the approximating function to the point $r$, so that this function runs above the trajectory of the approximated process $W$. Let

$$M_{T,r}^{\prime}:=\left\{h\in AC[0,T]\,\big{|}\,\forall t\in[0,T]:h(t)\geq W(t);h(0)=r\right\}.$$

Since the sets of functions $M_{T,r}$ and $M_{T,r}^{\prime}$ differ by a constant shift, it is easy to see that

$$I_{W}(T,r)=\inf\left\{|h|_{T}^{2}\>\big{|}\>h\in M_{T,r}^{\prime}\right\}.$$

Our main result asserts that, when $T$ grows, the quantity $I_{W}(T,r)$ grows merely logarithmically.

In Section 2 we establish a connection between the unilateral energy saving approximation of arbitrary continuous function with its minimal concave majorant. In Section 3 we establish the necessary properties of Wiener process minimal concave majorant using the results of Groeneboom [6]. Section 4 contains the proof of Theorem 1.

In Sections 5–6 we consider a class of adaptive Markovian (diffusion) approximation strategies based on the current and past values of $W$. We prove that in this class the optimal strategy is defined by the formula

$$h^{\prime}(t)=\frac{1}{h(t)-W(t)}\,.$$

For this strategy the energy consumption also has the logarithmic order but is two times larger than that for the optimal non-adaptive strategy using the information about the whole trajectory of $W$. Namely,

$$\frac{|h|_{2}^{2}}{\log T}\stackrel{{\scriptstyle\text{a.s.}}}{{\longrightarrow}}1,\qquad\textrm{as }T\to\infty.$$

It turns out that the optimal energy saving approximation under unilateral constraints may be described in terms of the minimal concave majorant (MCM) of the approximated function. Let $w:[0,T]\mapsto{\mathbb{R}}$ be a continuous function. Then the corresponding MCM $\overline{w}$ is the minimal concave function satisfying conditions

$$\overline{w}(t)\geq w(t),\qquad 0\leq t\leq T.$$

The optimal energy saving majorant $\chi_{*}$ is shown in Figure 1.

The uniqueness of the solution follows from the fact that the set of functions satisfying problem’s assumptions is convex and the functional $|\cdot|_{T}^{2}$ is strictly convex on this set.

Let us describe the solution.

Since case (a) is trivial, we consider case (b).

Let $\chi(\cdot)$ be the solution of our problem. We show first that $\chi$ is a convex non-decreasing function. Consider the function

$$\chi_{1}(t):=r+\int_{0}^{t}g(s)ds,\qquad 0\leq t\leq T,$$

where $g(\cdot)$ is the non-increasing monotone rearrangement of the function $\max\{\chi^{\prime}(\cdot),0\}$. Then $\chi_{1}$ is a concave non-decreasing function, $\chi_{1}(0)=r$ and $\chi_{1}(t)\geq\chi(t)\geq w(t)$ for all $t\in[0,T]$. Therefore, $\chi_{1}$ satisfies the problem’s constraints. On the other hand,

$$|\chi_{1}|_{T}^{2}=\int_{0}^{T}g(s)^{2}ds=\int_{0}^{T}\max\{\chi^{\prime}(\cdot),0\}^{2}ds\leq|\chi|_{T}^{2}.$$

Due to the problem solution uniqueness we obtain $\chi_{1}=\chi$. This equality proves that $\chi$ is concave and non-decreasing.

Since $\chi_{*}$ is the smallest concave non-decreasing function satisfying problem’s constraints, we have

$$\chi(t)\geq\chi_{*}(t),\qquad 0\leq t\leq T.$$

Furthermore, let us prove that $\chi(T)=\chi_{*}(T)=\max_{0\leq t\leq T}w(t)$. Indeed, in case (b) the function

$$\chi_{2}(t):=\min\{\chi(t),\chi_{*}(T)\},\qquad 0\leq t\leq T,$$

satisfies both problem constraints and $|\chi_{2}|_{T}^{2}\leq|\chi|_{T}^{2}$; by the uniqueness of the solution, we obtain $\chi=\chi_{2}$. In particular, $\chi(T)=\chi_{*}(T)$.

Finally, assume that the strict inequality $\chi(t_{0})>\chi_{*}(t_{0})$ holds for some $t_{0}\in[0,T]$. Then, since the function $\chi_{*}$ is concave and non-decreasing, there exists a non-decreasing affine function $\ell(\cdot)$ such that

$$\chi_{*}(t)\leq\ell(t),\qquad 0\leq t\leq T,$$

but $\ell(t_{0})<\chi(t_{0})$. However, at the endpoints of the interval $[0,T]$ the opposite inequality is true, because

$$\chi(0)=r=\chi_{*}(0)\leq\ell(0),\qquad\chi(T)=\chi_{*}(T)\leq\ell(0).$$

Therefore, there exists a non-degenerated interval $[t_{1},t_{2}]\subset[0,T]$ such that $t_{0}\in[t_{1},t_{2}]$, $\ell(t_{1})=\chi(t_{1})$, $\ell(t_{2})=\chi(t_{2})$.

Since $\ell^{\prime}(\cdot)$ is a constant, while $\chi^{\prime}(\cdot)$ is not a constant on $[t_{1},t_{2}]$, it follows from Hölder inequality that

			$\displaystyle(t_{2}-t_{1})\int_{t_{1}}^{t_{2}}\chi^{\prime}(t)^{2}dt>\left(\int_{t_{1}}^{t_{2}}\chi^{\prime}(t)dt\right)^{2}=(\chi(t_{2})-\chi(t_{1}))^{2}$
		$\displaystyle=$	$\displaystyle(\ell(t_{2})-\ell(t_{1}))^{2}=\left(\int_{t_{1}}^{t_{2}}\ell^{\prime}(t)dt\right)^{2}=(t_{2}-t_{1})\int_{t_{1}}^{t_{2}}\ell^{\prime}(t)^{2}dt.$

We obtain

$$\int_{t_{1}}^{t_{2}}\chi^{\prime}(t)^{2}dt>\int_{t_{1}}^{t_{2}}\ell^{\prime}(t)^{2}dt.$$

It follows that the function

$$\chi_{3}(t):=\min\{\chi(t),\ell(t)\},\qquad 0\leq t\leq T,$$

satisfies the problem’s constraints and $|\chi_{3}|_{T}^{2}<|\chi|_{T}^{2}$ but this is impossible by the definition of $\chi$. Therefore, the assumption $\chi(t_{0})>\chi_{*}(t_{0})$ brought us to a contradiction. $\square$

We recall some notation and results from the article [6] that will be used in the sequel. Denote

$$\tau(a):=\sup\left\{t>0\,\big{|}\,W(t)-t/a=\sup_{u>0}\left(W(u)-u/a\right)\right\}.$$

The function $a\mapsto\tau(a)$ is non-decreasing.

According to [6, Corollary 2.1] for every $a>0$ the random variable $\frac{\tau(a)}{a^{2}}$ has the distribution density

$$q(t)=2\,{\mathbb{E}\,}\left(\frac{X}{\sqrt{t}}-1\right)_{+},\qquad t>0,$$

where $x_{+}:=x{\mathbf{1}}_{\{x>0\}}$, $X$ is a standard normal random variable.

Next, let $\overline{W}$ be the global MCM for the Wiener process $W(t),t\geq 0$. Define a random process $L$ as

$$L(a,b):=\int_{\tau(a)}^{\tau(b)}\overline{W}^{\prime}(t)^{2}dt.$$

Our study is essentially based on the following result due to Groeneboom.

Moreover, there is an explicit description of the Lévy measure of $X$ in [6] but we do not need it here. We are only interested in the Kolmogorov’s strong law of large numbers for $X$ which asserts that

$$\frac{X(t)}{t}\stackrel{{\scriptstyle\text{a.s.}}}{{\longrightarrow}}1,\qquad\textrm{as }t\to\infty.$$

Using the definition of $X$, letting $a_{0}=0$, and making the variable change $V=e^{t}$, we may reformulate this result as

$$\frac{L(1,V)}{\log V}\stackrel{{\scriptstyle\text{a.s.}}}{{\longrightarrow}}1,\qquad\textrm{as }V\to\infty.$$

(2)

Let $T_{n}:=n$, then the events $D_{n}:=\left\{\tau\left(T_{n}^{\frac{1}{2}-\delta}\right)\geq\frac{T_{n}}{2}\right\}$ satisfy

$$\sum\limits_{n=1}^{\infty}{\mathbb{P}}(D_{n})<\infty.$$

Therefore, by Borel–Cantelli lemma, with probability $1$ for all sufficiently large $n$ the event $D_{n}$ does not hold, i.e. with probability $1$ for all sufficiently large $n$ we have

$$\tau\left(T_{n}^{\frac{1}{2}-\delta}\right)<\frac{T_{n}}{2}.$$

(4)

Let $n\geq 2$ be such that (4) holds for $T_{n}$ and let $T\in[T_{n-1},T_{n}]$. Since $\tau(\cdot)$ is non-decreasing, we have

$$\tau\left(T^{\frac{1}{2}-\delta}\right)\leq\tau\left(T_{n}^{\frac{1}{2}-\delta}\right)<\frac{T_{n}}{2}=\frac{n}{2}\leq n-1=T_{n-1}\leq T.$$

This provides us with a required lower bound for sufficiently large $T$.

In the same way, for the upper bound we have

	$\displaystyle{\mathbb{P}}\left(\tau\left(T^{\frac{1}{2}+\delta}\right)\leq 2T\right)$	$\displaystyle=$	$\displaystyle\int_{0}^{2T^{-2\delta}}2\,{\mathbb{E}\,}\left(\frac{X}{\sqrt{t}}-1\right)_{+}dt$
		$\displaystyle\leq$	$\displaystyle C_{3}\int^{2T^{-2\delta}}_{0}e^{-t/2}t^{-1/2}dt\leq C_{4}T^{-\delta},$

where $C_{3},C_{4}$ are some positive absolute constants.

Consider the sequence $T^{\prime}_{n}:=2^{n}$. For the events $D^{\prime}_{n}:=\left\{\tau\left(T_{n}^{\frac{1}{2}+\delta}\right)\leq 2T_{n}\right\}$ we have

$$\sum\limits_{n=1}^{\infty}{\mathbb{P}}(D^{\prime}_{n})<\infty.$$

Therefore, by Borel–Cantelli lemma with probability $1$ for all sufficiently large $n$ the event $D^{\prime}_{n}$ does not hold, i.e. with probability $1$ for all sufficiently large $n$ it is true that

$$\tau\left(T_{n}^{\frac{1}{2}+\delta}\right)>2\,T_{n}.$$

(5)

Let (5) be satisfied for some $T_{n}$ and let $T\in[T_{n},T_{n+1}]$. Since $\tau(\cdot)$ is non-decreasing, we have

$$\tau\left(T^{\frac{1}{2}+\delta}\right)\geq\tau\left(T_{n}^{\frac{1}{2}+\delta}\right)>2\,T_{n}=T_{n+1}\geq T.$$

This provides us with a required upper bound for all sufficiently large $T$.

$\square$

The next theorem describes the asymptotic behavior of the energy of the minimal concave majorant for a Wiener process. For some $r>0$ let $\overline{W}^{(r)}$ denote MCM of a Wiener process $W$ on the whole real line, starting from the height $r$. Then the majorant $\overline{W}^{(r)}$ is an affine function on some initial interval $[0,\theta(r)]$, its graph contains the point $(0,r)$ and is a tangent to the graph of $\overline{W}$, while on $[\theta(r),\infty)$ the functions $\overline{W}^{(r)}$ and $\overline{W}$ coinсide.

By using Lemma 4 for comparing the upper integration limits, we obtain

	$\displaystyle\liminf\limits_{T\to\infty}\frac{|\overline{W}^{(r)}|_{T}^{2}}{\log{T}}$	$\displaystyle\geq$	$\displaystyle\liminf\limits_{T\to\infty}\frac{L\left(1,T^{1/2-\delta}\right)}{\log{T}};$
	$\displaystyle\limsup\limits_{T\to\infty}\frac{|\overline{W}^{(r)}|_{T}^{2}}{\log{T}}$	$\displaystyle\leq$	$\displaystyle\liminf\limits_{T\to\infty}\frac{L\left(1,T^{1/2+\delta}\right)}{\log{T}}.$

Taking into account the law of large numbers (2) we have

$$\frac{1}{2}-\delta\leq\liminf\limits_{T\to\infty}\frac{|\overline{W}^{(r)}|_{T}^{2}}{\log{T}}\leq\limsup\limits_{T\to\infty}\frac{|\overline{W}^{(r)}|_{T}^{2}}{\log{T}}\leq\frac{1}{2}+\delta.$$

Letting $\delta\searrow 0$ yields the required result. $\square$

Upper bound. The restriction of the global MCM starting from the height $r$ onto the interval [0,T] belongs to the set of admissible functions: $\overline{W}^{(r)}\in M_{T,r}^{\prime}$. We derive from Theorem 5 that

$$\limsup\limits_{T\to\infty}\frac{I_{W}(T,r)}{\log{T}}\leq\limsup\limits_{T\to\infty}\frac{|\overline{W}^{(r)}|_{T}^{2}}{\log{T}}\leq\frac{1}{2}\qquad\textrm{a.s.}$$

Lower bound. For $r>0,T>0$ let $\overline{W}^{(r,T)}$ denote the local MCM of the Wiener process $W(t),t\in[0,T]$, starting from the height $r$. Let $\chi$ be the unique solution of the problem we are interested in, $|h|_{T}^{2}\to\min,h\in M^{\prime}_{T,r}$. Recall that its structure is described in Proposition 2. Since for large $T$ it is true that $\max_{0\leq s\leq T}W(s)>r$, for such $T$ the assumption of case (b) of that proposition is verified. In particular, it follows that

$$\chi(t)=\overline{W}^{(r,T)}(t),\qquad 0\leq t\leq t_{\max},$$

where

$$t_{\max}=t_{\max}(T):=\min\{t:W(t)=\max_{0\leq s\leq T}W(s)\}.$$

Notice that the function $\tau(\cdot)$ can not take values from the interval $(t_{\max},T)$. Therefore, if for some $a$ it is true that $\tau(a)<T$, then it is also true that $\tau(a)\leq t_{\max}$. In this case we have

$$\chi(t)=\overline{W}^{(r,T)}(t)=\overline{W}^{(r)}(t),\qquad 0\leq t\leq\tau(a).$$

It follows that

$$I_{W}(T,r)=|\chi|_{2}^{2}\geq\int_{0}^{\tau(a)}\chi^{\prime}(t)^{2}\,dt=|\overline{W}^{(r)}|_{\tau(a)}^{2}.$$

Let us fix $\delta\in(0,1/2)$. Let $a=a(T):=T^{1/2-\delta}$. Then by Lemma 4 we have

$$T^{\frac{1-2\delta}{1+2\delta}}<\tau(a)<T$$

a.s. for all sufficiently large $T$. Furthermore, it follows from Theorem 5 that, as $T\to\infty$,

$$|\overline{W}^{(r)}|_{\tau(a)}^{2}\geq\frac{\log\tau(a)}{2}\,(1+o(1))\geq\frac{1-2\delta}{2(1+2\delta)}\,\log T\,(1+o(1))\qquad\textrm{a.s.}$$

By combining these estimates, we obtain

$$I_{W}(T,r)\geq\frac{1-2\delta}{2(1+2\delta)}\,\log T\,(1+o(1))\qquad\textrm{a.s.}$$

Finally, by letting $\delta\searrow 0$, we arrive at

$$I_{W}(T,r)\geq\frac{1}{2}\,\log T\,(1+o(1))\qquad\textrm{a.s.,}$$

as required.

In practice, it is often necessary to arrange an approximation (a pursuit) in real time (adaptively), when the trajectory of the approximated process is known not on the entire time interval but only before the current time instant. In view of the Markov property of Wiener process, a reasonable strategy is to define the speed of a pursuit $h$ as a function of current positions of the processes $h$ and $W$, without taking past trajectories into account, i.e. let

$$h^{\prime}(t):=b(h(t),W(t),t).$$

(7)

On the qualitative level the function $b(x,w,t)$ must tend to infinity, as $x-w\searrow 0$, i.e. when the approximating process approaches the dangerous boundary it accelerates its movement trying to escape from a dangerous position. One has to optimize the function $b$ trying to reach the smallest average energy consumption. It is possible to reach the same logarithmic in time order of energy consumption as in the case of non-adaptive approximation but with somewhat larger coefficient. The difference of the coefficients represents the price we pay for not knowing the future of the process we try to approximate.

It is interesting to compare (7) with the form of the optimal adaptive strategy in the case of bilateral constraints [9] where

$$h^{\prime}(t)=b(h(t)-W(t)).$$

(8)

The latter strategy is more simple because the speed is governed only by the distance between the approximated and the approximating processes and does not depend on time.

Let us make a time and space change

	$\displaystyle U(\tau)$	$\displaystyle:=$	$\displaystyle e^{-\tau/2}W(e^{\tau}),$
	$\displaystyle z(\tau)$	$\displaystyle:=$	$\displaystyle e^{-\tau/2}h(e^{\tau}).$

Recall that $U(\cdot)$ is an Ornstein–Uhlenbeck process and therefore satisfies the equation

$$dU=-\frac{U\,d\tau}{2}+d\widetilde{W},$$

(9)

where $\widetilde{W}$ is a Wiener process. We have the following expression for the derivative of $z$

$$z^{\prime}(\tau)=-\frac{1}{2}z(\tau)+e^{\tau/2}h^{\prime}(e^{\tau}),$$

(10)

which yields

$$h^{\prime}(e^{\tau})=e^{-\tau/2}\left(z^{\prime}(\tau)+\frac{z(\tau)}{2}\right).$$

(11)

Let us consider the distance between the approximated and approximating processes

$\displaystyle Z(\tau)$

$\displaystyle:=$

$\displaystyle z(\tau)-U(\tau).$

(12)

We will study time-homogeneous diffusion strategies

$$dZ=b(Z)d\tau-d\widetilde{W}.$$

(13)

From equations (9) and (13) it follows that this is equivalent to

$$z^{\prime}(\tau)+\frac{U(\tau)}{2}=b(Z(\tau)),$$

which also implies

$$z^{\prime}(\tau)+\frac{z(\tau)}{2}=b(Z(\tau))+\frac{Z(\tau)}{2}.$$

(14)

Before proceeding to optimization, let us see how the diffusion strategies act in the initial framework. By (11) and (14) we have

	$\displaystyle h^{\prime}(e^{\tau})$	$\displaystyle=$	$\displaystyle e^{-\tau/2}\left(b(Z(\tau))+\frac{Z(\tau)}{2}\right):=e^{-\tau/2}\ \widetilde{b}(Z(\tau))$		(15)
		$\displaystyle=$	$\displaystyle e^{-\tau/2}\ \widetilde{b}\left(e^{-\tau/2}\left(h(e^{\tau})-W(e^{\tau})\right)\right),$		(16)

where $\widetilde{b}(x):=b(x)+\tfrac{x}{2}$. In other words, the form of the strategy is

$$h^{\prime}(t)=\frac{1}{\sqrt{t}}\ \widetilde{b}\left(\frac{1}{\sqrt{t}}\left(h(t)-W(t)\right)\right).$$

(17)

We see that this class of strategies is space-homogeneous but, in general, not time-homogeneous.

Now we proceed to the optimization of the shift coefficient $b(\cdot)$ determining the pursuit strategy. Let us use some basic facts about one-dimensional time-homogeneous diffusion, cf. [2, Ch.IV.11] and [5, Ch.2]. Let

	$\displaystyle B(x)$	$\displaystyle:=$	$\displaystyle 2\int^{x}b(u)du,$		(18)
	$\displaystyle p_{0}(x)$	$\displaystyle:=$	$\displaystyle e^{B(x)}.$		(19)

Assume that condition

$$\int_{0}\frac{dx}{p_{0}(x)}=\infty$$

(20)

is verified. Then, in Feller classification, the point $0$ is the entrance-boundary and not an exit-boundary for diffusion (13). This means that the diffusion $Z$ remains forever in $[0,\infty)$. Moreover, the function

$$p(x):=Q^{-1}p_{0}(x),$$

(21)

where $Q=\int_{0}^{\infty}p_{0}(x)dx$, is the density of the unique stationary distribution for $Z$. For the energy, by using (11) and (14), we obtain (a.s., as $T\to\infty$)

	$\displaystyle\int_{1}^{T}h^{\prime}(t)^{2}dt$	$\displaystyle=$	$\displaystyle\int_{0}^{\log T}h^{\prime}(e^{\tau})^{2}e^{\tau}d\tau=\int_{0}^{\log T}\left(z^{\prime}(\tau)+\frac{z(\tau)}{2}\right)^{2}d\tau$
		$\displaystyle=$	$\displaystyle\int_{0}^{\log T}\left(b(Z(\tau))+\frac{Z(\tau)}{2}\right)^{2}d\tau$
		$\displaystyle\sim$	$\displaystyle{\log T}\int_{0}^{\infty}\left(b(x)+\frac{x}{2}\right)^{2}p(s)dx$
		$\displaystyle=$	$\displaystyle{\log T}\int_{0}^{\infty}\left(\left(\frac{\log p}{2}\right)^{\prime}(x)+\frac{x}{2}\right)^{2}p(x)dx$
		$\displaystyle=$	$\displaystyle{\log T}\int_{0}^{\infty}\left(\frac{p^{\prime}(x)^{2}}{4p(x)}+\frac{xp^{\prime}(x)}{2}+\frac{x^{2}p(x)}{4}\right)dx$
		$\displaystyle=$	$\displaystyle{\log T}\left(-\frac{1}{2}+\int_{0}^{\infty}\left(\frac{p^{\prime}(x)^{2}}{4p(x)}+\frac{x^{2}p(x)}{4}\right)dx\right)$
		$\displaystyle:=$	$\displaystyle-\frac{\log T}{2}+\frac{\log T}{4}J(p).$

Taking into account condition (20), it remains to solve the variational problem

$$\min\left\{J(p)\Big{|}\int_{0}^{\infty}p(x)dx=1,p(0)=0\right\}$$

over the class of densities concentrated on $[0,\infty)$. After the variable change

$$y(x):=p(x)^{1/2},$$

the variational problem transforms into

$$\min\left\{\int_{0}^{\infty}\left(4y^{\prime}(x)^{2}+x^{2}y(x)^{2}\right)dx\ \Big{|}\int_{0}^{\infty}y(x)^{2}dx=1,y(0)=0\right\}.$$

We show in the next section that this minimum equals $6$; it is attained at the function

$$y(x)=(2/\pi)^{1/4}\,x\,\exp(-x^{2}/4).$$

It follows that the asymptotic energy behavior for the optimal strategy is

$$\int_{1}^{T}h^{\prime}(t)^{2}dt\sim\log T,\quad T\to\infty,$$

i.e. the optimal choice of the shift in the adaptive setting leads to two times larger energy consumption than for the optimal strategy in the non-adaptive setting.

In order to find the optimal shift, write

$$p(x)=y(x)^{2}=(2/\pi)^{1/2}\,x^{2}\,\exp(-x^{2}/2)$$

and we find from (18) – (21)

$$b(x)=\frac{1}{2}\,(\ln p)^{\prime}(x)=\frac{1}{x}-\frac{x}{2}.$$

Note that the density $p$ indeed satisfies the necessary condition (20).

Returning to the initial problem, we obtain the shift $\widetilde{b}(x)=\tfrac{1}{x}$, thus the strategy (17) takes the form

$$h^{\prime}(t)=\frac{1}{h(t)-W(t)}\,.$$

Curiously, the optimal diffusion strategy is not only space-homogeneous but also time-homogeneous, unlike arbitrary strategies of this class.