# Wong–Zakai approximation of regime-switching SDEs via rough path theory

Jasper Barr1111Corresponding author<EMAIL_ADDRESS>Giang T. Nguyen2,
Oscar Peralta3
1Mathematical Sciences Institute, Australian National University
2School of Mathematical Sciences, University of Adelaide
3School of Operations Research and Information Engineering, Cornell University

###### Abstract

This paper investigates the convergence of Wong–Zakai approximations to
regime-switching stochastic differential equations, generated by a collection
of finite-variation approximations to Brownian motion. We extend the results
of [NP21] to $\mathbb{R}^{d}$-valued RSSDE by utilising rough path theoretic
tools, acquiring the same modification of rate.

AMS 2020 Mathematics Subject Classification: 60H10, 60L90, 60F15.

Keywords: Wong–Zakai approximation, regime-switching stochastic differential
equations, rough path theory, strong convergence.

##  1 Introduction

The ability to incorporate uncertainty has become a standard aspect of modern
applied models. Indeed, the specific class of models known as stochastic
differential equations (SDE) has found a wide variety of applications in
modelling continuous phenomena across fields such as quantitative finance,
ecology, mathematical biology, dynamical systems, and so on. However, SDEs
suffer in that they only account for continuous sources of uncertainty,
typically driven by an $\mathbb{R}^{d}$-valued Brownian motion $B$. This
deficiency motivates the notion of regime-switching, in which the evolution of
the stochastic process $Y$ depends on an additional jump process $J$ with
countable state space $\mathcal{M}$:

$\mathrm{d}Y_{t}=\mu_{J_{t}}(t,Y_{t})\,\mathrm{d}t+\sigma_{J_{t}}(t,Y_{t})\,\mathrm{d}B_{t}\,,$

where $\\{\mu_{i}\\}_{i\in\mathcal{M}}$ and
$\\{\sigma_{i}\\}_{i\in\mathcal{M}}$ are a collection of drift and diffusion
vector fields indexed by the (countable) state space $\mathcal{M}$ of the jump
process $J$. Regime-switching SDEs (RSSDEs) allow one to account for discrete
changes in environment that influence the continuous dynamics of $Y$. Think of
a pressure sensor which is either operational or error-prone, or the
trajectory of a stock price in a “good” or “bad” market.

Regime-switching models have found a number of applications in quantitative
finance [DPR02, ZC13], in modelling rough volatility [YZ10], and more recently
in mean-field games [BDTY20].

Naturally, the introduction of the jump process complicates the process of
approximating such processes. Recently the first paper dealing with so-called
Wong–Zakai approximations of 1-dimensional RSSDEs emerged [NP21]. The
fundamental question addressed there is whether the strong convergence of
finite-variation approximations $\\{B^{\lambda}\\}_{\lambda\geq 0}$ to a
Brownian motion passes to convergence of the approximating RSSDEs
$\\{Y^{\lambda}\\}_{\lambda\geq 0}$ to $Y$, and how the rate of convergence is
affected. Recall that a collection of stochastic processes
$\\{B^{\lambda}\\}_{\lambda\geq 0}$ is said to converge strongly to $B$ with
rate function $\delta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ if for all $r$

$\mathbb{P}\big{(}\|B-B^{\lambda}\|_{\infty;[0,T]}\geq
k\delta(\lambda)\big{)}=o(\lambda^{-r})\,,$ (1)

where $k=k(r,T)$ is a constant dependent only on $r$ and $T$, and
$\|\cdot\|_{\infty;[0,T]}$ denotes the supremum norm over the interval
$[0,T]$. To address this question, Nguyen and Peralta [NP21] utilised a local
pathwise estimate of the form

$\|Y-Y^{\lambda}\|_{\infty;[0,T]}\leq L\|B-B^{\lambda}\|_{\infty;[0,T]}\,,$
(2)

where $L$ is a random variable with desirable probabilistic properties. The
estimate (2) allows one to locally control the error of the regime-switching
SDE approximation by the error in the driving processes, while the behaviour
of $L$ is exploited to extend the strong convergence of
$B^{\lambda}\rightarrow B$ to $Y^{\lambda}\rightarrow Y$ with a slightly worse
rate, in the sense that

$\mathbb{P}\left(\|Y-Y^{\lambda}\|_{\infty;[0,T]}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)=o(\lambda^{-r})$

for all $\varepsilon>0$ and $r>0$.

Unfortunately, (2) fails to hold in the multi-dimensional case, due to the
fact that the topology induced by the supremum norm is ill-suited to providing
such estimates in dimension $d>1$. In the higher-dimensional setting, one can
introduce oscillations at small scales to the approximations
$\\{B^{\lambda}\\}$ such that the limit
$\|B-B^{\lambda}\|_{\infty;[0,T]}\rightarrow 0$ remains unaffected, but it no
longer holds that $\|Y-Y^{\lambda}\|_{\infty;[0,T]}\rightarrow 0$. The
approximations $\\{Y^{\lambda}\\}_{\lambda\geq 0}$ can in fact converge to a
different limiting object given by $Y$, plus some correction term that
vanishes in the scalar case or when vector fields built from
$\\{\sigma_{i}\\}_{i\in\mathcal{M}}$ commute (see [IW92, Theorem 7.2] for
further discussion).

To resolve this problem we utilise rough path theory, introduced by Lyons
[Lyo98] in the mid 90s. Rough path theory avoids the aforementioned problem by
working with a stronger topology, induced by the so-called inhomogeneous rough
path metric $\rho_{p-var}$. This metric is built from a collection of path-
norms, one of which is the $p$-variation seminorm of a path
$X:[0,T]\rightarrow\mathbb{R}^{d}$

$\|X\|_{p-var;[0,T]}^{p}:=\sup_{\mathcal{P}\subset[0,T]}\sum_{t_{i}\in\mathcal{P}}|X_{t_{i}+1}-X_{t_{i}}|^{p}\,,\qquad
p\in[1,\infty),$ (3)

where $|\cdot|$ denotes the Euclidean norm and the supremum is taken over all
partitions $\mathcal{P}$ of $[0,T]$. Convergence of $B^{\lambda}$ to $B$ in
$p$-variation norm not only implies convergence in supremum norm, but it also
tracks small-scale oscillations. The payoff is that the local Lipschitz
continuity of the solution map is restored, albeit in the different function
space with a stronger topology that requires “higher order” information than
just the path $B$. We discuss this in further detail in Section 2.

To run a similar argument as in [NP21], we utilise a recent refinement
[CFRS16] of the Lipschitz estimate that applies to a class of Gaussian
processes (and approximations thereof), of which Brownian motion is included.
This leads us to the main result:

######  Theorem 1.1.

Under the conditions of Theorem 4.1 and Assumption 5.1, suppose that there
exists $\delta:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ with
$\lim_{\lambda\rightarrow\infty}\delta(\lambda)=0$ such that for all $r>0$

$\mathbb{P}\big{(}\rho_{p-var;[0,T]}(\mathbf{B}^{\lambda},\mathbf{B})\geq
k\delta(\lambda)\big{)}=o(\lambda^{-r})\,,$

where $k=k(r,T)>0$ is a constant dependent on $r$ and $T$ only. Then there
exists some constant $\beta=\beta(r,T)>0$ such that for all $\varepsilon>0$

$\mathbb{P}\left(\big{\|}Y-Y^{\lambda}\big{\|}_{\infty;[0,T]}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)=o(\lambda^{-r}).$
(4)

Assumption 5.1 matches Assumption 4 of [NP21], which ensures that the jumps of
$J$ are well-behaved by imposing bounds on the tail probability of the number
of jumps over the compact interval $[0,T]$. As noted in [NP21], this
assumption is not overly restrictive in that it allows for deterministic,
time-homogeneous Markov, time-inhomogeneous Markov, and semi-Markov processes.
The conditions of Theorem 4.1 are those required to guarantee the existence of
a unique solution to the corresponding rough regime-switching equation
associated to the regime-switching SDE of interest. Notably, this requires
much higher regularity than the standard assumption of local Lipschitz
continuity and linear growth from SDE theory.

The structure of the paper is as follows. We briefly recall in Section 2 some
main results and definitions from rough path theory. In Section 3 we formally
define the partitioning scheme introduced in [CLL13], and describe how the
functional of interest behaves under path concatenation and strong
convergence. In Section 4 we extend the improved Lipschitz estimate for RDEs
driven by Gaussian noise [CFRS16] to the regime-switching case. In Section 5,
we establish the strong convergence of approximations to regime-switching RDEs
by utilising tail behaviour of the greedy partition from Section 3 and the
pathwise estimates from Section 4. We conclude with a brief application to the
standard scheme of approximation via linear interpolation to a Brownian rough
path.

##  2 Preliminaries

In this section we provide some background information on rough path theory,
regime-switching SDEs and strong convergence.

###  2.1 Rough path theory

Rough path theory, first introduced in [Lyo98], provides a pathwise solution
theory for stochastic differential equations. To make rigorous the notion of
‘roughness’, we use the $p$-variation seminorm of a path
$X:[0,T]\rightarrow\mathbb{R}^{d}$ as defined in (3). If one is interested in
solving equations of the form

$\mathrm{d}Y_{t}=Y_{t}\,\mathrm{d}X_{t},\qquad Y_{0}=y,$ (5)

for paths $X:[0,T]\rightarrow\mathbb{R}^{d}$ and
$Y:[0,T]\rightarrow\mathcal{L}(\mathbb{R}^{d},\mathbb{R}^{e})$ (the space of
linear maps from $\mathbb{R}^{d}$ into $\mathbb{R}^{e}$), $p$-variation turns
out to be a natural scale with which to measure the roughness of paths. As
Young [You36] discovered, the integral $\int_{0}^{t}Y_{s}\,\mathrm{d}X_{s}$ is
defined classically via Riemann–Stieltjes integration if $Y$ and $X$ are of
bounded $p$ and $q$-variation, with $\frac{1}{p}+\frac{1}{q}>1$. This
condition is sharp in the sense that examples of paths exist satisfying
$p^{-1}+q^{-1}=1$ such that the Riemann–Stieltjes approximations of
$\int_{0}^{t}Y_{s}\,\mathrm{d}X_{s}$ fail to converge. In the simplified
setting where $Y_{t}=f(X_{t})$ for some smooth
$f:\mathbb{R}^{d}\rightarrow\mathcal{L}(\mathbb{R}^{d},\mathbb{R}^{e})$, this
implies that $X$ must be a path with finite $p$-variation for $p<2$.
Remarkably, this is the exact threshold which the sample path regularity of
Brownian motion fails to meet [FV10]. Thus, we call a path rough if it is of
finite $p$-variation only for some $p>2$.

To deal with this problem when the driving path $X$ is a Brownian motion,
stochastic calculus exploits the desirable probabilistic properties of
Brownian motion by considering convergence of the approximates
$\sum_{[u,v]\in\mathcal{P}}f(B_{u})(B_{v}-B_{u})$ in probability, rather than
almost surely, as the mesh size of the partition $\mathcal{P}$ tends to zero.
Rough path theory, on the other hand, takes a different approach.

The perspective of rough path theory is that the limit

$\lim\limits_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}f(X_{u})(X_{v}-X_{u})$

fails to converge because the zeroth order approximation $f(X_{r})\approx
f(X_{u})$ for $r\in[u,v]$ is not good enough to account for the roughness of
the driving path $X$. If instead we take the first order approximation
$f(X_{r})\approx f(X_{u})+Df(X_{u})(X_{r}-X_{u})$, the approximation of our
integral becomes

$\displaystyle\int_{0}^{t}f(X_{s})\,\mathrm{d}X_{s}$
$\displaystyle=\lim\limits_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}\int_{u}^{v}\left(f(X_{u})+Df(X_{u})\big{(}X_{r}-X_{u}\big{)}\right)\mathrm{d}X_{r}$
$\displaystyle=\lim\limits_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}f(X_{u})\big{(}X_{v}-X_{u}\big{)}+Df(X_{u})\int_{u}^{v}\big{(}X_{r}-X_{u}\big{)}\otimes\mathrm{d}X_{r},$
(6)

where $\otimes$ denotes the tensor product. When $X$ has finite $p$-variation
with $p<2$, so that $\int X\otimes\mathrm{d}X$ is classically defined, then
the above approximation converges to the usual Riemann–Stieltjes integral. In
the rough setting with $p\in[2,3)$, the iterated integral $\int
X\otimes\mathrm{d}X$ is no longer uniquely defined as a function of the path
$X$, and so we instead postulate values for the iterated integral by
specifying a two-parameter path
$\mathbb{X}:[0,T]^{2}\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{d}$. Any
choice of $\mathbb{X}$ satisfying an algebraic condition known as Chen’s
relation and having finite $p/2$-variation is a suitable candidate, with
different choices yielding different limits for (6). Equation (6) is then
referred to as the rough path integral with respect to the $p$-rough path
$\mathbf{X}=\big{(}X,\mathbb{X}\big{)}$. This approach works not only for
functions of the path $X$, but also for a class of paths known as _controlled
rough paths_.

######  Definition 2.1.

Let $X:[0,T]\rightarrow\mathbb{R}^{d}$ be a path of finite $p$-variation. We
say that $Y:[0,T]\rightarrow\mathbb{R}^{e}$ is _controlled by $X$_ if there
exists a path
$Y^{\prime}:[0,T]\rightarrow\mathcal{L}(\mathbb{R}^{d},\mathbb{R}^{e})$ such
that

$Y_{s,t}=Y_{s}^{\prime}X_{s,t}+R_{s,t}^{Y},$

where $Y$ and $Y^{\prime}$ have finite $p$-variation, and the remainder term
$R^{Y}$ has finite $p/2$-variation. We denote the space of all paths
controlled by $X$ by $\mathscr{D}^{p/2}_{X}$.

The space $\mathscr{D}^{p/2}_{X}$ is a Banach space under the norm

$(Y,Y^{\prime})\mapsto|Y_{0}|+|Y_{0}^{\prime}|+\|Y,Y^{\prime}\|_{X}^{p/2}$

with

$\|Y,Y^{\prime}\|_{X}^{p/2}:=\|Y\|_{p-var;[0,T]}+\|R^{Y}\|_{p/2-var;[0,T]}.$

This opens the door to fixed-point techniques, allowing us to establish the
existence and uniqueness of solutions to _rough_ differential equations by
finding fixed-points in $\mathscr{D}^{p/2}_{X}$.

Although this procedure of making a particular choice of $\mathbb{X}$ may seem
arbitrary, there are some natural choices when $X=B$ is a Brownian motion. In
this setting, we may simply enhance $B$ with its stochastic iterated integral
$\mathbb{B}$, in either the Itô or Stratonovich sense. Here, we work with the
Stratonovich iterated integrals $\int B\otimes\,\mathrm{d}B$, and refer to the
object $\mathbf{B}:=\big{(}B,\mathbb{B}\big{)}$ as the Brownian rough path, or
enhanced Brownian motion.

Given the importance of the $p$-variation of a path highlighted above, we
introduce the _inhomogeneous $p$-rough path metric_

$\rho_{p-var}(\mathbf{X},\mathbf{Y}):=\max\left\\{\|X-Y\|_{p-var;[0,T]},\|\mathbb{X}-\mathbb{Y}\|_{p/2-var;[0,T]}\right\\},$

with

$\|\mathbf{X}\|_{p/2-var;[0,T]}=\left(\sup_{\mathcal{P}}\sum_{[u,v]\in\mathcal{P}}|\mathbb{X}_{u,v}|^{p/2}\right)^{2/p}.$

Finally, we introduce the _$\gamma$ -Lipschitz_ norm of a map between two
normed spaces. In the following, let $\lfloor\gamma\rfloor$ be the largest
integer which is _strictly_ smaller than $\gamma$, so that we may always write
$\gamma=\lfloor\gamma\rfloor+\\{\gamma\\}$ with $\\{\gamma\\}\in(0,1]$.

######  Definition 2.2 ($\gamma$-Lipschitz norm, [FV10]).

A map $V:E\rightarrow F$ between two normed spaces $E$ and $F$ is called
$\gamma$-Lipschitz (in the sense of E. Stein), in symbols

$V\in\mathrm{Lip}^{\gamma}(E,F)\text{ or simply
}V\in\mathrm{Lip}^{\gamma}\text{ if }E=F,$

if $V$ is is $\lfloor\gamma\rfloor$ times continuously differentiable and such
that there exists a constant $0\leq M<\infty$ such that the supremum norm of
its $k$th-derivatives $k=0,...,\lfloor\gamma\rfloor$, and the Hölder norm of
its $\lfloor\gamma\rfloor$th derivative are bounded by $M$. The smallest $M$
satisfying the above conditions is the $\gamma$-Lipschitz norm of $V$ and
denoted $|V|_{\mathrm{Lip}^{\gamma}}$.

The payoffs to this approach are numerous. Whilst the solution map associated
with the SDE $\mathrm{d}Y_{t}=f(B_{t})\,\mathrm{d}B_{t}$, $Y_{0}=y_{0}$ is in
general only measurable, the solution map associated to a rough differential
equation $\mathrm{d}Y_{t}=V(Y_{t})\,\mathrm{d}\mathbf{X}_{t}$, $Y_{0}=y_{0}$
is locally Lipschitz continuous (under some standard conditions) with respect
to the initial condition, driving path, and the vector field $V$. This leads
to error estimates of the form

$\|Y-\widetilde{Y}\|_{\infty}\leq
C\big{(}|y_{0}-\widetilde{y}_{0}|+|V-\widetilde{V}|_{Lip^{\gamma}}+\rho_{p-var}(\mathbf{X},\widetilde{\mathbf{X}})\big{)}.$
(7)

In particular, this estimate may be applied when we consider approximations of
rough differential equations (RDEs) of the form (5). Given a rough path
$\mathbf{X}=(X,\mathbb{X})$ and a collection of finite-variation
approximations $\\{X^{\lambda}\\}_{\lambda\geq 0}$ of $X$, we can lift these
approximations to rough paths by enhancing them with the Riemann–Stieltjes
integrals
$\mathbb{X}^{\lambda}_{s,t}:=\int_{s}^{t}X_{s,r}^{\lambda}\otimes\mathrm{d}X^{\lambda}_{r}$.
Establishing convergence of
$\mathbf{X}^{\lambda}:=(X^{\lambda},\mathbb{X}^{\lambda})$ to $\mathbf{X}$ in
rough path metric then implies convergence of $Y^{\lambda}\rightarrow Y$ via
Equation (7), where $Y^{\lambda}$ is the solution of
$\mathrm{d}Y_{t}=f(Y_{t})\mathrm{d}\mathbf{X}_{t}$ and $Y^{\lambda}_{t}$ is
the solution of
$\mathrm{d}Y_{t}^{\lambda}=f(Y^{\lambda}_{t})\mathrm{d}\mathbf{X}^{\lambda}_{t}$.

######  Remark 2.3.

The supremum norm in Equation (7) can generally be replaced by the
$p$-variation rough path metric. However, we return to the weaker supremum
topology as it is a standard way to measure approximation error.

###  2.2 Gaussian rough paths

Unlike the semimartingale setting, there is no notion of stochastic
integration we could use to lift a general $\mathbb{R}^{d}$-valued Gaussian
process $X$ to a rough path $\mathbf{X}$. Recalling that the law of a
(centred) Gaussian process $X_{t}=(X^{1}_{t},...,X^{d}_{t})$ is completely
determined by its covariance function $R(s,t)=\mathbb{E}[X_{s}\otimes X_{t}]$,
it is perhaps not surprising that the question of finding a natural rough path
lift boils down to the regularity of the rectangular increments of
$\mathbb{R}^{d}$.

######  Definition 2.4.

Let $X$ be a centred $\mathbb{R}^{d}$-valued Gaussian process with covariance
function $R:[0,T]^{2}\rightarrow\mathbb{R}^{d\times d}$. The rectangular
increments of $R$ are given by

$R\begin{pmatrix}s&,&t\\\
s^{\prime}&,&t^{\prime}\end{pmatrix}:=\mathbb{E}[X_{s,t}\otimes
X_{s^{\prime},t^{\prime}}].$

Given a set $I\times I^{\prime}\subset[0,T]^{2}$, the 2D $\varrho$-variation
of $R$ is given by

$\|R\|_{\varrho,I\times
I^{\prime}}=\left(\sup\limits_{\begin{subarray}{c}\mathcal{P}\subset I,\\\
\mathcal{P}^{\prime}\subset
I^{\prime}\end{subarray}}\sum_{\begin{subarray}{c}[s,t]\in\mathcal{P},\\\
[s^{\prime},t^{\prime}]\in\mathcal{P^{\prime}}\end{subarray}}\left|R\begin{pmatrix}s&,&t\\\
s^{\prime}&,&t^{\prime}\end{pmatrix}\right|^{\varrho}\right)^{1/\varrho}$

Under the condition that $X$ is a Gaussian process with covariance $R$ of
finite 2D $\varrho$-variation for $\varrho<2$, we may define the integral of
$X^{i}$ against $X^{j}$ as the $L^{2}$-limit

$\int_{s}^{t}X^{i}_{s,r}\,\mathrm{d}X^{j}_{r}=\lim\limits_{|\mathcal{P}|\rightarrow
0}\sum_{[u,v]\in\mathcal{P}}X^{i}_{s,v}X^{j}_{u,v}.$ (8)

Further to this, we have an estimate of the form

$\mathbb{E}\left[\left(\int_{s}^{t}X^{i}_{s,r}\,\mathrm{d}X^{j}_{r}\right)^{2}\right]\leq
C\|R\|_{\varrho;[s,t]^{2}},$

where $C=C(\varrho)$ [Proposition 10.3, [HF14]]. Defining

$\mathbb{X}^{i,j}_{s,t}=\int_{s}^{t}X^{i}_{s,r}\
\mathrm{d}X_{r}^{j},\quad\mathbb{X}^{i,i}_{s,t}=\frac{1}{2}\left(X^{i}_{s,t}\right)^{2},\quad\text{and}\quad\mathbb{X}_{s,t}^{i,j}=-\mathbb{X}_{s,t}^{i,j}+X^{i}_{s,t}X^{j}_{s,t},$

it follows that if there exists $M$ and
$\varrho\in\big{[}1,\frac{3}{2}\big{)}$ such that for every
$i\in\\{1,...,d\\}$ and $0\leq s\leq t\leq T$

$\|R_{X^{i}}\|_{\varrho;[s,t]^{2}}\leq M|t-s|^{1/\varrho},$

then the process $(X,\mathbb{X})$ is almost surely a $p$-rough path for
$p>2\varrho$ [Theorem 10.4, [HF14]]. This result can be pushed further to
$\varrho\in\big{[}1,2\big{)}$, but requires a third level of the rough path to
be defined (see Chapter 15 of [FV10]).

######  Example 2.5 (Brownian motion).

Given a Brownian motion $B=(B^{1},...,B^{d})$, the covariance function has the
form $R(s,t)=(s\wedge t)I_{d}$. Since $B^{i}$ and $B^{j}$ are centred and
independent for $i\neq j$, $\mathbb{E}[X^{i}_{s,t}X^{j}_{s,t}]=0$. The on-
diagonal terms then take the form

$\displaystyle\mathbb{E}[X^{i}_{s,t}X^{i}_{s^{\prime},t^{\prime}}]$
$\displaystyle=\mathbb{E}[X^{i}_{t^{\prime}}X^{i}_{t}]-\mathbb{E}[X_{s^{\prime}}^{i}X^{i}_{t}]+\mathbb{E}[X_{s}^{i}X_{s^{\prime}}^{i}]-\mathbb{E}[X_{s}^{i}X_{t^{\prime}}^{i}]$
$\displaystyle=\min\\{t,t^{\prime}\\}-\min\\{s^{\prime},t\\}+\min\\{s,s^{\prime}\\}-\min\\{s,t^{\prime}\\}$
$\displaystyle=|[s^{\prime},t^{\prime}]\cap[s,t]|,$

showing that $B$ has finite 2D $\varrho$-variation for $\varrho=1$. Lifting
$B$ to a rough path $\mathbf{B}$ using the Gaussian framework will yield a
process $\mathbf{B}$ which is indistinguishable from the Stratonovich lift
$(B,\mathbb{B}^{\text{Strat}})$ [HF14].

######  Example 2.6 (Fractional Brownian motion).

Recalling fractional Brownian motion $B^{H}$ for
$H\in\big{(}0,\frac{1}{2}\big{]}$ is the $\mathbb{R}^{d}$-valued process with
covariance function

$R^{H}(s,t)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right),$

[FV10] show that $B^{H}$ has finite 2D $\varrho$-variation for
$\varrho=\frac{1}{2H}$.

###  2.3 Regime-switching SDE via rough paths

The consistency between stochastic and rough integration, when both are
defined, is a well-known feature of rough path theory [FV10]. Indeed, the
standard regularity requirements that are imposed to guarantee the existence
of a unique solution to a given RDE are enough to ensure the existence of a
unique strong solution to the corresponding SDE.

######  Definition 2.7 (Regime-switching SDE).

Let $J$ be a jump process with finite activity on compact intervals and
countable state space $\mathcal{E}$. Let
$\mu:\mathbb{R}_{+}\times\mathbb{R}^{m}\times\mathcal{E}\rightarrow\mathbb{R}^{m}$
and
$\sigma:\mathbb{R}_{+}\times\mathbb{R}^{m}\times\mathcal{E}\rightarrow\mathcal{L}(\mathbb{R}^{d},\mathbb{R}^{m})$,
and write $\mu(t,x,i)=\mu_{i}(t,x)$, $\sigma(t,x,i)=\sigma_{i}(t,x)$. Finally,
let $B=(B^{1},...,B^{d})$ be an $\mathbb{R}^{d}$-valued Brownian motion
independent of $J$ and $Y_{0}\in\mathbb{R}^{m}$. Then the equation

$Y_{t}=Y_{0}+\int_{0}^{t}\mu_{J_{s}}(s,Y_{s})\,\mathrm{d}s+\int_{0}^{t}\sigma_{J_{s}}(s,Y_{s})\,\mathrm{d}B_{s}$
(9)

is referred to as a regime-switching stochastic differential equation.

######  Remark 2.8.

One can express the solution as a coupled process
$\big{\\{}(Y_{t},J_{t})\big{\\}}_{t\geq 0}$ where $J$ is given by the
stochastic integral with respect to a Poisson random measure $\mathfrak{p}$
[YZ10]. This can be useful in refining some arguments when working with jump
processes $J$ with probabilistic behaviour depending on the trajectory of $Y$,
but for our purposes (9) will be more useful.

######  Remark 2.9.

Under the conditions on $J$ in Definition 2.7, with suitable conditions on
$\mu$ and $\sigma$, there exists a unique strong solution for the SDEs

$Y_{t}^{i}=Y_{0}^{i}+\int_{0}^{t}\mu_{i}(s,Y_{s}^{i})\,\mathrm{d}s+\int_{0}^{t}\sigma_{i}(s,Y_{s}^{i})\,\mathrm{d}B_{s}\,.$

Considering the stopping times $\tau_{0}=0$ and
$\tau_{k+1}=\inf\\{t>\tau_{k}:J_{t}\neq J_{\tau_{k}}\\}$, we may write for
$t\in[\tau_{k},\tau_{k+1}]$

$\displaystyle Y_{t}$
$\displaystyle=Y_{0}+\int_{0}^{t}\mu_{J_{s}}(s,Y_{s})\,\mathrm{d}s+\int_{0}^{t}\sigma_{J_{s}}(s,Y_{s})\,\mathrm{d}B_{s}$
$\displaystyle=Y_{0}+\int_{0}^{\tau_{k}}\mu_{J_{s}}(s,Y_{s})\,\mathrm{d}s+\int_{\tau_{k}}^{t}\mu_{J_{s}}(s,Y_{s})\,\mathrm{d}s+\int_{0}^{\tau_{k}}\sigma_{J_{s}}(s,Y_{s})\,\mathrm{d}B_{s}+\int_{\tau_{k}}^{t}\sigma_{J_{s}}(s,Y_{s})\,\mathrm{d}B_{s}$
$\displaystyle=Y_{\tau_{k}}+\int_{\tau_{k}}^{t}\mu_{J_{\tau_{k}}}(s,Y_{s})\,\mathrm{d}s+\int_{\tau_{k}}^{t}\sigma_{J_{\tau_{k}}}(s,Y_{s})\,\mathrm{d}B_{s}\,.$

The almost sure finite jump activity of $J$ then implies that
$\lim_{k\rightarrow\infty}\tau_{k}=\infty$, so that the above holds for all
$t\in[0,T]$, $T>0$. Thus, we can recover solutions of (9) by concatenating
solutions of SDEs within known regimes.

We now introduce the rough equivalent of regime-switching SDEs:

######  Definition 2.10.

Let $J$ be a jump process with a.s. finite activity on compact intervals and
countable state space $\mathcal{E}$. Let $p,\gamma$ be such that $2<p<\gamma$.
Assume that

  1. 1.

$V^{j}=(V_{i}^{j})_{1\leq i\leq d}$ is a collection of
$\mathrm{Lip}^{\gamma}(\mathbb{R}^{e})$ vector fields for $j\in\mathcal{E}$;

  2. 2.

$\sup_{j\in\mathcal{E}}|V^{j}|_{\mathrm{Lip}^{\gamma}}<\infty$; and

  3. 3.

$\mathbf{X}$ is a Gaussian rough path.

Let $(\tau_{k})_{k=0}^{N^{J}+1}$ be the jump times of $J$ on $[0,T]$ (with
$N^{J}$ denoting the number of jumps of $J$) and let
$\\{Y^{k}\\}_{k=0}^{N^{J}}$ be the RDE solutions to

$\displaystyle\mathrm{d}Y_{t}^{0}$
$\displaystyle=y_{0}+\int_{0}^{t}V_{J_{0}}(Y_{s}^{0})\,\mathrm{d}\mathbf{X}_{s},$
$\displaystyle t$ $\displaystyle\in[0,\tau_{1}],\quad y_{0}\in\mathbb{R}^{e},$
$\displaystyle\mathrm{d}Y_{t}^{k}$
$\displaystyle=Y_{\tau_{k}}^{k-1}+\int_{\tau_{k}}^{t}V_{J_{\tau_{k}}}(Y_{s}^{k})\,\mathrm{d}\mathbf{X}_{s}\,,$
$\displaystyle t$ $\displaystyle\in[\tau_{k},\tau_{k+1}]\,,\quad 1\leq k\leq
N.$

Then the path $\\{Y_{t}\\}_{t\geq 0}$ constructed by concatenating the
individual paths $\\{Y^{k}\\}_{k=0}^{N^{J}}$ is said to solve the regime-
switching rough differential equation driven by $\mathbf{X}$ and $J$.

It remains to prove consistency between stochastic and rough regime-switching
differential equations. Given that solutions to these equations are determined
via fixed-points of the rough and stochastic integral respectively, we prove
consistency for general regime-switching rough integration and regime-
switching stochastic integration.

######  Theorem 2.11.

Let $\mathbf{B}=\big{(}B,\mathbb{B}\big{)}$ denote the Itô Brownian rough
path, $J$ be a Markov chain with finite state space $\mathcal{E}$ with
$\mathbb{E}[N^{J}]<\infty$, and suppose
$\big{(}X^{i},(X^{i})^{\prime}\big{)}\in\mathscr{D}_{B(\omega)}^{p/2}$ almost
surely for each $i\in\mathcal{E}$. Then the regime-switching rough integral

$\int_{0}^{t}X^{J_{s}}\,\mathrm{d}\mathbf{B}_{s}:=\lim_{n\rightarrow\infty}\sum_{[u,v]\in\mathcal{P}_{n}}\left(X_{u}^{J_{u}}B_{u,v}+\big{(}X^{J_{u}}_{u}\big{)}^{\prime}\mathbb{B}_{u,v}\right),$
(10)

exists, with the limit taken along any sequence
$\\{\mathcal{P}_{n}\\}_{n=1}^{\infty}$ with mesh size tending to 0. In the
case that $X^{i}$ and $(X^{i})^{\prime}$ are adapted for every
$i\in\mathcal{E}$, then

$\int_{0}^{t}X^{J_{s}}\,\mathrm{d}\mathbf{B}_{s}=\int_{0}^{t}X^{J_{s}}\,\mathrm{d}B_{s},$

almost surely.

###### Proof.

In the following we assume that

$\mathbb{E}\left[\sup_{i\in\mathcal{E}}\|X^{i},\big{(}X^{i}\big{)}^{\prime}\|_{B}^{p/2}\right]<\infty,$

noting that if this is not the case we may proceed by localisation. Having
defined the regime-switching rough integral for a fixed trajectory of
$J(\omega)$ and $\mathbf{B}(\omega)$ via concatenation of the rough integral
over periods when $J$ is in a fixed regime, we must show that the
approximation in Equation (10) converges to the same path. To this end, let
$I^{c}_{t}$ denote the integral achieved by concatenation and $I_{t}$ by the
proposed limit (10). Let $(\tau_{k})_{k=0}^{N^{J}+1}$ denote the jump times of
$J$ on the interval $[0,T]$. Noting that $I_{t}=I_{t}^{c}$ for
$t\in[0,\tau_{1}]$, we consider $t\in[\tau_{1},\tau_{2})$. We must deal with
the fact that for any sequence of partitions
$\\{\mathcal{P}_{n}\\}_{n=1}^{\infty}$, there will generally be consecutive
points $u^{n},v^{n}\in\mathcal{P}_{n}$ with $u^{n}<\tau_{1}<v^{n}$ such that
we will be approximating the regime-switching rough integral by the wrong
control. That is, if $J_{u^{n}}=i$ and $J_{v^{n}}=j$, we will be approximating
$I_{t}$ over the interval $[u^{n},v^{n})$ as if the jump process was in state
$i$ over the whole interval. Adding and subtracting the ‘correct’ control we
see that

$\displaystyle\bigg{|}\sum_{[u,v]\in\mathcal{P}_{n}}$
$\displaystyle\left(X_{u}^{J_{u}}B_{u,v}+\big{(}X^{J_{u}}_{u}\big{)}^{\prime}\mathbb{B}_{u,v}\right)\pm\left(X_{\tau_{1}}^{J_{\tau_{1}}}B_{\tau_{1},v^{n}}+\big{(}X_{\tau_{1}}^{J_{\tau_{1}}}\big{)}^{\prime}\mathbb{B}_{\tau_{1},v^{n}}\right)\bigg{|}$
$\displaystyle\leq\bigg{|}\sum_{[u,v]\in\mathcal{P}_{n}^{\prime}}X^{J_{u}}_{u}B_{u,v}+\big{(}X^{J_{u}}_{u}\big{)}^{\prime}\mathbb{B}_{u,v}\bigg{|}+\left|\big{(}X_{u^{n}}^{i}-X_{v^{n}}^{j}\big{)}B_{\tau_{1},v^{n}}\right|$
$\displaystyle\qquad\qquad+\left|\left(\big{(}X^{i}_{u^{n}}\big{)}^{\prime}-\big{(}X^{j}_{\tau_{1}}\big{)}^{\prime}\right)\mathbb{B}_{\tau_{1},v^{n}}\right|+\left|\big{(}X^{i}_{u^{n}}\big{)}^{\prime}B_{u^{n},\tau_{1}}\otimes
B_{\tau_{1},v^{n}}\right|,$

where $\mathcal{P}_{n}^{\prime}=\mathcal{P}_{n}\cup\\{\tau_{1}\\}$ and we have
used the fact that $B$ is additive and $\mathbb{B}$ satisfies Chen’s relation
to split the terms. Since $\tau_{1}\in\mathcal{P}_{n}^{\prime}$ for every $n$,
the first term converges to $I^{c}_{t}$. The other three terms do not affect
the limit, which we see from the bounds

$\displaystyle\left|\big{(}X_{u^{n}}^{i}-X_{v^{n}}^{j}\big{)}B_{\tau_{1},v^{n}}\right|$
$\displaystyle\leq
C\|B\|_{\alpha}|v^{n}-\tau_{1}|^{\alpha}=O\big{(}|\mathcal{P}_{n}^{\prime}|^{\alpha}\big{)}$
$\displaystyle\left|\left(\big{(}X^{i}_{u^{n}}\big{)}^{\prime}-\big{(}X^{j}_{\tau_{1}}\big{)}^{\prime}\right)\mathbb{B}_{\tau_{1},v^{n}}\right|$
$\displaystyle\leq
C\|\mathbb{B}\|_{2\alpha}|v^{n}-\tau_{1}|^{2\alpha}=O\big{(}|\mathcal{P}_{n}^{\prime}|^{\alpha}\big{)}$
$\displaystyle\left|\big{(}X^{i}_{u^{n}}\big{)}^{\prime}B_{u^{n},\tau_{1}}\otimes
B_{\tau_{1},v^{n}}\right|$ $\displaystyle\leq
C\|B\|_{\alpha}^{2}|v^{n}-\tau_{1}|^{\alpha}|\tau_{1}-u^{n}|^{\alpha}=O\big{(}|\mathcal{P}_{n}^{\prime}|^{\alpha}\big{)},$

for

$C:=2\max\left\\{\sup_{i\in\mathcal{E}}\|X^{i}\|_{\infty},\,\sup_{i\in\mathcal{E}}\|(X^{i})^{\prime}\|_{\infty}\right\\}<\infty.$

We repeat this process for $[\tau_{2},\tau_{3})$, $[\tau_{3},\tau_{4})$ and so
on, noting that since $N^{J}<\infty$ almost surely we can cover $[0,T]$ in a
finite number of steps, showing that $I_{t}=I_{t}^{c}$ almost surely for any
$t\in[0,T]$.

Next, we show that the rough path and stochastic integrals coincide almost
surely. To do so, we use the standard technique of showing that our rough path
approximation converges in $L^{2}$ to the Itô integral
$\int_{0}^{t}X^{J_{s}}\mathrm{d}B_{s}$. Then, since the approximation
converges to the rough path integral almost surely and the stochastic integral
in $L^{2}$, these limits must coincide. We consider the specific partition
$\mathcal{P}_{n}=\\{tk/n\\}_{k=0}^{n}$ of $[0,t]$. Consider the $L^{2}$ error

$\displaystyle\mathbb{E}[I_{t}^{n}]$
$\displaystyle=\mathbb{E}\bigg{[}\bigg{|}\int_{0}^{t}X^{J_{s}}\mathrm{d}B_{s}-\sum_{k=0}^{n-1}X_{t_{k}^{n}}^{J_{t_{k}}}B_{t_{k}^{n},t_{k+1}^{n}}+(X_{t_{k}^{n}}^{J_{t_{k}}})^{\prime}\mathbb{B}_{t_{k}^{n},t_{k+1}^{n}}\bigg{|}^{2}\bigg{]}$
$\displaystyle=\mathbb{E}\left[\left|\sum_{k=0}^{n-1}\int_{t_{k}^{n}}^{t_{k+1}^{n}}\left(X^{J_{s}}_{s}-X_{t_{k}^{n}}^{J_{t_{k}}}-(X_{t_{k}^{n}}^{J_{t_{k}}})^{\prime}B_{t_{k}^{n},s}\right)\mathrm{d}B_{s}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\left|\sum_{k=0}^{n-1}\int_{t_{k}}^{t_{k+1}}\left(X^{J_{s}}_{s}-X_{t_{k}^{n}}^{J_{t_{k}}}-(X_{t_{k}^{n}}^{J_{t_{k}}})^{\prime}B_{t_{k}^{n},s}\right)\mathrm{d}B_{s}\right|^{2}\big{|}\mathcal{F}_{t}^{J}\right]\right],$

where in the final step we condition on the $\sigma$-algebra
$\mathcal{F}^{J}_{t}:=\sigma(J_{s},0\leq s\leq t)$ generated by
$\\{J_{t}\\}_{t\in[0,T]}$. Noting that the product of the stochastic integral
over disjoint intervals has expectation zero and applying the (multivariate)
Itô isometry to the remaining terms, we get

$\displaystyle\mathbb{E}[I_{t}^{n}]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\sum_{k=0}^{n-1}\int_{t_{k}^{n}}^{t_{k+1}^{n}}\left|X_{s}^{J_{s}}-X_{t_{k}^{n}}^{J_{t_{k}^{n}}}-(X_{t_{k}^{n}}^{J_{t_{k}^{n}}})^{\prime}B_{t_{k}^{n},s}\right|^{2}\mathrm{d}s\big{|}\mathcal{F}_{t}^{J}\right]\right].$

Now, if $J_{s}=i$ for all $s\in[t_{k}^{n},t_{k+1}^{n}]$, then the integrand is
controlled by $B$ for the fixed regime over the whole interval, meaning that
$X^{i}_{t_{k}^{n},s}=(X^{i}_{t_{k}^{n}})^{\prime}B_{t_{k}^{n},s}+R_{t_{k}^{n},s}^{i}$,
where $R^{i}\in C^{2\alpha}$ is the remainder term. In such a case, we can
bound the integral by writing

$\displaystyle\int_{t_{k}^{n}}^{t_{k+1}^{n}}\left|X_{s}^{i}-X_{t_{k}^{n}}^{i}-(X_{t_{k}^{n}}^{i})^{\prime}B_{t_{k}^{n},s}\right|^{2}\mathrm{d}s$
$\displaystyle=\int_{t_{k}^{n}}^{t_{k+1}^{n}}|R_{t_{k}^{n},s}^{i}|^{2}\mathrm{d}s$
$\displaystyle\leq\|R^{i}\|_{2\alpha}\int_{t_{k}^{n}}^{t_{k+1}^{n}}|s-t_{k^{n}}|^{4\alpha}\mathrm{d}s$
$\displaystyle\leq\sup_{i\in\mathcal{E}}\|R^{i}\|_{2\alpha}\cdot\frac{1}{4\alpha+1}|t^{n}_{k+1}-t_{k}^{n}|^{1+4\alpha}$
$\displaystyle\leq\sup_{i\in\mathcal{E}}\|R^{i}\|_{2\alpha}\cdot\frac{1}{4\alpha+1}\left(\frac{t}{n}\right)^{1+4\alpha}$

In the case that a jump occurs at time $\tau$ to state $j$ in the interval
$[t_{k}^{n},t_{k+1}^{n}]$, we decompose the integral via

$\displaystyle\int_{t_{k}^{n}}^{t_{k+1}^{n}}\left|X_{s}^{J_{s}}-X_{t_{k}^{n}}^{i}-(X_{t_{k}^{n}}^{i})^{\prime}B_{t_{k}^{n},s}\right|^{2}\mathrm{d}s$
$\displaystyle=\int_{t_{k}^{n}}^{\tau}\left|X_{s}^{i}-X_{t_{k}^{n}}^{i}-(X_{t_{k}^{n}}^{i})^{\prime}B_{t_{k}^{n},s}\right|^{2}\mathrm{d}s$
$\displaystyle\qquad+\int_{\tau}^{t_{k+1}^{n}}\left|X_{s}^{j}-X_{t_{k}^{n}}^{i}-(X_{t_{k}^{n}}^{i})^{\prime}B_{t_{k}^{n},s}\right|^{2}\mathrm{d}s.$

The first term can be bounded as before, while for the second we observe that

$\displaystyle\int_{\tau}^{t_{k+1}^{n}}\left|X_{s}^{j}-X_{t_{k}^{n}}^{i}-(X_{t_{k}^{n}}^{i})^{\prime}B_{t_{k}^{n},s}\right|^{2}\mathrm{d}s$
$\displaystyle=\int_{\tau}^{t_{k+1}^{n}}\left|(X_{s}^{j}-X_{s}^{i})+R_{t_{k}^{n},s}^{i}\right|^{2}\mathrm{d}s$
$\displaystyle\leq
2\sup_{i\in\mathcal{E}}\|X^{i}\|_{\infty}^{2}|t_{k+1}^{n}-\tau|$
$\displaystyle\qquad\quad+\sup_{i\in\mathcal{E}}\|R^{i}\|_{p/2}\frac{|t_{k+1}^{n}-\tau|^{4\alpha+1}}{4\alpha+1}$
$\displaystyle\leq
C\left(\frac{t}{n}+\frac{1}{4\alpha+1}\left(\frac{t}{n}\right)^{4\alpha+1}\right).$

Since this occurs in at most $N^{J}$ of the $n$ intervals, we can bound
$\mathbb{E}[I_{n}]$ (absorbing constants dependent on $\alpha$ and $t$) by

$\displaystyle\mathbb{E}[I_{n}]$ $\displaystyle\leq
C\mathbb{E}\left[\sup_{i\in\mathcal{E}}\|R^{i}\|_{2/p}\left(n^{-4\alpha}+\frac{N^{J}}{n}+\frac{N^{J}}{n^{4\alpha+1}}\right)\right]$
$\displaystyle\leq
C\mathbb{E}\left[\sup_{i\in\mathcal{E}}\|X^{i},(X^{i})^{\prime}\|_{B}^{p/2}\right]\left(n^{-4\alpha}+\mathbb{E}[N^{J}]\big{(}n^{-1}+n^{-(1+4\alpha)}\big{)}\right)\rightarrow
0,$

provided $\mathbb{E}[N^{J}]<\infty$ and
$\mathbb{E}\left[\sup_{i\in\mathcal{E}}\|X^{i},(X^{i})^{\prime}\|_{B}^{p/2}\right]<\infty$.
∎

###  2.4 Strong convergence

Let $(M,d)$ be a metric space, and suppose that
$\\{X^{\lambda}\\}_{\lambda\geq 0}$ is a family of $M$-valued random variables
with time horizon $[0,T]$. We say that $X^{\lambda}$ converges strongly to an
$M$-valued random variable $X$ if

$\mathbb{P}\left(\lim\limits_{\lambda\rightarrow\infty}d(X^{\lambda},X)=0\right)=1\,.$

Further, we say that $X^{\lambda}$ converges with rate $\delta(\lambda)$ if
there exists a function $\delta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ and
constant $\alpha(q,T)$ such that

$\mathbb{P}\left(d(X^{\lambda},X)\geq\alpha\delta(\lambda)\right)=o(\lambda^{-q})\,,$

for all $q>0$. Of particular interest is the case when we take $M$ to be the
path space of a given stochastic process. For example, we may take the usual
choice $M=C\big{(}[0,T],\mathbb{R}\big{)}$ with
$d(X,Y)=\sup_{t\in[0,T]}|X_{t}-Y_{t}|$ to investigate the strong convergence
of stochastic processes with continuous sample paths, as in [RW85, NP21]. In
the sequel, we consider strong convergence in the rough path space
$C^{p-\text{var}}\big{(}[0,T],\mathbb{R}^{d}\big{)}$ equipped with the
$p$-variation rough path metric.

There are two approaches we will take to prove strong convergence. The first
is a rudimentary application of the Markov inequality, while the second
utilises Lipschitz estimates on bounded sets in $M$. The second method is well
suited to extend strong convergence of processes that drive regime-switching
SDEs to the solutions of said RSSDEs.

##  3 The greedy partition

We introduce the greedy partition first introduced in [CLL13] by Cass,
Litterer and Lyons. In the following, let $\Delta_{[s,t]}$ be shorthand for
the 2-simplex on $[s,t]$:

$\Delta_{[s,t]}:=\big{\\{}(u,v):s\leq u\leq v\leq t\big{\\}}.$

We first recall the definition of a control.

######  Definition 3.1.

A function $\omega:\Delta_{[0,T]}\rightarrow[0,\infty)$ is called a control
if:

  1. 1.

$\omega$ is superadditive, so that for all $s\leq u\leq t\in[0,T]$

$\omega(s,u)+\omega(u,t)\leq\omega(s,t),$

  2. 2.

$\omega(s,s)=0$ for all $s\in[0,T]$, and

  3. 3.

$\omega$ is continuous.

######  Definition 3.2.

Let $\omega:\Delta_{[s,t]}\rightarrow[0,\infty)$ be a control. For $\alpha>0$,
set

$\displaystyle\tau_{0}(\alpha)$ $\displaystyle=s\,,$
$\displaystyle\tau_{i+1}(\alpha)$
$\displaystyle=\inf\\{u:\omega(\tau_{i},u)\geq\alpha,\tau_{i}(\alpha)<u\leq
t\\}\wedge t\,,\qquad i\geq 0$

and define

$N_{\alpha,[s,t]}(\omega)=\sup\\{n\in\mathbb{N}\cup\\{0\\}:\tau_{n}(\alpha)<t\\}\,.$

The sequence $(\tau_{i}(\alpha))_{i=0}^{\infty}$ is then called the greedy
sequence, with $N_{\alpha,[s,t]}(\omega)+1$ counting the number of distinct
elements in $(\tau_{i}(\alpha))_{i=0}^{\infty}$.

Lemma 4.9 and Corollary 4.10 in [CLL13] establish that
$N_{\alpha,[s,t]}(\omega)$ is well-defined for the particular choice
$\omega(s,t)=\|\mathbf{X}\|_{p-var;[s,t]}^{p}$ whenever $\mathbf{X}$ is a
$p$-rough path, and that the greedy sequence (with the trivial tail removed)
forms a partition of $[s,t]$. As discussed in [FR13], the random functional
$N_{\alpha,[s,t]}(\|\mathbf{X}\|_{p-var;[s,t]}^{p})$ enjoys far better
probabilistic tail estimates than $\|\mathbf{X}\|_{p-var;[s,t]}^{p}$. This,
combined with the Lipschitz estimate of Theorem 4 in [CFRS16], is the primary
tool we will utilise in the proof of our main theorem.

Figure 1: The greedy partition for a path $X$ with control $\|X\|_{p-var}^{p}$
and error $\alpha=\varepsilon$.

Specifically, we use the tail estimate from Corollary 2 of [FR13],

$\mathbb{P}\left(N_{\alpha,[0,T]}\big{(}\|\mathbf{X}\|_{p-var;[0,T]}^{p}\big{)}>u\right)\leq\exp\left\\{-\frac{1}{2}\left(\hat{a}+\frac{\alpha^{1/p}u^{1/q}}{CK}\right)^{2}\right\\},$

which applies to a class of Gaussian $p$-rough paths $\mathbf{X}$ including
Brownian motion, and where $C>0,K>0,q>0,\hat{\alpha}>-\infty$ are constants
dependent on the type of process $\mathbf{X}$ chosen. Since for every
polynomial $\sum_{i=1}^{n}a_{i}x^{i}$ there exists some $x_{0},C_{1},C_{2}>0$
such that

$C_{1}x^{n}\leq\sum_{i=1}^{n}a_{i}x^{i}\leq C_{2}x^{n},\text{ for }x>x_{0},$

there exists some constant $c$ such that

$\mathbb{P}\left(N_{\alpha,[0,T]}\big{(}\|\mathbf{X}\|_{p-var;[0,T]}^{p}\big{)}>u\right)\leq\exp\left\\{-cu^{2/q}\right\\}$

for all $u>u^{\prime}$, where $u^{\prime}$ depends on $c$ and
$c=c(\hat{a},C,K,p,q)$ depends on the previous constants.

######  Remark 3.3.

For the Brownian rough path case $\mathbf{X}=\mathbf{B}$, we may choose $q=1$,
yielding Gaussian tails.

We now investigate the behaviour of $N_{\alpha,[\cdot,\cdot]}$ under
concatenation.

######  Lemma 3.4.

Let $\\{s=p_{0}<p_{1}<\cdots<p_{M+1}=t\\}$ be a partition of $[s,t]$. Then

$N_{\alpha,[s,t]}(\omega)\geq\sum_{k=0}^{M}N_{\alpha,[p_{k},p_{k+1}]}(\omega)\,.$

###### Proof.

Let $(\tau_{i}(\alpha))$ be the greedy sequence over $[s,t]$ for the control
$\omega$. If there exist $i,k$ such that $\tau_{i}<p_{k}<p_{k+1}<\tau_{i+1}$,
then $N_{\alpha,[p_{k},p_{k+1}]}(\omega)=0$, which follows from

$N_{\alpha,[p_{k},p_{k+1}]}(\omega)\leq
N_{\alpha,[\tau_{i},\tau_{i+1}]}(\omega)=0\,.$

Thus, without loss of generality we assume that there is at most one $p_{k}$
between any given pair $\tau_{i}<\tau_{i+1}$. Next, we set
$k_{i}:=\sup\\{k:\tau_{k}\leq p_{i}\\}$ for $i\geq 0$ and write

$\displaystyle\alpha N_{\alpha,[s,t]}$
$\displaystyle=\sum_{k=0}^{N_{\alpha,[s,t]}(\omega)-1}\omega(\tau_{k},\tau_{k+1})$
$\displaystyle=\sum_{i=1}^{M+1}\sum_{j=k_{i-1}}^{k_{i}-1}\omega(\tau_{j},\tau_{j+1})$
$\displaystyle=\alpha\sum_{i=1}^{M+1}N_{\alpha,[\tau_{k_{i-1}},\tau_{k_{i}+1}]}(\omega)$
$\displaystyle\geq\alpha\sum_{i=1}^{M+1}N_{\alpha,[p_{i-1},p_{i}]}(\omega)$

with the last inequality following from the fact that
$[p_{i-1},p_{i}]\subseteq[\tau_{k_{i-1}},\tau_{k_{i}+1}]$.

∎

We now investigate the tail behaviour of an approximation
$\\{\mathbf{X}^{\lambda}\\}_{\lambda>0}$ to $\mathbf{X}$, under the assumption
of strong convergence. We begin with a simple set inclusion:

######  Lemma 3.5.

Let $\mathbf{X}$ and $\mathbf{X}^{\lambda}$ be $p$-rough paths, and take
$\alpha>0$. Then

$\left\\{N_{\alpha,[s,t]}\big{(}\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq
u\right\\}\subset\left\\{N_{\frac{\alpha}{2^{p-1}},[s,t]}\left(\|\mathbf{X}\|_{p-var}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\right)\geq
u\right\\}.$

###### Proof.

By the triangle inequality and Jensen’s inequality, we have

$\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\leq
2^{p-1}\left(\|\mathbf{X}\|_{p-var}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\right)\,.$

Then, if $\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\geq\alpha$ it follows that

$\|\mathbf{X}\|_{p-var}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\geq\frac{\|\mathbf{X}^{\lambda}\|_{p-var}^{p}}{2^{p-1}}\geq\frac{\alpha}{2^{p-1}}\,,$

yielding

$\left\\{\|\mathbf{X}^{\lambda}\|_{p-var;[s,t]}^{p}\geq\alpha\right\\}\subset\left\\{\|\mathbf{X}\|_{p-var;[s,t]}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var;[s,t]}^{p}\geq\frac{\alpha}{2^{p-1}}\right\\}\,.$

For simplicity of notation, let us write

$\displaystyle\omega^{1}(s,t)$
$\displaystyle=\|\mathbf{X}^{\lambda}\|^{p}_{p-var;[s,t]},$
$\displaystyle\omega^{2}(s,t)$
$\displaystyle=\|\mathbf{X}\|_{p-var;[s,t]}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var;[s,t]}^{p}\,.$

Consider the greedy sequence $\tau_{i}^{1}(\alpha)$ (resp.
$\tau_{j}^{2}(\alpha/2^{p-1})$) associated with the control $\omega^{1}$
(resp. $\omega^{2}$). Since $\omega^{1}(\tau_{i}^{1},\tau_{i+1}^{1})=\alpha$
for all $i=0,...,N_{\alpha,[s,t]}(\omega^{1})-1$, we have
$\omega^{2}(\tau_{i}^{1},\tau_{i+1}^{1})\geq\alpha/2^{p-1}$. Thus, given
consecutive times $\tau_{i}^{1}<\tau_{i+1}^{1}$, one can always find a value
of $j\in\\{0,...,N_{\alpha/2^{p-1},[s,t]}(\omega^{2})\\}$ such that
$\tau_{i}^{1}\leq\tau_{j}^{2}\leq\tau^{1}_{i+1}$. To see this, observe that
taking $k_{i}=\sup\\{\ell:\tau_{\ell}^{2}\leq\tau_{i}^{1}\\}$, we have by the
superadditivity of $\omega^{1}$ and $\omega^{2}$ that

$\omega^{2}(\tau_{k_{i}}^{2},\tau_{i+1}^{1})\geq\omega^{2}(\tau_{k_{i}}^{2},\tau_{i}^{1})+\omega^{2}(\tau_{i}^{1},\tau_{i+1}^{1})\geq\omega^{2}(\tau_{k_{i}}^{2},\tau_{i}^{1})+\frac{\alpha}{2^{p-1}}\,.$

It follows that setting
$\tau_{i}^{1}\leq\tau_{k_{i}+1}^{2}\leq\tau_{i+1}^{1}$, and also that $i\leq
k_{i}$. Thus we have

$\tau_{i}^{2}\leq\tau_{k_{i}}^{2}\leq\tau_{i}^{1}\,,$

for $i\in\\{0,...,N_{\alpha,[s,t]}(\omega^{1})\\}$. Finally we have

$\displaystyle N_{\alpha,[s,t]}(\omega^{1})$
$\displaystyle=\sup\\{n\in\mathbb{N}\cup\\{0\\}:\tau^{1}_{n}(\alpha)<t\\}$
$\displaystyle\leq\sup\\{n\in\mathbb{N}\cup\\{0\\}:\tau^{2}_{n}(\alpha/2^{p-1})<t\\}$
$\displaystyle=N_{\frac{\alpha}{2^{p-1}},[s,t]}(\omega^{2})\,,$

which completes the proof. ∎

Next, we investigate how the strong convergence of
$\mathbf{X}^{\lambda}\rightarrow\mathbf{X}$ in the inhomogeneous $p$-rough
path metric affects the tails of
$N_{\alpha,[s,t]}\big{(}\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}$ and
$N_{\alpha,[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}$.

######  Lemma 3.6.

Suppose that $\mathbf{X}^{\lambda}\rightarrow\mathbf{X}$ strongly in the
inhomogeneous $p$-rough path metric $\rho_{p-var}$, with rate
$\delta(\lambda)$. Then

$\mathbb{P}\left(N_{\alpha,[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}>0\right)=o(\lambda^{-r})\,,$

for all $r>0$.

###### Proof.

We note that $N_{\alpha,[s,t]}(\omega)>0$ if and only if there exists some
$\tau<t$ such that $\omega(s,\tau)\geq\alpha$. Since $\omega$ is
superadditive, this implies $\omega(s,t)\geq\alpha$. Thus,

$\displaystyle\mathbb{P}\left(N_{\alpha,[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}>0\right)$
$\displaystyle\leq\mathbb{P}\left(\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\geq\alpha\right).$

Bounding the homogeneous norm by

$\displaystyle\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}$
$\displaystyle=\|X-X^{\lambda}\|_{p-var}^{p}+\|\mathbb{X}-\mathbb{X}^{\lambda}\|_{p/2-var}^{p/2}$
$\displaystyle\leq
2\max\left\\{\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})^{p},\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})^{p/2}\right\\},$

we see that

$\displaystyle\mathbb{P}\left(N_{\alpha,[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}>0\right)$
$\displaystyle\leq\mathbb{P}\left(\max\left\\{\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})^{p},\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})^{p/2}\right\\}\geq\frac{\alpha}{2}\right)$
$\displaystyle\leq\mathbb{P}\left(\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})\geq\left(\frac{\alpha}{2}\right)^{1/p}\right)+\mathbb{P}\left(\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})\geq\left(\frac{\alpha}{2}\right)^{2/p}\right)$
$\displaystyle\leq
2\mathbb{P}\left(\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})\geq\varepsilon\right),$

with $\varepsilon=\max\\{(\alpha/2)^{1/p},(\alpha/2)^{2/p}\\}$. Finally, we
pick a $\lambda^{\prime}$ such that $k\delta(\lambda)\leq\varepsilon$ for all
$\lambda>\lambda^{\prime}$. This yields

$2\mathbb{P}\left(\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})\geq\varepsilon\right)\leq
2\mathbb{P}\left(\rho_{p-var}(\mathbf{X},\mathbf{X}^{\lambda})\geq
k\delta(\lambda)\right)=o(\lambda^{-r}),$

as required. ∎

######  Lemma 3.7.

Let $\mathbf{X}^{\lambda},\mathbf{X}$ be Gaussian rough paths of finite mixed
$(1,\rho)$ variation, and suppose that
$\mathbf{X}^{\lambda}\rightarrow\mathbf{X}$ strongly in inhomogeneous
$p$-rough path metric with rate $\delta(\lambda)$, and $\alpha>0$. Then there
exists some constant $c$ such that

$\mathbb{P}\left(N_{\alpha,[s,t]}\big{(}\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq
u\right)\leq\exp\left\\{-c_{1}u^{2/q}\right\\}+o(\lambda^{-r}).$

###### Proof.

By Lemma 3.5, we have

$\mathbb{P}\left(N_{\alpha,[s,t]}\big{(}\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq
u\right)\leq\mathbb{P}\left(N_{\alpha_{p},[s,t]}\left(\|\mathbf{X}\|_{p-var}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\right)\geq
u\right)\,.$ (11)

Lemma 5 of [CFRS16] states that

$N_{\alpha_{p},[s,t]}\left(\|\mathbf{X}\|_{p-var}^{p}+\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\right)\leq
2N_{\alpha_{p},[s,t]}\big{(}\|\mathbf{X}\|_{p-var}^{p}\big{)}+2N_{\alpha_{p},[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}+2\,,$

which when substituted into (11) yields

$\displaystyle\mathbb{P}\big{(}N_{\alpha,[s,t]}$
$\displaystyle\big{(}\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq
u\big{)}\leq\mathbb{P}\left(N_{\alpha_{p},[s,t]}\big{(}\|\mathbf{X}\|_{p-var}^{p}\big{)}+N_{\alpha_{p},[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq\frac{u-2}{2}\right)$
$\displaystyle\leq\mathbb{P}\left(N_{\alpha_{p},[s,t]}\big{(}\|\mathbf{X}\|_{p-var}^{p}\big{)}\geq\frac{u}{4}\right)+\mathbb{P}\left(N_{\alpha_{p},[s,t]}\big{(}\|\mathbf{X}-\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq\frac{u-4}{4}\right)\,.$

Applying the Weibull tails of
$N_{\alpha,[s,t]}\big{(}\|\mathbf{X}\|_{p-var}^{p}\big{)}$ and Lemma 3.6, we
see that

$\mathbb{P}\left(N_{\alpha,[s,t]}\big{(}\|\mathbf{X}^{\lambda}\|_{p-var}^{p}\big{)}\geq
u\right)\leq\exp\left\\{-c_{1}u^{2/q}\right\\}+o(\lambda^{-r}).$

∎

##  4 Lipschitz estimates for rough RSSDE

In this section, we extend the (local) Lipschitz estimates for rough
differential equations to the regime-switching case.

######  Theorem 4.1.

Let $\mathbf{X}$ be a Gaussian rough path,
$\\{\mathbf{X}^{\lambda}\\}_{\lambda>0}$ a family of rough path lifts of
finite-variation processes, and $J$ be a Markov process independent of
$\mathbf{X}$, $\mathbf{X}^{\lambda}$ with state space $\mathcal{E}$ and a.s.
finite jump activity on compact intervals. Let $\gamma>p>2$, and
$\\{V_{i}\\}_{i\in\mathcal{E}}$ be a family of vector fields such that
$|V_{i}|_{Lip^{\gamma}}$ is uniformly bounded by some $\nu$, that is,
$\sup_{i\in\mathcal{E}}|V_{i}|_{Lip^{\gamma}}\leq\nu$.

Now, let $\\{0=t_{0}^{J}<t_{1}^{J}<\cdots<t_{N^{J}+1}^{J}=T\\}$ be a partition
of $[0,T]$ given by the jump times of $J$. Define $Y^{k}$ and $Y^{k,\lambda}$
to be the RDE solutions to

$\mathrm{d}Y^{k}_{t}=V_{J_{t_{k}}}(Y^{k}_{t})\,\mathrm{d}\mathbf{X}_{t}\,,\quad
Y^{k}_{t_{k}}=Y^{k-1}_{t_{k}}\,,\quad t\in[t_{k},t_{k+1}],$

and

$\mathrm{d}Y^{k,\lambda}_{t}=V_{J_{t_{k}}}(Y^{k,\lambda}_{t})\,\mathrm{d}\mathbf{X}^{\lambda}_{t}\,,\quad
Y^{k,\lambda}_{t_{k}}=Y^{k-1,\lambda}_{t_{k}}\,,\quad t\in[t_{k},t_{k+1}],$

respectively, and define $Y$ (resp. $Y^{\lambda}$) to be the concatenation of
the $\\{Y^{k}\\}_{k=0}^{N^{J}}$ (resp. $\\{Y^{k,\lambda}\\}_{k=0}^{N^{J}}$).
Then, there exists some constant $C=C(\gamma,p,\nu,\alpha)$ such that

$\|Y-Y^{\lambda}\|_{\infty;[0,T]}\leq
K_{1}C^{N^{J}}\rho_{p-var;[0,T]}(\mathbf{X},\mathbf{X}^{\lambda})\exp\left\\{C\left(N_{\alpha,[0,T]}(\mathbf{X})+N_{\alpha,[0,T]}(\mathbf{X}^{\lambda})\right)\right\\},$

where $K_{1}=C^{2}/(C-1)$.

###### Proof.

Under the proposed conditions each $Y^{k}$ and $Y^{k,\lambda}$ exists uniquely
and is continuous, and these properties extend to the concatenation $Y$ and
$Y^{\lambda}$. Theorem 4 of [FR13] yields the estimate

$\displaystyle\|Y^{k}-Y^{k,\lambda}\|_{\infty;[t_{k},t_{k+1}]}$
$\displaystyle\leq
C\left[|Y_{t_{k}}^{k}-Y^{k,\lambda}_{t_{k}}|+\rho_{p-var;[t_{k},t_{k+1}]}(\mathbf{X},\mathbf{X}^{\lambda})\right]$
(12)
$\displaystyle\qquad\times\exp\left\\{C\big{(}N_{\alpha;[t_{k},t_{k+1}]}(\mathbf{X})+N_{\alpha;[t_{k},t_{k+1}]}(\mathbf{X}^{\lambda})\big{)}\right\\}$

for each $k=0,\dots,J$. Noting that

$\displaystyle|Y_{t_{k}}^{k}-Y_{t_{k}}^{k,\lambda}|$
$\displaystyle\leq\|Y^{k}-Y^{k,\lambda}\|_{\infty;[t_{k-1},t_{k}]}$
$\displaystyle\leq
C\left[|Y_{t_{k-1}}^{k-1}-Y^{k-1,\lambda}_{t_{k-1}}|+\rho_{p-var;[t_{k-1},t_{k}]}(\mathbf{X},\mathbf{X}^{\lambda})\right]$
$\displaystyle\qquad\times\exp\left\\{C\big{(}N_{\alpha;[t_{k-1},t_{k}]}(\mathbf{X})+N_{\alpha;[t_{k-1},t_{k}]}(\mathbf{X}^{\lambda})\big{)}\right\\}\,,$

we can rewrite Equation (12) as

$\displaystyle\|Y^{k}-Y^{k,\lambda}$
$\displaystyle\|_{\infty;[t_{k},t_{k+1}]}\leq
C\rho_{p-var;[t_{k},t_{k+1}]}(\mathbf{X},\mathbf{X}^{\lambda})\exp\left\\{C\big{(}N_{\alpha;[t_{k},t_{k+1}]}(\mathbf{X})+N_{\alpha;[t_{k},t_{k+1}]}(\mathbf{X}^{\lambda})\big{)}\right\\}$
$\displaystyle+C^{2}\left[|Y_{t_{k-1}}^{k-1}-Y^{k-1,\lambda}_{t_{k-1}}|+\rho_{p-var;[t_{k-1},t_{k}]}(\mathbf{X},\mathbf{X}^{\lambda})\right]$
$\displaystyle\times\exp\left\\{C\big{(}N_{\alpha;[t_{k-1},t_{k}]}(\mathbf{X})+N_{\alpha;[t_{k},t_{k+1}]}(\mathbf{X})+N_{\alpha;[t_{k-1},t_{k}]}(\mathbf{X}^{\lambda})+N_{\alpha;[t_{k},t_{k+1}]}(\mathbf{X}^{\lambda})\big{)}\right\\}\,.$

Iterating back to $[0,t_{1}]$ yields

$\displaystyle\|Y^{k}-Y^{k,\lambda}\|_{\infty;[t_{k},t_{k+1}]}$
$\displaystyle\leq\sum_{j=0}^{k}C^{j+1}\rho_{p-var;[t_{k-j},t_{k+1-j}]}(\mathbf{X},\mathbf{X}^{\lambda})$
$\displaystyle\times\exp\left\\{C\left(\sum_{i=0}^{j}N_{\alpha;[t_{k-i},t_{k+1-i}]}(\mathbf{X})+N_{\alpha;[t_{k-i},t_{k+1-i}]}(\mathbf{X}^{\lambda})\right)\right\\}\,.$

By Lemma 3.4 and the fact that
$\rho_{p-var;[u,v]}(\cdot,\cdot)\leq\rho_{p-var;[s,t]}(\cdot,\cdot)$ if
$[u,v]\subseteq[s,t]$, we have

$\|Y^{k}-Y^{k,\lambda}\|_{\infty;[t_{k},t_{k+1}]}\leq\left(\sum_{j=1}^{k+1}C^{j}\right)\rho_{p-var;[0,t_{k+1}]}(\mathbf{X},\mathbf{X}^{\lambda})\exp\left\\{C\left(N_{\alpha,[0,t_{k+1}]}(\mathbf{X})+N_{\alpha,[0,t_{k+1}]}(\mathbf{X}^{\lambda})\right)\right\\}\,.$

Finally, noting that

$\|Y-Y^{\lambda}\|_{\infty;[0,T]}=\max_{k=0,...,J}\|Y^{k}-Y^{k,\lambda}\|_{\infty;[t_{k},t_{k+1}]}\,,$

we arrive at the estimate

$\displaystyle\|Y-Y^{\lambda}\|_{\infty;[0,T]}$
$\displaystyle\leq\left(\sum_{j=1}^{J+1}C^{j}\right)\rho_{p-var;[0,T]}(\mathbf{X},\mathbf{X}^{\lambda})\exp\left\\{C\left(N_{\alpha,[0,T]}(\mathbf{X})+N_{\alpha,[0,T]}(\mathbf{X}^{\lambda})\right)\right\\}$
$\displaystyle\leq
K_{1}K_{2}^{J}\rho_{p-var;[0,T]}(\mathbf{X},\mathbf{X}^{\lambda})\exp\left\\{C\left(N_{\alpha,[0,T]}(\mathbf{X})+N_{\alpha,[0,T]}(\mathbf{X}^{\lambda})\right)\right\\}$

with $K_{1}=C^{2}/(C-1)$ and $K_{2}=C$. ∎

##  5 Strong convergence

To establish the strong convergence of $Y^{\lambda}\rightarrow Y$ in supremum
norm, we apply the methodology in [NP21] of utilising the probabilistic
properties of the local Lipschitz coefficient of the solution map (Theorem
4.1). As in [NP21], we impose a constraint on the jump process $J$ to control
the growth of the constant $K_{2}^{J}$ appearing in Theorem 4.1.

######  Assumption 5.1.

There exists some $\gamma_{0}>0$ such that
$\mathbb{P}\big{(}N^{J}>j)=o\big{(}\mathrm{e}^{-j(\log j-\gamma_{0})}\big{)}$.

Assumption 5.1 clearly holds if $J$ is deterministic. We refer to Lemma 4.3 of
[NP21] which implies that Assumption 5.1 holds whenever $J$ is of bounded jump
intensity.

######  Lemma 5.2.

Let $J$ be a jump process satisfying Assumption 5.1. Then the number of jumps
$N^{J}$ has finite expectation.

###### Proof.

Using the tail sum formula for expectation, we see that

$\mathbb{E}\left[N^{J}\right]=\sum_{j=0}^{\infty}\mathbb{P}(N^{J}>j).$

Since $\mathbb{P}(N^{J}>j)=o(\mathrm{e}^{-j(\log j-\gamma_{0})})$, for large
enough $j_{0}$ we have

$\mathbb{E}[N^{J}]=\sum_{j=0}^{j_{0}-1}\mathbb{P}(N^{J}>j)+\sum_{j=j_{0}}^{\infty}\mathrm{e}^{-j(\log
j-\gamma_{0})}\leq
C_{j_{0}}+k\sum_{j=j_{0}}^{\infty}\mathrm{e}^{-j/2}<\infty.$

∎

We are now in a position to show that $Y^{\lambda}\rightarrow Y$ strongly in
supremum norm with rate slightly worse than
$\mathbf{B}^{\lambda}\rightarrow\mathbf{B}$ in rough path metric.

######  Theorem 5.3.

Under the conditions of Theorem 4.1 and Assumption 5.1, suppose that there
exists $\delta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ with
$\lim_{\lambda\rightarrow\infty}\delta(\lambda)=0$ such that for all $r>0$

$\mathbb{P}\big{(}\rho_{p-var;[0,T]}(\mathbf{B}^{\lambda},\mathbf{B})\geq
k\delta(\lambda)\big{)}=o(\lambda^{-r})\,,$

where $k=k(r,T)>0$ is a constant dependent on $r$ and $T$ only. Then there
exists some constant $\beta=\beta(r,T)$ such that for all $\varepsilon>0$

$\mathbb{P}\big{(}\|Y-Y^{\lambda}\|_{\infty;[0,T]}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\big{)}=o(\lambda^{-r}).$
(13)

###### Proof.

Using the estimate in Theorem 4.1, we see that

$\displaystyle\mathbb{P}\left(\|Y-Y^{\lambda}\|_{\infty;[0,T]}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)$
$\displaystyle\leq\mathbb{P}\bigg{(}K_{1}K_{2}^{J}\rho_{p-var;[0,T]}(\mathbf{B},\mathbf{B}^{\lambda})$
$\displaystyle\qquad\quad\times\exp\left\\{C\left(N_{\alpha,[0,T]}(\mathbf{B})+N_{\alpha,[0,T]}(\mathbf{B}^{\lambda})\right)\right\\}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\bigg{)}\,.$

Let $f_{\gamma_{0},r}(x):=\exp\big{(}x(\log x-\gamma_{0})/r\big{)}$ and
$A_{\lambda}:=f^{-1}_{\gamma_{0},r}(\lambda)$ as in [NP21]. Write

$E_{\lambda}=\left\\{K_{1}K_{2}^{J}\rho_{p-var;[0,T]}(\mathbf{B},\mathbf{B}^{\lambda})\exp\left\\{C\left(N_{\alpha,[0,T]}(\mathbf{B})+N_{\alpha,[0,T]}(\mathbf{B}^{\lambda})\right)\right\\}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right\\},$

and introduce the events

$\displaystyle F_{1}$ $\displaystyle=\left\\{J\leq A_{\lambda}\right\\},$
$\displaystyle F_{2}$
$\displaystyle=\left\\{N_{\alpha,[0,T]}(\mathbf{B})\leq\sqrt{\frac{(r+\varepsilon)\log\lambda}{c_{1}}}\right\\},$
$\displaystyle F_{3}$
$\displaystyle=\left\\{N_{\alpha,[0,T]}(\mathbf{B}^{\lambda})\leq\sqrt{\frac{(r+\varepsilon)\log\lambda}{c_{1}}}\right\\}.$

As

$\mathbb{P}(E_{\lambda})\leq\mathbb{P}(E_{\lambda}\cap F_{1}\cap F_{2}\cap
F_{3})+\mathbb{P}(F_{1}^{\mathrm{c}})+\mathbb{P}(F_{2}^{\mathrm{c}})+\mathbb{P}(F_{3}^{\mathrm{c}})\,,$
(14)

we can show strong convergence by showing that every term on the RHS of (14)
is $o(\lambda^{-r})$. Lemma 3.7 shows that

$\displaystyle\mathbb{P}(F_{3}^{\mathrm{c}})$
$\displaystyle=\mathbb{P}\left(N_{\alpha,[0,T]}(\mathbf{B}^{\lambda})\geq\sqrt{\frac{(r+\varepsilon)\log\lambda}{c_{1}}}\right)$
$\displaystyle\leq\exp\left\\{-\frac{c_{1}(r+\varepsilon)\log\lambda}{c_{1}}\right\\}+o(\lambda^{-r})$
$\displaystyle=\frac{1}{\lambda^{r+\varepsilon}}+o(\lambda^{-r})$
$\displaystyle=o(\lambda^{-r})\,,$

while the Gaussian tails of $N_{\alpha,[0,T]}(\mathbf{B})$ yield

$\displaystyle\mathbb{P}(F_{2}^{\mathrm{c}})$
$\displaystyle=\mathbb{P}\left(N_{\alpha,[0,T]}(\mathbf{B})\geq\sqrt{\frac{(r+\varepsilon)\log\lambda}{c_{1}}}\right)$
$\displaystyle\leq\exp\left\\{-\frac{c_{1}(r+\varepsilon)\log\lambda}{c_{1}}\right\\}$
$\displaystyle=\frac{1}{\lambda^{r+\varepsilon}}=o(\lambda^{-r}).$

The proof of Theorem 4.6 of [NP21] shows
$\mathbb{P}\big{(}F_{1}^{\mathrm{c}}\big{)}=\mathbb{P}\big{(}N^{J}>A_{\lambda}\big{)}=o(\lambda^{-r})$.
Thus, (14) becomes

$\displaystyle\mathbb{P}(E_{\lambda})$
$\displaystyle\leq\mathbb{P}\bigg{(}K_{1}K_{2}^{A_{\lambda}}\rho_{p-var[0,T]}(\mathbf{B},\mathbf{B}^{\lambda})$
$\displaystyle\qquad\exp\left\\{C\left(\sqrt{\frac{(r+\varepsilon)\log\lambda}{c_{1}}}+\sqrt{\frac{(r+\varepsilon)\log\lambda}{c_{1}}}\right)\right\\}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\bigg{)}+o(\lambda^{-r})\,.$

Collecting constants that are independent of $\lambda$, the above becomes

$\mathbb{P}(E_{\lambda})\leq\mathbb{P}\left(\widetilde{K}K_{2}^{A_{\lambda}}\exp\left\\{\sqrt{\log\lambda}\right\\}\lambda^{-\varepsilon_{2}}\rho_{p-var;[0,T]}(\mathbf{B},\mathbf{B}^{\lambda})\geq\beta\delta(\lambda)\right)+o(\lambda^{-r}),$

for some constant $\widetilde{K}$. We will now show that

$K_{2}^{A_{\lambda}}\exp\left\\{\sqrt{\log\lambda}\right\\}\lambda^{-\varepsilon}\rightarrow
0.$

We have that $K_{2}^{A_{\lambda}}\lambda^{-\varepsilon/2}\rightarrow 0$ from
[NP21] via the asymptotic decomposition of Lambert–W functions. Next,

$\exp\\{\sqrt{\log{\lambda}}\\}\lambda^{-\varepsilon/2}=\exp\left\\{\sqrt{\log\lambda}-\frac{\varepsilon}{2}\log\lambda\right\\}\rightarrow
0\,.$

As a result, we may choose some $\lambda^{\prime}$ such that
$K_{2}^{A_{\lambda}}\exp\left\\{\sqrt{\log\lambda}\right\\}\lambda^{-\varepsilon/2}\leq
1$, for all $\lambda>\lambda^{\prime}$. Thus, for all
$\lambda>\lambda^{\prime}$,

$\displaystyle\mathbb{P}(E_{\lambda})\leq\mathbb{P}(\tilde{K}\rho_{p-var;[0,T]}(\mathbf{B},\mathbf{B}^{\lambda})\geq\beta\delta(\lambda))\,.$

Finally, setting $\beta:=k/\tilde{K}$, we see that

$\mathbb{P}(E_{\lambda})\leq\mathbb{P}\big{(}\rho_{p-var;[0,T]}(\mathbf{B},\mathbf{B}^{\lambda})\geq
k\delta(\lambda)\big{)}=o(\lambda^{-r})\,,$

as required. ∎

######  Theorem 5.4.

Let $\mathbf{X}$ and $\\{\mathbf{X}^{\lambda}\\}_{\lambda\geq 0}$ be Gaussian
rough paths, and $J$ a jump process independent of $\mathbf{X}$ and
$\mathbf{X}^{\lambda}$ satisfying Assumption 5.1. Suppose there exists
$\delta:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ with
$\lim_{\lambda\rightarrow\infty}\delta(\lambda)=0$ such that for all $r>0$

$\mathbb{P}\left(\rho_{p-var;[0,T]}(\mathbf{X}^{\lambda},\mathbf{X})\geq
k\delta(\lambda)\right)=o(\lambda^{-r}),$

where $k=k(r,T)>0$ is a constant dependent on $r$ and $T$ only. Then there
exists some constant $\beta=\beta(r,T)$ such that for all $\varepsilon>0$,

$\mathbb{P}\left(\|Y-Y^{\lambda}\|_{\infty;[0,T]}\geq\beta\delta(\lambda)\lambda^{\varepsilon}\right)=o(\lambda^{-r})$

###### Proof.

The same proof for Theorem 5.3 holds with the sets $F_{2}$ and $F_{3}$
replaced with

$\displaystyle F_{2}$
$\displaystyle=\left\\{N_{\alpha,[0,T]}(\mathbf{X})\leq\left(\frac{(r+\varepsilon)\log\lambda}{c_{1}}\right)^{q/2}\right\\},$
$\displaystyle F_{3}$
$\displaystyle=\left\\{N_{\alpha,[0,T]}(\mathbf{X}^{\lambda})\leq\left(\frac{(r+\varepsilon)\log\lambda}{c_{1}}\right)^{q/2}\right\\}.$

∎

##  6 Approximation schemes

Having proved Theorem 5.3, it remains to provide approximations of enhanced
Gaussian processes that converge strongly. The following lemma will prove
useful in this pursuit.

######  Lemma 6.1.

Let $(M,d)$ be a complete metric space, $X$ a $M$-valued random variable and
$\\{X^{\lambda}\\}_{\lambda\geq 0}$ a collection of $M$-valued random
variables. If there exist constants $C=C(q)>0$ and $\eta>0$ such that
$\|d(X,X^{\lambda})\|_{L^{q}}\leq C\lambda^{-\eta}$ for all $q>q^{\prime}$,
then $X^{\lambda}\rightarrow X$ strongly with rate function
$\delta(\lambda)=\lambda^{-\gamma}$, for $\gamma<\eta$.

###### Proof.

By Markov’s inequality, we write

$\displaystyle\mathbb{P}\left(d(X,X^{\lambda})\geq k\delta(\lambda)\right)$
$\displaystyle\leq\frac{\mathbb{E}[d(X,X^{\lambda})^{q}]}{k^{q}\delta(\lambda)^{q}}$
$\displaystyle\leq\frac{C^{q}\lambda^{-q\eta}}{k^{q}\lambda^{-q\gamma}}$
$\displaystyle=\left(\frac{C}{k}\right)^{q}\lambda^{q(\gamma-\eta)}.$

To show that the RHS is $o(\lambda^{-r})$, we simply set
$q=\frac{r+\varepsilon}{\eta-\gamma}$ provided $q>q^{\prime}$. ∎

The class of Gaussian rough path we work with are those introduced in
[CFRS16]:

######  Condition 6.2 (Condition 10, [CFRS16]).

Let $X=(X^{1},...,X^{d})$ be a centred, continuous Gaussian process with
independent components. Assume that the covariance of every component has
_Hölder dominated_ finite mixed $(1,\rho)$-variation for some $\rho\in[1,2)$
on $[0,T]^{2}$, that is, there exists $K<\infty$ such that, for $k=1,...,d$
and uniformly over $s<t$ in $[0,T]$,

$\|X\|_{(1,\rho)-var;[s,t]}:=\sup_{(t_{i}),(t_{j}^{\prime})\in\mathcal{D}([s,t])}\left(\sum_{t_{j}^{\prime}}\left(\sum_{t_{i}}\left|\mathbb{E}\left[X_{t_{i},t_{i+1}}^{k}X_{t_{j}^{\prime},t_{j+1}^{\prime}}^{k}\right]\right|\right)^{\rho}\right)^{1/\rho}\leq
K\left(|t-s|^{1/\rho}\right).$

The notion of mixed variation is a more refined notion than $p$-variation.
Specifically, we note that fractional Brownian motion satisfies condition 6.2
with $\rho=\frac{1}{2H}$. The class of approximations to these processes that
we work with are those satisfying the following condition, as presented in
[CFRS16].

######  Condition 6.3.

Let $X$ be a Gaussian path satisfying condition 6.2. Let
$\\{X^{h}\\}_{h\in(0,1]}$ be a collection of centred, Gaussian processes with
independent components for every $h\in(0,1]$ such that

  1. 1.

$(X^{h},X):[0,T]\rightarrow\mathbb{R}^{2d}$ is jointly Gaussian,
$(X^{h;i},X^{i})$ and $(X^{h;j},X^{j})$ are independent for $i\neq j$, and

$\left|\left|(X^{h},X)\right|\right|_{(1,\rho)-var;[0,T]}=:K<\infty,\qquad\rho\in[1,2)$

as in Condition 6.2.

  2. 2.

The second moments converge uniformly, so that

$\sup_{t\in[0,T]}\mathbb{E}\left[\big{|}X_{t}^{h}-X_{t}\big{|}^{2}\right]=:\delta(h)^{1/\rho}\rightarrow
0\,\text{ for }h\rightarrow 0.$

Condition 6.3 implies the convergence of the rough path lift
$\mathbf{X}^{h}\rightarrow\mathbf{X}$ in $p$-variation rough path metric for
$p>2\rho$. Further to this, one can show that taking $X^{h}$ to be the linear
interpolation of $X$ with mesh-size at most $h$, that $\delta(h)=h$.

Approximating the rough path lift $\mathbf{B}^{H}$ of a fractional Brownian
motion $B^{H}$ via linear interpolation, Friz and Riedel [FR14] provide the
estimate

$\left|\left|\rho_{p-var;[0,T]}\big{(}\mathbf{B}^{H;n},\mathbf{B}^{H}\big{)}\right|\right|_{L^{r}}\leq
Cn^{-\eta}$

with $\eta<H$. Applying Lemma 6.1, we see that
$\mathbf{B}^{n}\rightarrow\mathbf{B}$ strongly with rate
$\delta(n)=n^{-\gamma}$, for all $\gamma<H$ and
$H\in\big{(}\frac{1}{4},\frac{1}{2}\big{]}$.

## Acknowledgments

GTN and OP gratefully acknowledge the support of the Australian Research
Council DP180103106 grant. Additionally, OP acknowledges financial support
from the Swiss National Science Foundation Project 200021-191984.

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# Millimeter-VLBI observations of low-luminosity active galactic nuclei with
source-frequency phase-referencing

Wu Jiang Shanghai Astronomical Observatory, Chinese Academy of Sciences,
Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, Nanjing 210008, China  Zhiqiang Shen Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China  Key
Laboratory of Radio Astronomy, Chinese Academy of Sciences, Nanjing 210008,
China  Ivan Martí-Vidal Department of Astronomy and Astrophysics, University
of Valencia, E-46100, Burjassot, Spain  Xuezheng Wang  Shanghai Astronomical
Observatory, Chinese Academy of Sciences, Shanghai 200030, China  ShanghaiTech
University, 393 Middle Huaxia Road, Pudong, Shanghai, 201210, China  Dongrong
Jiang  Shanghai Astronomical Observatory, Chinese Academy of Sciences,
Shanghai 200030, China  Key Laboratory of Radio Astronomy, Chinese Academy of
Sciences, Nanjing 210008, China  Noriyuki Kawaguchi National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan

(Received Sep. 10, 2021; Revised Nov. 07, 2021; Accepted Nov. 8, 2021)

###### Abstract

We report millimeter-VLBI results of low-luminosity active galactic nuclei (M
84 and M 87) up to 88 GHz with source-frequency phase-referencing
observations. We detected the weak VLBI core and obtained the first image of M
84 at 88 GHz. The derived brightness temperature of M 84 core was about
7.2$\times 10^{9}$ K, which could serve as a lower limit as the core down to
30 Schwarzschild radii was still un-resolved in our 88 GHz observations. We
successfully determined the core-shifts of M 87 at 22-44 GHz and 44-88 GHz
through source-frequency phase-referencing technique. The jet apex of M 87
could be deduced at $\sim$46 $\mu$as upstream of the 43 GHz core from core-
shift measurements. The estimated magnetic field strength of the 88 GHz core
of M 87 is $4.8\pm 2.4$ G, which is at the same magnitude of 1-30 G near the
event horizon probed by the Event Horizon Telescope.

LLAGN, M 84, M 87 — VLBI

##  1 Introduction

The low-luminosity active galactic nuclei (LLAGNs) classified by their low
bolometric luminosities and sub-Eddington accretion rate commonly exist in
nearby galaxies (Nagar et al., 2002; Ho, 2008). Unlike their bright cousins,
the broad-band spectral energy distributions prefer the model of an inner
advection-dominated accretion flow and an outer truncated thin disc (Yu et
al., 2011; Nemmen et al., 2014). Meanwhile compact flat-spectrum radio cores
were detected in LLAGNs (Nagar et al., 2002) and suggested to be scaled-down
versions of AGN jets (Falcke & Biermann, 1999). Hosting a large mass of
central supermassive black hole and its proximity make LLAGN approachable to
the launching and accelaration region of inner jet, even to its event horizon
at millimeter or sub-millimeter wavelength.
According to the inhomogeneous model of relativistic jet, the position of the
VLBI core is frequency-dependent. This frequency-dependent shift in the
location of the core (core-shift) can be used to estimate the magnetic field
strength and electron number density of the jet (Lobanov, 1998; Kovalev et
al., 2008). However, only a few active galactic nuclei (AGNs) have reliable
core-shift measurements at millimeter wavelengths (O’Sullivan & Gabuzda,
2009). The core-shift measurements are mostly obtained at low frequencies due
to limited sensitivity at high frequencies (Pushkarev et al., 2012). Although
millimeter-VLBI can approach the inner region of jet as the plasma turns
optically thin at high frequency, the core-shift is difficult to obtain due to
the rapid phase flutuations of atmosphere and thus limited coherent
intergration time for the conventional VLBI phase-referencing observations.
Fortunately, a newly proposed VLBI phase-referencing technique called source-
frequency phase-referencing (SFPR) can be used to measure the core-shift
effect (Rioja & Dodson, 2011), essentially being of great advantadge at
millimeter wavelengths. We successfully obtained the first VLBI image of the
LLAGN M 81* at 88 GHz and measured the shift between 7 mm and 3 mm wavelengths
in the compact jet (Jiang et al., 2018).
In this paper, we will present the applications of SFPR to two LLAGNs, M 84
and M 87. The observation summary and data reduction are presented in section
2. The results are in section 3, followed by the conclusion in section 4.

##  2 Observations and data reduction

###  2.1 Observations

The observations of M 84 and M 87 were carried out with the VLBA in SFPR mode
on June 22, 2019 with 22 and 44 GHz frequency pair. The observations with 44
and 88 GHz frequency pair were firstly performed on June 18, 2019 but most
stations failed at 88 GHz. Therefore it was reobserved on March 31, 2021 for
satisfied weather conditions at most stations. The on-source time of each
source per scan was $\sim$30 seconds at a frequency. A fast frequency-
switching cycle of two frequencies on a source was taken as a loop. M 87
($\sim 1^{\circ}.5$ apart from M 84 in the sky) was interleaved every 5 loops
of M 84 for the 22/44 GHz epoch and 9 loops for the 44/88 GHz epoch. Several
loops of blazar 1219+044 were observed at the beginning and the end of
observations but not included here, due to rapid phase flutuations of low
antenna elevation angles. The 2019 epoch was recorded at 2048 Mbps by
splitting the 512 MHz total bandwidth into 16 intermediate frequency (IF)
bands while the 2021 epoch was at 4096 Mbps by splitting 1024 MHz bandwidth
into 8 IFs.

###  2.2 Data reduction

The data calibration and reduction followed the procedures in Rioja & Dodson
(2011) and Jiang et al. (2018). Firstly we performed standard amplitude and
phase calibrations in AIPS for both M 84 and M 87 at the reference frequencies
(22 GHz of 2019 epoch and 44 GHz of 2021 epoch), respectively. Their images
were obtained by further clean and self-calibration in Difmap. Secondly the
corresponding phase solutions of the AIPS task FRING after taking into account
clean models of M 84 and M 87 were multiplied by a factor of two, while the
delay and delay rate solutions were unchanged. These revised solutions were
then applied to the target frequencies (44 GHz in 2019 epoch and 88 GHz in
2021 epoch, respectively), which is called the frequency phase transfer (FPT)
calibration. The phase flutuations in proportion to the observing frequency
such as the unmodeled tropspheric and geometric errors were eliminated in this
step. Thirdly the SFPR-ed images of M 84 at target frequencies were obtained
by further phase referencing to the FRING solutions of the FPT calibrated data
of M 87 at the same target frequencies, which refined the unmodeled dispersive
ionospheric and instrumental errors. The corresponding core-shift measurements
were derived from the SFPR-ed images using JMFIT in AIPS. Finally high
resolution VLBI images of M 84 were obtained by clean and self-calibration in
Difmap (Jiang et al., 2018).
Since the brightness-peak position of the image is usually referred as the
core position, the prominent jet of M87 in right ascension (RA) direction
would cause the peak position to be slightly offset from the “true core”
position towards the downstream side, due to the blending of near-core jet
emission within the finite beam size. To evaluate this effect, we used the
method in Hada et al. (2014), the M 87 structure was convolved by different
beam size with diameters ranging from the minor axis of the nominal
synthesized beam (shown in Table 1), whose direction was almost in RA
direction in our observations, to about four times larger. Then, we plotted
systematic changes of the brightness-peak position as a function of beam size.
In the case of 22 GHz, the image were restored with beam sizes ranging from
0.3 to 1.2 mas in incremental steps of 0.1 mas. We found a progressive
position shift of the brightness peak toward the downstream to be 10 $\mu$as
per 0.1 mas in RA direction. In the case of 44 GHz, the image was restored
with beam sizes ranging from 0.2 to 0.8 mas in incremental steps of 0.1 mas.
The position shift of the brightness peak toward the downstream was 6.8
$\mu$as per 0.1 mas in RA direction. In the case of 88 GHz, the image was
restored with beam sizes ranging from 0.15 to 0.6 mas in incremental steps of
0.05 mas. The position shift of the brightness peak toward the downstream was
2.5 $\mu$as per 0.1 mas in RA direction. That means the “true core” position
would be shifted to upstream, with respect to the brightness-peak position
when convolved with a nominal beam. At 22 GHz, the upstream shift in RA
direction would be about 10$\times$0.39 mas/0.1 mas$=39$ $\mu$as. At 44 GHz,
it would be about 6.8 $\times$ 0.22 mas/0.1 mas $=$15 $\mu$as for the 2019
epoch, and about 6.8$\times$0.24 mas/0.1 mas $=16$ $\mu$as for the 2021 epoch.
At 88 GHz, it was about 2.5$\times$0.16 mas/0.1 mas ${=}$4 $\mu$as. Where,
0.39 mas, 0.22 mas, 0.24 mas and 0.16 mas were the nominal beam sizes in the
RA direction (see Table 1). Consequently, the measured core shift in RA
direction from the peak-positions would be about 39$-15=24$ $\mu$as larger
than the “true core shift” in magnitude at 22-44 GHz and, 16$-4=$12 $\mu$as
larger at 44-88 GHz.

###  2.3 Error analysis

The error budgets of SFPR mainly include the dynamic tropospheric error, the
core identification error of M 87 and the statistical error from images. These
errors are independently to each other and the total errors can be calculated
as the root-sum-square of them. We adopt 11 $\mu$as for the dynamic
tropospheric error among frequencies under relatively stable weather
conditions, assuming 0.01 m uncanceled error by the water vapor fluctuation as
that in Hada et al. (2011). The absolute tropospheric position error for a
single frequency can be significantly larger than this value, while most of
the error can be canceled out by the FPT calibration in SFPR observations. We
also performed the error analysis for the core identification error of M 87 as
in Hada et al. (2011), using the core position differences between two
methods. One defined the centroid of the elliptical Gaussian fitting M 87 core
region as the core position, the other is the brightness-peak position of the
images convolved with a circular Gaussian beam of about a half of synthesized
beam in the core-jet direction. The uncertainties of core identification of M
87 in RA direction are 7 $\mu$as, 6 $\mu$as at 22 GHz and 44 GHz for the 2019
epoch, respectively. It was 5 $\mu$as, 3 $\mu$as at 44 GHz and 88 GHz for the
2021 epoch, respectively. Since the SFPR-ed images coupled the errors from
both frequencies, the statistical error at each frequency was 1/$\sqrt{2}$ of
the beam size divided by the signal-to-noise ratios of SFPR-ed image. It was
12 $\mu$as at 22 GHz and 44 GHz for the 2019 epoch, 14 $\mu$as at 44 GHz and
88 GHz for the 2021 epoch. The intrinsic structural uncertainty of M84 would
also affect the core-shift of M 87. In our observations, the position angles
of fitting to the core region of M 84 at 22 and 44 GHz with an elliptical
Gaussian were within 5 degrees in the north. Therefore the uncertainty of
core-shift effect in M 84 in RA direction was 5 $\mu$as, 3 $\mu$as and 2
$\mu$as at 22, 44 and 88 GHz, respectively. Other minor errors such as the
ionospheric residuals and geometric errors at each frequency were taken the
empirical values (Hada et al., 2011). The total errors by root-sum-square all
the above uncertainties give out 19 $\mu$as and 18 $\mu$as at 22 and 44 GHz
for the 2019 epoch, 19 $\mu$as and 18 $\mu$as at 44 and 88 GHz for the 2021
epoch, respectively.

##  3 Results and discussions

###  3.1 VLBI core of M 84 at 3 mm

The nearby elliptical galaxy M 84 is located in the center of Virgo Cluster at
a distance of 18.5 Mpc ($z=0.00339$) and has a central supermassive black hole
weighing $\sim 8.5\times 10^{8}M_{\odot}$. The combination of its proximity
and a large black hole mass yields a privileged linear resolution conversion
factor down to 1 micro-arcsecond ($\mu$as) $\sim$ 1 Schwarzschild radii
($R_{S}$), allowing us to investigate its close vicinity of the supermassive
black hole with VLBI. Two side jets are seen at a large viewing angle of
$\sim$74∘ (Meyer et al., 2018). The image of M 84 at 88 GHz (Figure 1) was
obtained by performing further clean and self-calibration in phase only to the
SFPR-ed visibility data, which were scan averaged. The MODELFIT task in Difmap
was used to fit the calibrated visibility with circular Gaussian components.
The VLBI core of M 84 could be fitted by a circular Gaussian with a flux
density of 38.0$\pm 5.7$ mJy and a diameter of 29$\pm 11$ $\mu$as well. The
apparent brightness temperature of the core (Kim et al., 2018), $T_{b,app}$,
can be calculated by

$T_{b,app}=1.22\times
10^{12}\frac{S_{core}(1+z)}{\nu^{2}\theta_{core}^{2}}\,K,$ (1)

where $S_{core}$ is the core flux density in Jy, $\nu$ is the observing
frequency in GHz, $\theta_{core}$ is the equivalent size in milli-arcsecond
(mas) and $z$ is the redshift. The $T_{b,app}$ of M 84 core at 88 GHz is $\sim
7.2\times 10^{9}$ K. Similar to other LLAGNs (Kim et al., 2018), the
$T_{b,app}$ is generally quite low. The derived brightness temperature could
serve as a lower limit as the core size of 29 $\mu$as (30 $R_{S}$) was only
one-tenth of the beam size and still un-resolved in our 88 GHz observations.

###  3.2 Core-shift of M 87

M87 is the most prominent elliptical galaxy within the Virgo Cluster, located
at a distance of 16.8$\pm$0.8 Mpc away. Its central supermassive black hole
($\sim 6.5\times 10^{9}M_{\odot}$) and jet have been well-studied in almost
every wave band from radio to $\gamma$-rays (Event Horizon Telescope
Collaboration et al., 2019; EHT MWL Science Working Group et al., 2021). The
core-shift of M 87 up to 43 GHz has been measured through conventional phase-
referencing to M 84 (Hada et al., 2011, 2013). The core-shift in RA direction
$r_{RA}(\nu)$ followed $\nu^{-0.94}$ and indicated that the black hole was
located at $\sim$41 $\mu$as eastwards of the 43 GHz core (Hada et al., 2011).
The core-shifts at 22-44 GHz and 44-88 GHz could be obtained from the SFPR-ed
images (Rioja & Dodson, 2011; Jiang et al., 2018) as shown in Figure 2. Since
the jet extended structure of M 84 is toward the north direction, the core-
shift of M 84 in RA direction could be negligible. The core-shift of M 87 in
RA direction could be obtained from the SFPR-ed images by Equation (2) in
Jiang et al. (2018). As a result, we obtained a position shift in RA direction
of $-64\pm 8$ $\mu$as at the 22 GHz core with regard to the 44 GHz core, and
of $-33\pm 11$ $\mu$as at 44-88 GHz. The error in 1$\sigma$ was given by JMFIT
in AIPS. After taking into account of the blending effect of near-core jet
emission in M 87 (see Section 2.2), the “true core shift” would be about
$-(64-24)$=$-40$ $\mu$as at 22-44 GHz and $-(33-12)$=$-21$ $\mu$as at 44-88
GHz. Incorporating the uncertainties given by error analysis (see Section
2.3), the core positions relative to that of the 44 GHz core in RA was $-40\pm
19$ $\mu$as at 22 GHz for the 2019 epoch and $21\pm 18$ $\mu$as at 88 GHz for
the 2021 epoch, respectively.
Using the same formula $r_{RA}(\nu)=A\nu^{-k}+B$ in Hada et al. (2011), the
above-mentioned two core-shifts in RA direction could be solved with solutions
$A=-1.45$, $k=0.92$ and $B=0.045$. As presented in the bottom-right corner of
Figure 3, assuming a jet position angle of $-69^{\circ}$ with respect to north
in M 87 (Kim et al., 2018), $B$ value indicates that the jet apex is located
at $\sim$48 $\mu$as upstream of the 43 GHz core. It was consistent with
previous result of $44\pm 13$ $\mu$as in Hada et al. (2011). Since there was
no strong flare event to cause the core-shift variations (Plavin et al., 2019)
during our observations as well as in Hada et al. (2011) session, it would be
reasonable to align the 43 GHz core positions of M 87 among these sessions.
The core-shift in RA at 22-43 GHz even during the elevated very high energy
Gamma-ray state in 2012 was found to be also at a similar level ($\sim$10
$\mu$as larger) (Hada et al., 2014). Furthermore, aligning the 43 GHz cores
among Hada et al. (2011) and the two epochs of this work, we fitted these
combined core-shift measurements with weighted least square method. It gave
out $A=-1.36\pm 0.15$, $k=0.92\pm 0.06$ and $B=0.043\pm 0.007$ as shown in
Table 2. The results are consistent and it implies the jet apex is 43$\pm 7$
$\mu$as in RA (as indicated by black dashed line in Figure 3) or at $\sim$46
$\mu$as upstream of the 43 GHz core.
Following the Equation (4) in Lobanov (1998), we could estimate the core-shift
measure $\Omega_{r,22-44}=0.13\pm 0.06$ [pc GHz] and $\Omega_{r,44-88}=0.12\pm
0.10$ [pc GHz], using the resultant power law index $k_{r}=1.09$. The magnetic
field strength at the 88 GHz core is estimated to be $4.8\pm 2.4$ G using the
mean value of $\Omega_{r}$, by Equation (B.2) in Paraschos et al. (2021) with
a spectral index of $-0.5$, a Doppler factor of 2, a jet viewing angle of 18∘
and an intrinsic jet opening angle of $63^{\circ}.6$ at 88 GHz (Kim et al.,
2018). This is consistent with the estimated 1-30 G near event horizon by the
Event Horizon Telescope at 230 GHz (Event Horizon Telescope Collaboration et
al., 2021).

##  4 Conclusions

We have successfully demonstrated that the SFPR technique could be applied to
the mm-VLBI observations of LLAGNs. It helps to overcome the limited coherent
intergration time and has great advantages in detecting the weak VLBI core as
well as measuring the core-shift at millimeter-wavelengths. The VLBI core of M
84 at 88 GHz was detected and the lower limit of its apparent brightness
temperature $\sim 7.2\times 10^{9}$ K was obtained. By means of SFPR to M 84,
the core-shift of M 87 in RA direction was determined at a precision of
$\sim$20 $\mu$as, which further constrained the jet apex at $\sim$46 $\mu$as
upstream of the 43 GHz core. With the aid of simultaneous multi-frequency
receiving system and more stations available (Zhao et al., 2019; Rioja &
Dodson, 2020), SFPR will be a very powerful tool to investigate the
compactness of the jet base at high frequencies as well as physical parameters
such as core structure, brightness temperature and magnetic field of the inner
region of jet.

The authors thank the anonymous referee for very critical and constructive
suggestions. This work was supported in part by the National Natural Science
Foundation of China (grant Nos. 11803071, 11590780, 11590784, and 11933007)
and Key Research Program of Frontier Sciences, CAS (grant No.QYZDJ-SSW-
SLH057). VLBA is operated by the National Radio Astronomy Observatory, which
is a facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc.

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Figure 1: Natural-weighted SFPR-ed VLBI image of M 84 at 88 GHz after self-
calibration. The peak flux density is 38.1 mJy beam-1. The contour levels are
at 2.2$\times(-1,1,2,4,8,16)$ mJy beam-1. The lowest level is three times the
rms noise of the image. The beam indicated in the left bottom corner is 0.29
mas $\times$ 0.16 mas at $-11^{\circ}$.

Figure 2: SFPR-ed images of M 84 at 44 (left) and 88 GHz (right) on 2019 June
22 and 2021 March 31, respectively. The peak intensity is 4.1 and 8.2 mJy
beam-1, respectively. The contour levels are at $-3$, 3, 6, 9 and 12 times the
rms of the images (0.3 and 1.0 mJy beam-1). The full-width at half-maximum of
the convolving beams (0.63 mas $\times$ 0.22 mas at $-15^{\circ}.6$ and 0.29
mas $\times$ 0.16 mas at $-11^{\circ}$.3) are shown in grey at the bottom-left
corners of each image. Figure 3: Offsets in right ascension (RA) with respect
to observing frequency from 43 GHz core in M 87. The data in hollow circles
are from Hada et al. (2011). The solid circles are the measurements in this
work. The fitting to the these two datasets with the same formula as Hada et
al. (2011) is the green curve, indicating the jet apex position of $43$
$\mu$as in RA (black dashed line). The fitting to the data of this work only
is the dashed curve in magenta. The subplot in the bottom-right corner shows
the core positions at 22, 44, 88 GHz and the jet apex, which are marked as
diamond, circle, triangle and star, respectively, overlapping with the VLBI
image of M 87 at 44 GHz in 2021. The contour levels are at 12.0 mJy beam-1
($27\sigma$ noise level of the image) stepping by a factor of 3, the beam in
grey is 0.53 mas $\times$ 0.22 mas at $-22^{\circ}.7$.  Table 1: Summary of
VLBI image parameters of M 84 and M 87 in the observations

|  |  | Synthesized beam on M 84 / M 87
---|---|---|---
No. | Epoch | Frequency | Major axis | Minor axis | Position angle | RA component
|  | GHz | mas | mas | degree | mas
1 | 2019-06-22 | 21.9 | 1.49 / 1.26 | 0.41 / 0.37 |  $-14^{\circ}.5$ / $-20^{\circ}.5$ | 0.42 / 0.39
2 | 2019-06-22 | 43.8 | 0.63 / 0.57 | 0.22 / 0.21 |  $-15^{\circ}.6$ / $-20^{\circ}.4$ | 0.23 / 0.22
3 | 2021-03-31 | 43.8 | 0.60 / 0.53 | 0.24 / 0.22 |  $-18^{\circ}.0$ / $-22^{\circ}.7$ | 0.25 / 0.24
4 | 2021-03-31 | 87.5 | 0.29 / 0.27 | 0.16 / 0.15 |  $-11^{\circ}.0$ / $-19^{\circ}.6$ | 0.16 / 0.16

Note. — Col.(1) Number of VLBI images at different epochs and observing
frequencies; Cols.(2)-(3) Observing date and frequency; Cols.(4)-(6) The
parameters (major axis, minor axis and position angle) of the nominal
synthesized beam at each frequency; Col.(7) The beam size in the RA direction
(RA component) of the nominal synthesized beam at each frequency.

Table 2: Fitting the location of jet apex in M 87

|  | $r_{RA}=A\nu^{-k}+B$
---|---|---
No. | Core-shift data used | A | k | B
1 | this work | $-1.45$ | 0.92 | $0.045$
2 |  Hada et al. (2011) and this work | $-1.36\pm 0.15$ | $0.92\pm 0.06$ | $0.043\pm 0.007$

Note. — Col.(1) Number of the fitting; Col.(2) Core-shift measurements used;
Cols.(3)-(5) Fitting parameters $A$, $k$ and $B$. $B$ is the offset of the jet
apex in RA direction from the 43 GHz core.