A Taylor method for stochastic differential equations with time-dependent delay via the polynomial condition

Abstract

The subject of this article is an analytic approximate method for a class of stochastic differential equations with time-dependent delay, with coefficients that do not necessarily satisfy the Lipschitz condition nor the linear growth condition but they satisfy the polynomial condition. More precisely, it is assumed that equations from the observed class have unique solutions with bounded moments, their coefficients satisfy the polynomial condition and some derivatives of the specific order of the drift and diffusion coefficients are uniformly bounded. Approximate equations are defined on partitions of the time interval and their coefficients are Taylor approximations of the coefficients of the initial equation up to arbitrary derivatives. It is shown that the solutions of the approximate equations converge in the $L^p$ sense and almost surely toward the solution of the initial equation and the rate of the convergence is given.

Keywords:

• Almost sure convergence
• L^p convergence
• polynomial condition
• stochastic differential equations with time-dependent delay
• Taylor approximation

MSC2020—Mathematical Sciences Classification System::

• 41A25
• 41A58
• 60H10

  Appendix

Let us assume that the Borel measurable delay function $\delta :[t_0,T]\to [0,\tau ]$ is Lipschitz continuous. In other words, let there be a positive constant L such that (41) $$|\delta (s_1)-\delta (s_2)|\le L|s_1-s_2|,\hspace{1em}{s}_1,s_2\in [t_0,T].$$(41)

This implies that function ${\delta }_1:[t_0,T]\to [t_0-\tau ,T],{\delta }_1(t)=t-\delta (t),$ is also Lipschitz continuous with the Lipschitz constant $1+L.$

The main reason for employing more restrictive assumption on δ is to determine the points in which the derivatives of the coefficients of Equation (8)(8) $$x(t)=\eta (0)+{\int }_{t_0}^{t}a(x(u),x(u-\delta (u)),u)du+{\int }_{t_0}^{t}b(x(u),x(u-\delta (u)),u)dw(u),$$(8) with respect to the second argument will be defined and which will be fixed on each partitioning interval. These points are of the form $x^n(t_{k-\lfloor \delta (t_k)/{\delta }_n\rfloor }),$ where $\lfloor ·\rfloor$ represents the greatest integer function. For simpler notation, let us introduce a new function ${\delta }_2:\{0,\dots ,n-1\}\to \{-n\prime ,\dots ,n-1\}$ defined as ${\delta }_2(k)=k-\lfloor \delta (t_k)/{\delta }_n\rfloor .$

This time, Equation (13)(13) $$\begin{array}{c}{x}^n(t)=x^n(t_k)+{\int }_{t_k}^t\sum _{i=0}^{m_1}\frac{d^{i}a(x^n(t_k),{\tilde{x}}^n(s),s)}{i!}ds\\ +{\int }_{t_k}^t\sum _{i=0}^{m_2}\frac{d^{i}b(x^n(t_k),{\tilde{x}}^n(s),s)}{i!}dw(s),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\},\end{array}$$(13) have following form (42) $$\begin{array}{c}\text{}{x}^n(t)=x^n(t_k)+{\int }_{t_k}^t\sum _{i=0}^{m_1}\frac{d^{i}a(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}ds\\ \text{}+{\int }_{t_k}^t\sum _{i=0}^{m_2}\frac{d^{i}b(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}dw(s),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\}.\end{array}$$(42)

Instead of Proposition 1 we have following assertions.

Proposition 2.

I Let $\{x^n(t) | t\in [t_k,t_{k+1}]\},k\in \{0,\dots ,n-1\}$, be the solutions of Equation (42)(42) $$\begin{array}{c}\text{}{x}^n(t)=x^n(t_k)+{\int }_{t_k}^t\sum _{i=0}^{m_1}\frac{d^{i}a(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}ds\\ \text{}+{\int }_{t_k}^t\sum _{i=0}^{m_2}\frac{d^{i}b(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}dw(s),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\}.\end{array}$$(42) and let the assumptions ${\mathcal{B}}_1-{\mathcal{B}}_5$ be satisfied. Then, for every $0<r\le (M+1)\overline{p}$ $$E\underset{s\in [t_k,t]}{\sup }|x^n(s)-x^n(t_k){|}^r\le \overline{C}\prime {\delta }_n^{r/2},\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\},$$ where $C\prime$ represents a positive generic constant independent of n.

II Under the assumptions from part I, assumption ${\mathcal{B}}_6$ and Lipschitz condition (41), for every $0<r\le (M+1)\overline{p}$ $$\underset{s\in [t_k,t]}{\sup }E|x^n({\delta }_1(s))-x^n(t_{{\delta }_2(k)}){|}^r\le C″{\delta }_n^{r/2},\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\},$$ where $C″$ stands for a positive generic constant independent of n.

Proof.

The proof of the part I is analogous to the proof of part I from Proposition 1. The only difference occurs in the coefficients of Equation (13)(13) $$\begin{array}{c}{x}^n(t)=x^n(t_k)+{\int }_{t_k}^t\sum _{i=0}^{m_1}\frac{d^{i}a(x^n(t_k),{\tilde{x}}^n(s),s)}{i!}ds\\ +{\int }_{t_k}^t\sum _{i=0}^{m_2}\frac{d^{i}b(x^n(t_k),{\tilde{x}}^n(s),s)}{i!}dw(s),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\},\end{array}$$(13) which contain $x^n(t_{{\delta }_2(k)})$ instead of ${\tilde{x}}^n(s).$

II This part is proved completely differently from the part II of Proposition 1. Let $s\in [t_k,t],t\in [t_k,t_{k+1}]$ and $k\in \{0,\dots ,n-1\}$ be arbitrary numbers. Since our goal is to estimate the term $E|x^n({\delta }_1(s))-x^n(t_{{\delta }_2(k)}){|}^r,$ we consider the difference (43) $${\delta }_1(s)-t_{{\delta }_2(k)}={\delta }_1(s)-(t_0+k{\delta }_n-\mathrm{ }\lfloor \frac{\delta (t_k)}{{\delta }_n}\rfloor {\delta }_n)=s-t_k-\delta (s)+\lfloor \frac{\delta (t_k)}{{\delta }_n}\rfloor {\delta }_n.$$(43)

The triangle inequality, Lipschitz continuity (41)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) and $$-\delta (s)+\delta (t_k)-{\delta }_n\le -\delta (s)+\lfloor \frac{\delta (t_k)}{{\delta }_n}\rfloor {\delta }_n\le -\delta (s)+\delta (t_k)$$ imply (44) $$|s-t_k-\delta (s)+\lfloor \frac{\delta (t_k)}{{\delta }_n}\rfloor {\delta }_n|\le s-t_k+{\delta }_n+|\delta (s)-\delta (t_k)|\le (3+\lfloor L\rfloor ){\delta }_n.$$(44)

Thus, (43) and (44) give (45) $$|{\delta }_1(s)-t_{{\delta }_2(k)}|\le (3+\lfloor L\rfloor ){\delta }_n.$$(45)

Consequently, there are no more than $N=3+\lfloor L\rfloor$ division points of the partition (9)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) –(10)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) between ${\delta }_1(s)$ and $t_{{\delta }_2(k)}$ (excluding the ${\delta }_1(s)$ if it is one of those division points) and the number of such points N does not depend of n nor $n\prime .$ This is very convenient since n (and therefore $n\prime$) is going to tend to $+\infty .$

Let us first consider the case when ${\delta }_1(s)\ge t_{{\delta }_2(k)}.$ For the following estimation it is of our interest to note the index $i=\max \{j\in \{-n\prime ,\dots ,k\} | t_j\le {\delta }_1(s)\}.$ Then, bearing in mind (45)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) and the inequality (2)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) , we compute (46) $$\begin{array}{c}\text{}E|x^n({\delta }_1(s))-x^n(t_{{\delta }_2(k)}){|}^r\\ \text{}=E|x^n({\delta }_1(s))-x^n(t_i)+x^n(t_i)-\cdots -x^n(t_{{\delta }_2(k)+1})+x^n(t_{{\delta }_2(k)+1})-x^n(t_{{\delta }_2(k)}){|}^r\\ \text{}\le (1\vee {(3+\lfloor L\rfloor )}^{r-1})\sum _{j={\delta }_2(k)}^{i}E\underset{u\in [t_j,t_{j+1}\wedge s]}{\sup }|x^n(u)-x^n(t_j){|}^r.\end{array}$$(46)

If $j\le -1$ in the previous sum, then ${\mathcal{B}}_6$ provides $$\begin{array}{cc}E\underset{u\in [t_j,t_{j+1}\wedge s]}{\sup }|x^n(u)-x^n(t_j){|}^r\hfill & =E\underset{u\in [t_j,t_{j+1}]}{\sup }|\eta (u-t_0)-\eta (t_j-t_0){|}^r\hfill \\ \hfill & \le {(D\prime )}^{r/2}E\mathrm{}\underset{u\in [t_j,t_{j+1}]}{\sup }\mathrm{}{(1+|u-t_0{|}^{q\prime }+|t_j-t_0{|}^{q\prime })}^{r/2}|u-t_j{|}^r\hfill \\ \hfill & \le {(D\prime )}^{r/2}{(1+2{\tau }^{q\prime })}^{r/2}{(1\wedge \tau )}^{r/2}{\delta }_n^{r/2}\hfill \end{array}$$ and if $j\ge 0,$ the part I yields (48) $$E\underset{u\in [t_j,t_{j+1}\wedge s]}{\sup }|x^n(u)-x^n(t_j){|}^r\le \overline{C}\prime {\delta }_n^{r/2}.$$(48)

Hence, (47)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) and (48)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) substituted into (46)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) imply (49) $$E|x^n({\delta }_1(s))-x^n(t_{{\delta }_2(k)}){|}^r\le C″{\delta }_n^{r/2},$$(49) where $$C″=(1\vee {(3+\lfloor L\rfloor )}^{r-1})(3+\lfloor L\rfloor )({(D\prime )}^{r/2}{(1+2{\tau }^{q\prime })}^{r/2}{(1\wedge \tau )}^{r/2}\vee \overline{C}\prime )$$ is a constant independent of n. This estimate holds whether ${\delta }_1(s)$ is the division point of (9)(9) $$t_0<t_1<\cdots <t_n=T,$$(9) –(10)(10) $$t_0-\tau =t_{-n\prime }<t_{-n\prime +1}<\cdots <t_{-1}<t_0,$$(10) or not.

Let us note that the estimate (49)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) is obtained similarly in the case when ${\delta }_1(s)<t_{{\delta }_2(k)}$ and since $s\in [t_k,t]$ is arbitrary, the proof is complete.□

Analogously to Theorems 3.1 and 3.2 we state following theorems.

Theorem 5.1.

Let all of the assumptions from Proposition 2 hold and let x be the solution of Equation (8)(8) $$x(t)=\eta (0)+{\int }_{t_0}^{t}a(x(u),x(u-\delta (u)),u)du+{\int }_{t_0}^{t}b(x(u),x(u-\delta (u)),u)dw(u),$$(8) with initial condition (7)(32) $$\begin{array}{c}E{\left[\right|x(s){|}^{\overline{p}\overline{q}/2}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}/2}]}^2|x(s)-x^n(s){|}^{\overline{p}}\\ \le 2^{\overline{p}}E\left[\right|x(s){|}^{\overline{p}\overline{q}}+|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}]\left[\right|x(s){|}^{\overline{p}}+|x^n(s){|}^{\overline{p}}]\\ \le 2^{\overline{p}}\left[E\right|x(s){|}^{\overline{p}(1+\overline{q})}+E|x(s){|}^{\overline{p}}|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}+E|x(s){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}+E|x({\delta }_1(s)){|}^{\overline{p}\overline{q}}|x^n(s){|}^{\overline{p}}].\end{array}$$(32) and let $x^n$ be the solution to Equation (42)(42) $$\begin{array}{c}\text{}{x}^n(t)=x^n(t_k)+{\int }_{t_k}^t\sum _{i=0}^{m_1}\frac{d^{i}a(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}ds\\ \text{}+{\int }_{t_k}^t\sum _{i=0}^{m_2}\frac{d^{i}b(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}dw(s),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\}.\end{array}$$(42) . Then $$E\underset{t\in [t_0-\tau ,T]}{\sup }|x(t)-x^n(t){|}^{\overline{p}}\le C{\delta }_n^{(m+1)\overline{p}/2},$$ where $m=m_1\wedge m_2$ and C is a generic constant independent of n.

Theorem 5.2.

Let the assumptions from Theorem 5.1 hold. Then the sequence ${(x^n)}_{n\in \mathbb{N}}$ of the solutions to Equation (42)(42) $$\begin{array}{c}\text{}{x}^n(t)=x^n(t_k)+{\int }_{t_k}^t\sum _{i=0}^{m_1}\frac{d^{i}a(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}ds\\ \text{}+{\int }_{t_k}^t\sum _{i=0}^{m_2}\frac{d^{i}b(x^n(t_k),x^n(t_{{\delta }_2(k)}),s)}{i!}dw(s),\mathrm{\hspace{1em}}\mathrm{\hspace{1em}}t\in [t_k,t_{k+1}],k\in \{0,\dots ,n-1\}.\end{array}$$(42) converges almost surely to the solution x of Equation (8)(8) $$x(t)=\eta (0)+{\int }_{t_0}^{t}a(x(u),x(u-\delta (u)),u)du+{\int }_{t_0}^{t}b(x(u),x(u-\delta (u)),u)dw(u),$$(8) .

The proofs of these theorems are omitted since they are not different from the ones in previous section.

Notes

1 This is a corollary of the Hölder inequality for r > 1, in a way that

$${(\sum _{i=1}^n{b}_i)}^r\le {({(\sum _{i=1}^n{b}_i^r)}^{1/r}{(\sum _{i=1}^{n}1)}^{1-1/r})}^r=n^{r-1}\sum _{i=1}^n{b}_i^r,$$

since $x\mapsto x^r$ is an increasing function, $x\ge 0.$ Case $r\in \{0,1\}$ is trivial and the case when $r\in (0,1)$ follows from

$$\sum _{i=1}^n{b}_i={(\sum _{i=1}^n{b}_i^1)}^1\le {(\sum _{i=1}^n\sum _{i=1}^n)}^{1/r}.$$

Additional information

Funding

The research is supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, No. 451-03-9/2021-14/200124.