# Magnetic Suboxides as a Source of Two-Level System Losses in Superconducting
Niobium

D. Bafia<EMAIL_ADDRESS>Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  A. Grassellino  Fermi National Accelerator Laboratory,
Batavia, Illinois 60510, USA  A. Romanenko  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA

###### Abstract

We identify one potential source of quantum decoherence in three-dimensional
superconducting radio-frequency (SRF) resonators and two-dimensional transmon
qubits that utilize oxidized niobium: magnetic suboxides which drive two-level
system (TLS) losses. By probing the effect of sequential in situ vacuum baking
treatments on the RF peformance of bulk Nb SRF resonators and on the oxide
structure of a representative Nb sample within a time-of-flight secondary ion
mass spectrometer (TOF-SIMS), we find a non-monotonic evolution of cavity
quality factor $Q_{0}$ which correlates with the interplay of TLS-hosting
magnetic suboxide generation and oxide dissolution. We localize this effect to
the oxide itself and present the non-role of diffused interstitial oxygen in
the underlying Nb by regrowing a new oxide via chemistry and wet oxidation
which reveals a mitigation of aggravated TLS losses.

††preprint: APS/123-QED

##  I Introduction

The realization of a computationally useful superconducting quantum computer
requires the maximization of quantum coherence lifetimes to unlock quantum
memory, high-fidelity gate operations, and better entanglement among qubits
[1]. In superconducting systems, quantum coherence is limited by dissipative
decay channels present in both the linear resonator and the transmon qubit
that contains the nonlinear Josephson junction necessary to create uniquely
addressable energy levels. Current decoherence candidates include quasi-
particles [2, 3], radiation-induced losses [4, 5], and dielectric losses such
as two-level systems (TLS) [6], to name a few. Dielectric loss arises from
each material and interface used in the construction of superconducting qubits
and dictates the total decay rate of the system according to

$\Gamma_{1}=\frac{1}{T_{1}}=\frac{\omega}{Q_{0}}=\omega\sum_{i}\frac{F_{i}}{Q_{i}}+\Gamma_{0},$
(1)

where $T_{1}$ is the photon lifetime, $\omega$ is angular frequency, $Q_{0}$
is the quality factor, $F_{i}$ and $Q_{i}$ are the filling and quality factors
of the $i^{th}$ loss channel, and $\Gamma_{0}$ captures losses driven by non-
dielectric mechanisms [7, 8]. The quality factor is inversely related to the
dielectric loss tangent $Q_{i}=1/\text{tan}\delta_{i}$. As a result, decay
rate minimization requires the mitigation of every loss channel down to
tolerable levels. While straightforward for a single interface, this task is
difficult in complex multi-layer systems like those used in modern qubit
fabrication designs. Many such designs utilize thin films of aluminum and
niobium atop dielectric or semiconducting substrates, leading to multiple
potentially deleterious materials and interfaces.

Recent work on three-dimensional (3-D) niobium superconducting radio-frequency
(SRF) resonators in the low GHz frequency range has highlighted the amorphous
native niobium oxide as a dramatic source of TLS loss, implying an inherent
limitation in qubits fabricated with oxidized niobium [9, 10]. By removing the
native niobium oxide via in situ vacuum baking at 340-450 ∘C for several
hours, a ten-fold improvement in quality factor $Q_{0}$ was observed,
demonstrating intra-resonator photon lifetimes of $T_{1}\approx$ 2 s, a
200-fold increase over the previous state-of-the-art [11]. By fitting
temperature dependent $Q_{0}$ data of bulk niobium resonators with the
standard TLS model [12, 13], Romanenko et al. found the dielectric loss
tangent of the native niobium oxide at 5 GHz to be $\delta_{0}$ = 0.08. This
value is in agreement with values obtained from niobium-on-silicon lumped
element resonators post sequential oxidation [14] and niobium-on-silicon
coplanar-waveguides post sequential etching [15]. This consistency between
dramatically different architectures demonstrates that observations of the Nb-
air interface made in simplified, single interface systems such as 3-D Nb SRF
resonators are directly translatable to more complex systems such as those
used in transmon qubits, allowing for a one-to-one correspondence between thin
film and bulk material measurements.

While there exists an empirical mitigation strategy to minimize their
deleterious effect in superconducting quantum systems, the precise origin and
nature of the TLS losses within the niobium oxide is not yet fully understood.
A complete picture of the underlying mechanisms would aid in the further
development of mitigation strategies that may instead be stable to air. There
exist a few possible origins for these TLS losses. Oxygen vacancies within the
amorphous oxide have the potential of creating magnetic impurities [16, 17]
and driving impedance, as supported by X-ray absorption spectroscopy studies
[18] and density functional theory calculations [18, 19]. These oxygen
vacancies may serve as OH tunneling sites [20]. Another possibility stems from
the conducting or magnetic nature of the NbOx oxides [10]. Here, we present
results that support the role of the latter in driving TLS losses in oxidized
niobium.

One avenue for identifying the origins of loss in the amorphous native Nb
oxide is by studying its evolution with in situ baking, allowing for the
investigation of the role of various oxides on $Q_{0}$. There exists extensive
literature on this topic as it is relevant for high $Q_{0}$ and high electric
field applications of Nb SRF resonators in particle accelerators. It is well
known that native niobium oxides exist in a graded structure composed of Nb2O5
(insulator [21]), NbO2 (semiconductor [22]), NbO (metallic [23]), and the
metal substrate [21, 24]. In situ vacuum baking dissolves the dominant
pentoxide and grows the suboxides, reducing according to Nb2O5 $\rightarrow$
NbO2 $\rightarrow$ NbO [24, 25, 26, 27]. Ma et al. showed that in situ vacuum
baking at 150∘C for 40 hours reduced the pentoxide to NbO2; continued baking
at 280∘C for 8 hours showed further reduction to metallic NbxO (x$\sim$2)
[25]. Further studies by Ma et al. revealed the reduction of the pentoxide to
NbO2 and NbO post vacuum baking at 250∘C for several hours [27]. These results
are corroborated by the synchrotron measurements of Delheusy et al. [26] and
ARXPS measurements by Dacca et al. [23]. Moreover, at the expense of
dissolving the oxide, in situ vacuum baking drives inward oxygen diffusion
from the oxide over the oxide/metal interface rather than into the vacuum [25,
28, 29, 26, 30]. Subsequent reoxidation studies via exposure to ambient air
shows that a partially dissolved oxide grows at the oxide/metal interface and
is morphologically equivalent to the native grown oxide without prior heat
treatment [27].

In this paper, by investigating the role of vacuum baking on niobium SRF
resonators that contain a $\sim$5 nm full wet-grown native amorphous oxide, we
find evidence that suggests that potentially magnetic niobium suboxides may
serve as a source of TLS that limits the performance of 3-D Nb resonators and
2-D transmon qubits which utilize oxidized Nb. Modest in situ vacuum baking
treatments (150 ∘C-200 ∘C) for durations as short as 25 minutes aggravate TLS-
like losses that are eventually suppressed with increasing bake time. We
recreate the in situ vacuum baking treatments within a time-of-flight
secondary ion mass spectrometer (TOF-SIMS) on a Nb SRF resonator cutout and
confirm that the oxide reduces considerably, suggesting the formation of
possibly magnetic suboxides. We hypothesize that the non-monotonic evolution
of $Q_{0}$ with successive baking is due to the interplay of TLS-hosting
magnetic suboxide generation and oxide dissolution.

##  II Experimental Method

For our studies, we used two 1.3 GHz TESLA-shaped [31] niobium single-cell SRF
resonators that have undergone a bulk 120 $\mu$m removal from the inner RF
surface via electropolishing, followed by 800 ∘C degassing, and an additional
40 $\mu$m removal from the inner surface via electropolishing [32]. The
resonators were then exposed to air and high pressure rinsed with ultra-pure
water, forming a $\sim$5 nm amorphous native oxide. Afterwards, the resonators
were assembled for testing and evacuated to a vacuum level of approximately
1E-5 Torr. Typically, resonators tested at this point are called
electropolished (EP) resonators. The resonators then underwent sequential
testing and treatment to various low temperature vacuum bakes [33] for
increasing durations while actively pumping to maintain the vacuum level
within the resonators. Such treatments have been shown to dissolve the native
niobium oxide and introduce oxygen interstitial into the underlying niobium
[25, 28, 29, 26, 30].

One resonator, TE1AES019, was first subjected to an in situ low temperature
vacuum bake at 90 ∘C for 384 hours followed by sequential in situ baking steps
at 200 ∘C for increasing durations. For the final treatment, the resonator was
baked at 340 ∘C for 5 hours to ensure complete removal of the niobium
pentoxide [10]. The resonator was RF tested after each step and maintained
vacuum throughout the entire course of the study.

The second resonator, TE1AES021, was used to probe baking treatments that more
closely resembled those typically utilized in transmon qubit fabrication and
to better identify the source of additional RF loss. It underwent sequential
in situ vacuum baking treatments at 150 ∘C with resonator RF measurements
performed after each step. After the RF test post baking at 150 ∘C for a net
duration of 15 hours and 25 minutes, resonator TE1AES021 was subjected to an
ex situ vacuum bake at 200 ∘C for 19.5 hours. This involved first venting the
resonator to atmosphere followed by vacuum baking. Afterwards, the resonator
was once again high pressure rinsed, reforming the amorphous $\sim$5 nm native
oxide, but this time with a larger concentration of subsurface O interstitial
present compared to the baseline measurement post EP. For the final treatment,
the resonator underwent an HF acid rinse, which dissolved the native oxide,
and upon subsequent high pressure rinsing, reformed a new native oxide at the
expense of consuming approximately 2 nm of niobium [34].

To test our resonators, we utilized methods similar to those outlined in [9,
35]. Resonators were installed in large vertical helium dewars and cooled
through the superconducting transition temperature at $\sim$9.25 K while
employing methods to minimize the possibility of trapping magnetic flux [36,
37]. Resonators were then driven at their resonant frequency while using a
phase-locked loop to track and maintain resonance. We used a signal analyzer
set to a resolution bandwidth of $\sim$220 Hz to perform single shot, zero
span decay measurements of the transmitted power $P_{t}(t)$ from the resonator
after shutting off the incident RF power. By fitting the resulting $P_{t}(t)$
data with the procedure outlined in [9], we obtained a measure of $Q_{0}$ down
to fields of $E_{acc}\sim$ 1 kV/m with experimental error never exceeding 10
$\%$. The fields were then converted into a measure of the photon number via
$n=U/\hbar\omega$, where $U$ is the stored energy in the resonator, which is
related to $P_{t}(t)$. We performed the tests at temperatures of 1.4-1.6 K,
where the population of quasi-particles was virtually non-existent, and their
contribution to the surface resistance was negligible. As a result, the
quality factor in these tests was dominated by losses stemming from material
properties. We note that while most TLS remain thermally saturated at these
temperatures, there remains a small fraction of unsaturated TLS which drive
the characteristic response of $Q_{0}$ with intra-resonator photon number [9,
10].

##  III Results

###  III.1 Resonator RF measurements

The upper panel in Fig. 1 presents data acquired on resonator TE1AES019 after
an initial 90 ∘C $\times$ 384 hour in situ vacuum bake followed by sequential
in situ vacuum bakes at various temperatures and durations. Shown also for
reference is data from Romanenko and Schuster taken on an EP resonator which
contains a full 5 nm native wet-grown oxide and serves as our baseline test
[9]. For all tests, we observe the expected saturation of the quality factor
at electric fields $<$ 0.01 MV/m (i.e. photon numbers $<$ 2E18) that is
characteristic of two-level systems [9, 10, 6]. We find that the 90 ∘C
$\times$ 384 hours $+$ 200 ∘C $\times$ 1 hour in situ vacuum baking treatment
produces a resonator with a low field quality factor of 9E9, roughly three
times lower than the baseline test. An additional 200 ∘C $\times$ 5 hour in
situ vacuum bake produces identical results at fields below 0.1 MV/m. After a
second round of 200 ∘C $\times$ 5 hour vacuum bake, a slight improvement in
$Q_{0}$ up to 1.2E10 occurs. For the final treatment, we subjected the
resonator to a 340 ∘C $\times$ 5 hour vacuum baking treatment which is known
to fully remove the native Nb2O5 layer [10, 38]. The resulting measurements
display a $Q_{0}$ at low fields of greater than 7E10, the highest achieved in
this study, and in line with previous work [10, 38]. Overall, these tests show
a non-monotonic dependence of resonator $Q_{0}$ with bake duration.

Figure 1: Measurements of quality factor vs electric field at temperatures of
1.4 K-1.6 K for the 1.3 GHz 3-D Nb resonators TE1AES019 (upper) and TE1AES021
(lower) after sequential baking and chemical treatments. Upper axes plot the
converted intra-resonator photon number. For reference, the gray left pointed
triangles depict data obtained on an EP resonator which contained a full 5 nm
native wet grown oxide from Romanenko and Schuster [9]. Solid black lines show
fits with Eq. 2.

The lower panel in Fig. 1 depicts data taken on resonator TE1AES021 post
various baking and chemical treatments. Surprisingly, we find that a modest in
situ vacuum bake at 150 ∘C for 25 minutes reduces the resonator $Q_{0}$ by a
factor of two when compared to the baseline EP test from Romanenko and
Schuster. Subjecting the resonator to an additional round of in situ vacuum
baking at 150 ∘C for 15 hours produces a low field $Q_{0}$ of 9E9, identical
to the results obtained on TE1AES019 after the first and second tests. The
results post ex situ bake at 200 ∘C for 19.5 hours, for which we re-iterate
received subsequent high pressure rinsing with ultra-pure water, yield a
$Q_{0}$ that is similar in value to the baseline EP at fields $<$ 4E-3 MV/m.
Subsequent HF acid rinsing and high pressure water rinsing reveals a $Q_{0}$
vs $E_{acc}$ curve that is nearly identical to the baseline EP data.

We fit the data in Fig. 1 with the standard TLS model with a quasi-particle
term built in. The electric field dependence of a niobium SRF resonator at low
fields may be modelled as a single interface system (metal/air) containing TLS
according to

$\frac{1}{Q_{0}}=\frac{F\delta_{0}}{\big{(}1+\big{(}\frac{E_{acc}}{E_{c}}\big{)}^{2}\big{)}^{\beta}}+\frac{1}{Q_{qp}}$
(2)

where $\beta$ is a fitting parameter, $E_{c}$ is the characteristic saturation
field, $\delta_{0}$ is the dielectric loss, and $Q_{qp}$ is the quasi-particle
contribution to RF loss [13, 9]. Fig. 2 plots the product of the filling
factor and dielectric loss tangent for each curve presented in Fig. 1. We
choose to present this convolved product as it is agnostic to the precise
values of the filling factor and loss tangent and gives a qualitative measure
of the TLS contribution to RF loss. Resonator TE1AES019 shown in the upper
panel of Fig. 2 displays a non-monotonic dependence of the TLS-driven losses
with in situ vacuum baking, achieving as low as 3.63E-12 after the final round
of treatment, consistent with previous results [10]. TE1AES021 presented in
the lower panel suggests that while in situ vacuum baking increases TLS-driven
RF losses, treatments which utilize high pressure water rinsing returns them
to the baseline EP values.

Figure 2: Fitted $F\delta_{0}$ values and fit error of TE1AES019 (upper) and
TE1AES021 (lower) using the data presented in Fig. 1. The 90 ∘C $\times$ 384
hours in situ vacuum bake that precedes the sequential in situ vacuum baking
treatments for resonator TE1AES019 has been omitted from the horizontal axis
for brevity. Most error bars are smaller than the symbol size.

###  III.2 TOF-SIMS studies

As the amorphous niobium oxide is known to be a host of TLS, it is likely that
its dynamics dictate the performance evolution observed in Figs. 1 and 2. To
investigate this possibility, we studied the effect of sequential vacuum
baking at 205 ∘C on an electropolished niobium SRF resonator cutout in a time-
of-flight secondary ion mass spectrometer (TOF-SIMS). We used a liquid bismuth
ion Bi+ beam to perform secondary ion mass measurements while sputtering with
a 1 keV cesium ion Cs+ gun to obtain depth profiles of the oxide. The sample
was sequentially vacuum baked in steps of 205 ∘C in situ followed by cool down
to and measurement at room temperature. Two depth profiles were obtained at
different spots on the sample after each step of treatment. All measurements
were performed in a vacuum better than 1E-10 Torr. Fig. 3 presents the
resulting averaged Nb2O5- data.

Figure 3: TOF-SIMS depth profiles of the Nb2O5- signal acquired on an as-
received electropolished resonator cutout subjected to sequential rounds of in
situ vacuum baking. Durations presented in the legend represent integrated
bake times.

The as-received electropolished resonator cutout, which is denoted as “EP” in
Fig. 3, displays a fully intact 5 nm native pentoxide. After the first round
of in situ baking at 205 ∘C for 3 hours, we find that this Nb2O5- layer is
reduced slightly. In addition, there is an anomalous tail which begins at a
depth of 4.1 nm and extends further toward the bulk. After baking the sample
for a total of 19 hours, we find a nearly identical Nb2O5- signal compared to
the preceding treatment but without the presence of the anomalous tail. Baking
for a net duration of 43.5 hours shows a further reduction in the thickness of
the Nb2O5- layer.

The evolution of the SIMS profiles presented in Fig. 3 confirms the gradual
dissolution of the native Nb2O5 layer with sequential in situ vacuum baking at
205 ∘C. This confirms the observations of previous works and suggests that
this dissolution is accompanied by the growth of the NbO2 and NbO suboxides
[24, 25, 26, 27, 25, 27, 26, 23] as well as the inward diffusion of oxygen
over the oxide/metal interface [25, 28, 29, 26, 30].

##  IV Discussion

The prominent increase in TLS-driven losses combined with the reduction in
Nb2O5 and subsequent growth of suboxides after in situ baking at 205 ∘C or 150
∘C suggests the role of niobium suboxides as a source of TLS. Indeed, previous
tunneling and XPS measurements on cavity-grade niobium baked at various
temperatures and durations show that, when compared to an unbaked niobium
sample, the 250 ∘C $\times$ 2 hr bake not only causes Nb2O5 to reduce into its
suboxides but also yields an increased level of subgap states which is likely
driven by magnetic moments formed due to the presence of oxygen vacancies
within the oxide [17]. Moreover, other work has shown that these magnetic
moments increase with oxygen vacancy concentration [16]. As such, in situ
vacuum baking may promote the growth of magnetic suboxides that drive TLS-like
losses in 3-D niobium resonators. The aggravation and subsequent mitigation of
TLS-driven losses post in situ vacuum baking is likely due to the interplay of
magnetic suboxide generation and oxide dissolution.

This hypothesis is further corroborated by the fact that treatments which re-
expose the resonator to air and reform the native oxide show nearly identical
TLS-driven losses when compared to the baseline EP test. Moreover, despite the
fact that TE1AES021 post the additional HF acid rinse contained a larger
concentration of interstitial oxygen in the underlying niobium material, it
gave nearly identical performance to the test post baseline EP in Figs. 1 and
2. This localizes the TLS losses to the oxide and asserts the non-role of
interstitial oxygen in niobium. This further supports the conclusions made in
[9].

We note that while TE1AES021 post the additional 200 ∘C for 19.5 hour ex situ
bake yields similar results to the baseline EP test, they vary slightly.
Previous work suggests that a partially dissolved oxide readily forms a new
oxide upon re-oxidation that is morphologically identical to the native grown
oxide without an intermediate heat treatment [27]. However, our results
suggest there may exist some minute variations at the interface which may
drive the differing RF performance. This will be a subject of future study.

Fig. 4 presents the additional surface resistance due to TLS-induced
dissipation for resonator TE1AES019 post the 90 ∘C $\times$ 384 hours $+$ 200
∘C $\times$ 1 hour vacuum baking treatment. This was obtained using the
relation $R_{s}=G/Q_{0}$ ($G$ is a known geometry factor) and by subtracting
off the contribution from non-TLS losses. Surprisingly, the surface resistance
at low fields levels off at 22.5 n$\Omega$, roughly two times larger than what
was reported for a 100 nm thick niobium oxide grown via anodization [9]. This
is also well explained by the magnetic suboxide hypothesis as oxides grown
through anodization are known to contain fewer oxygen vacancies than wet grown
oxides [39]. As a result, while the partially depleted oxide post vacuum
baking is significantly thinner than 100 nm, it may host a larger
concentration of magnetic moments within the suboxides and thus introduce a
greater level of TLS losses.

Figure 4: Surface resistance introduced by TLS for resonator TE1AES019 post
90∘C $\times$ 384 hour + 200∘C $\times$ 1 hour shown in the upper panel in
Fig. 1.

While the present work focuses on the effect of vacuum baking on 3-D niobium
SRF resonator performance in the quantum regime, the observations made here
are applicable to transmon qubits which utilize oxidized niobium. As many
qubit fabrication processes perform vacuum baking treatments after the
deposition and oxidation of niobium thin films, there exists the potential of
generating additional magnetic suboxides which may further limit transmon
qubit quantum coherence times.

##  V Conclusion

In summary, we studied the effect of low temperature vacuum baking on the
quality factor of two 1.3 GHz 3-D niobium SRF resonators and found evidence
which suggests that magnetic suboxides are a potential source of TLS losses
that limit performance in both Nb resonators and transmon qubits which utilize
oxidized Nb. In situ vacuum baking produces an aggravation and subsequent
mitigation of these losses due to the interplay of magnetic suboxide
generation and oxide dissolution.

##  VI Acknowledgments

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, National Quantum Information Science Research Centers,
Superconducting Quantum Materials and Systems Center (SQMS) under contract
number DE-AC02-07CH11359.

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# Lyapunov-type conditions and stochastic differential equations driven by
$G$-Brownian motion

Xinpeng LI  ,  Xiangyun LIN  and  Yiqing LIN  Xinpeng LI
Institute for advanced research and school of mathematics
Shandong University
250100 Jinan, China
Xiangyun LIN
College of Mathematics and Systems Science
Shandong University of Science and Technology
266590 Qingdao, China
Yiqing LIN
Fakultät für Mathematik, Universität Wien
1090 Wien, Austria
_Corresponding author:_ Xiangyun LIN<EMAIL_ADDRESS>
###### Abstract.

This paper studies the solvability and the stability of stochastic
differential equations driven by $G$-Brownian motion (GSDEs). In particular,
the existence and uniqueness of the solution for locally Lipschitz GSDEs is
obtained by localization methods, also the stability of such GSDEs are
discussed with Lyapunov-type conditions.

###### Key words and phrases:

$G$-Brownian motion; stopping times; $G$-stochastic differential equations;
Lyapunov functions; stability.

###### 2000 Mathematics Subject Classification:

60H05; 60H10

Xipeng Li research is supported by Fundamental Research Funds of Shandong
University under grant No. 2014GN007 and by China Postdoctoral Science
Foundation under grant No. 2014M561907; Xiangyun Lin research is supported by
the National Natural Science Foundation of China under grant No. 11271007;
Yiqing LIN research is supported by European Research Council (ERC) under
grant FA506041.

##  1\. Introduction

Recently, Peng [12] set up a framework of a time consistent sublinear
expectation associated with a new type of Brownian motion that draws a lot of
attention. This sublinear expectation, named $G$-expectation, is primarily
defined as a sublinear functional $\mathbb{E}[\cdot]$ on a well-posed space
$C_{b,lip}(\Omega)$, which is the collection of all bounded and Lipschitz
functions on the canonical space $\Omega:=\mathcal{C}[0,\infty)$. Under the
$G$-expectation, the canonical process $B$ is seen to be a $G$-Brownian motion
whose increments are zero-mean, independent, and $G$-normal distributed.
Observed in a later work of Denis-Hu-Peng [4], this type of sublinear
expectations can be regarded as the supreme of linear expectations over a
weakly compact subset of martingale measures and thus, induces a capacity on
$\Omega$. Accordingly, the notion of “quasi-sure” (q.s.) related to this
capacity is brought to this framework.

As the foundation of the $G$-stochastic calculus, Peng also introduces the
$G$-Itô integral in his pioneer work [12]. Indeed, this integral is first
defined pathwisely as the Riemann-Stieltjes sum on $[0,T]$ for all “adapted”
step process formed by random variables in $C_{b,lip}(\Omega)$. And then,
owing to the $G$-Itô inequality, this definition can be extended to the
$M^{2}_{G}$-norm (similar to the classical $M^{2}$-norm) closure of the step
process space. Following Peng’s work, related topics in the $G$-stochastic
analysis are systematically developed by many authors.

In particular, some authors are interested in the stochastic differential
equation driven by $G$-Brownian motion (GSDE), which has a similar form as its
counterpart in the classical framework, however, holds in a q.s. sense:

(1.1) $X_{t}=x+\int^{t}_{0}f(s,X_{s})ds+\int^{t}_{0}h(s,X_{s})d\langle
B\rangle_{s}+\int^{t}_{0}g(s,X_{s})dB_{s},\ 0\leq t\leq T,\ q.s.,$

where $\langle B\rangle$ is the quadratic variation of the $G$-Brownian
motion. Under the Lipschitz assumptions on the coefficients, Peng [12] and Gao
[5] have proved the wellposedness of such equation with the fixed-point
iteration. Moreover, Bai-Lin [2] has studied the case when coefficients are
integral-Lipschitz and Lin [9, 10] considers the reflected GSDEs with some
good enough boundaries. We observe in these papers the Lipschitz structure of
the coefficients and the condition on the reflecting boundary make sure that
the solution $X$ and $g(\cdot,X_{\cdot})$ live in the space of good integrands
defined by Peng [12].

In contrast to the existing results on GSDEs, the focus of the present paper
is on (1.1) with local Lipschitz coefficients under a Lyapunov-type condition,
while detailed discussion to this kind of classical stochastic differential
equations (SDEs) can be found, for example, in Has’minskiǐ [6]. Following a
standard procedure, we shall apply the localization technique to approximate
the solution of this equation, however, one can see that the localized
coefficient will be eliminated from the integrand space defined in [12], since
it may be no longer “continuous” in $\omega$ somehow. Thanks to the work of
Li-Peng [8], the space of suitable integrands are essentially expanded, which
requires less regularity and only “local integrability”. Indeed, Li-Peng first
extend the definition of the $G$-Itô integral for all step process formed by
bounded random variable, and verify that in their case the $G$-Itô inequality
still holds true, so that this definition can be extended by the completion
under the $M^{2}_{G}$-norm. Due to this extension, one can consider the
$G$-Itô integral for such process ${\bf 1}_{[0,\tau]}(\cdot)$, where $\tau$ is
an $\mathcal{F}^{B}$-stopping time taking values in $[0,T]$ and thus, the
$G$-Itô integral is said to be well defined in [8] on an even larger class
$M^{2}_{w}([0,T])$ by stopping techniques. Taking advantage of these results,
we deduce that this kind of GSDEs are well posed on $M^{2}_{w}([0,T])$.
Particularly, we define the truncated GSDEs that are uniform Lipschitz, and
carefully choose the stopping times in order to construct a consistent
localized sequence. Finally, the Lyapunov-type condition ensures that the
solution can be approximated pathwisely.

Another task of this paper is to consider the stability of the GSDEs under
perturbations of the initial value, which shows a rough picture of asymptotic
behavior of GSDEs under certain conditions. We recall Has’minskiǐ [6] and Mao
[11] for details in this topic for SDEs. In the $G$-framework, quasi-sure
exponential stability of a linear GSDE has been considered in Zhang-Chen [14]
and the (exponential) moment stability of GSDEs has been studied by Hu-Ren-Xu
[7] with a particular Lyapunov function (independent of $h$ and $g$). In what
follows, we obtain the sufficient conditions for the moment stability by means
of a general Lyapunov function and investigate the relation between the moment
stability and the quasi-sure stability for GSDEs. Moreover, a property of
$G$-Brownian motion is obtained at the end of this paper.

This paper is organized as follows: Section 2 provides preliminaries in the
$G$-framework for our further discussion; in Section 3 we conclude the main
idea in [8] and explain the stopping techniques in the $G$-framework; Section
4 studies the solvability of local Lipschitz GSDEs while Section 5 shows the
stability results for such equations; in the appendix, Proposition 1.4 in IV-1
of [12] is generalized, which is essential for our argument.

##  2\. Preliminaries

We first recall some notions and results of $G$-expectation and the related
spaces of random variables and stochastic processes. The reader interested in
more detailed description of these notions and results is referred to Denis-
Hu-Peng [4], Gao [5], Li-Peng [8] and Peng [12].

Let $\Omega$ be the space of all $\mathbb{R}^{d}$-valued continuous paths with
$\omega_{0}=0$ equipped with the distance

$\rho(\omega^{1},\omega^{2}):=\sum^{\infty}_{N=1}2^{-N}((\max_{t\in[0,N]}|\omega^{1}_{t}-\omega^{2}_{t}|)\wedge
1),$

and let $B_{t}(\omega):=\omega_{t}$ be the canonical process.

Set

$L_{ip}(\Omega):=\\{\varphi(B_{t_{1}},\cdots,B_{t_{n}}):n\geq 1,\ 0\leq
t_{1}\leq\cdots\leq t_{n}<\infty,\ \varphi\in C_{b,lip}(\mathbb{R}^{d\times
n})\\},$

where $C_{b,lip}(\mathbb{R}^{d\times n})$ the collection of all bounded and
Lipschitz functions on $\mathbb{R}^{d\times n}$.

Fixing a sublinear monotone function $G$:
$\mathbb{S}^{d}\rightarrow\mathbb{R}$, the related $G$-expectation on
$(\Omega,L_{ip}(\Omega))$ can be constructed in the following way: for each
$\xi\in L_{ip}(\Omega)$ with the form

$\xi=\varphi(B_{t_{1}},B_{t_{2}}-B_{t_{1}},\cdots,B_{t_{n}}-B_{t_{n-1}}),\ \
0\leq t_{1}<t_{2}<\cdots<t_{n},$

define

$\mathbb{E}[\xi]:=u_{1}(0,0),$

where $u_{1}(0,0)\in\mathbb{R}$ is obtained via the following procedure: for
$k=n,\cdots,1$, $u_{k}:=u_{k}(t,x;x_{1},\cdots,x_{k-1})$ is a function of
$(t,x)$ with the parameter
$(x_{1},\cdots,x_{k-1})\in\mathbb{R}^{d\times(k-1)}$, which is the solution of
the following $G$-heat equation defined on
$[t_{k-1},t_{k})\times\mathbb{R}^{d}$:

$\frac{\partial u_{k}}{\partial t}-G(D^{2}u_{k})=0$

with terminal condition

$u_{k}(t_{k},x;x_{1},\cdots,x_{k-1})=u_{{k+1}}(t_{k},x;x_{1},\cdots,x_{k-1},x),$

where
$u_{n+1}(t_{n},x;x_{1},\cdots,x_{n-1},x):=\varphi(x_{1},\cdots,x_{n-1},x)$.
Note that under this $\mathbb{E}[\cdot]$, the canonical process
$(B_{t})_{t\geq 0}$ is a $G$-Brownian motion (see §III-2 in [12]).

We denote by $L_{G}^{p}(\Omega)$ the completion of $Lip(\Omega)$ under the
natural norm $||\cdot||_{p}:=\mathbb{E}[|\cdot|^{p}]^{\frac{1}{p}}$. For each
$t\in[0,\infty)$, we list the following notations:

  * $\bullet$

$\Omega_{t}:=\\{\omega_{\cdot\wedge t}:\omega\in\Omega\\}$;

  * $\bullet$

$\mathcal{F}_{t}:=\mathcal{B}(\Omega_{t})$;

  * $\bullet$

$L^{0}(\Omega)$: the space of all $\mathcal{B}(\Omega)$-measurable real
functions;

  * $\bullet$

$L^{0}(\Omega_{t})$: the space of all $\mathcal{B}(\Omega_{t})$-measurable
real functions;

  * $\bullet$

$B_{b}(\Omega)$: all bounded elements in $L^{0}(\Omega)$;
$B_{b}(\Omega_{t}):=B_{b}(\Omega)\cap L^{0}(\Omega_{t})$;

  * $\bullet$

$C_{b}(\Omega)$: all continuous elements in $B_{b}(\Omega)$;
$C_{b}(\Omega_{t}):=C_{b}(\Omega)\cap L^{0}(\Omega_{t})$.

We can extend the domain of $G$-expectation $\mathbb{E}[\cdot]$ from
$L_{ip}(\Omega)$ to $L^{0}(\Omega)$ by the procedure introduced in [4], i.e.,
constructing an upper expectation $\bar{\mathbb{E}}[\cdot]$:

$\bar{\mathbb{E}}[X]:=\sup_{\mathbb{P}\in\mathcal{P}_{G}}E^{\mathbb{P}}[X],\
X\in L^{0}(\Omega),$

where $\mathcal{P}_{G}$ is a weakly compact family of martingale measures on
$(\Omega,\mathcal{B}(\Omega))$. This upper expectation coincides with the
$G$-expectation $\mathbb{E}[\cdot]$ on $L_{ip}(\Omega)$ and thus, on its
completion $L^{1}_{G}(\Omega)$. Naturally, a Choquet capacity can be defined
by:

$\bar{C}(A):=\sup_{\mathbb{P}\in\mathcal{P}_{G}}\mathbb{P}(A),\
A\in\mathcal{B}(\Omega),$

and the notation of “quasi-surely” (q.s.) is introduced as follows:

######  Definition 2.1.

A set $A\in\mathcal{B}(\Omega)$ is called polar if $\bar{C}(A)=0.$ A property
is said to hold quasi-surely if it holds outside a polar set.

We note that the following Borel-Cantelli Lemma, Markov’s inequality and
Upwards convergence theorem also holds for such Choquet capacity and upper
expectation (Lemma 5, Lemma 13 and Theorem 10 in [4])

######  Lemma 2.2.

Let $X\in L^{0}(\Omega)$ satisfying $\bar{\mathbb{E}}[|X|^{p}]<\infty$, for
$p>0$. Then, for each $a>0$,

$\bar{C}(\\{|X|>a\\})\leq\frac{\bar{\mathbb{E}}[|X|^{p}]}{a^{p}}.$

######  Lemma 2.3.

Let $\\{A_{n}\\}_{n\in\mathbb{N}}\subset\mathcal{B}(\Omega)$ such that

$\sum^{\infty}_{n=0}\bar{C}(A_{n})<\infty.$

Then, $\limsup_{n\rightarrow\infty}A_{n}$ is a polar set.

######  Theorem 2.4.

Let $\\{X^{n}\\}_{n\in\mathbb{N}}\subset L^{0}(\Omega_{T})$ be a sequence such
that $X^{n}\uparrow X$, q.s., and there exists a
$\mathbb{P}\in\mathcal{P}_{G}$ with $E^{\mathbb{P}}[X^{0}]>-\infty$, then
$\bar{\mathbb{E}}[X^{n}]\uparrow\bar{\mathbb{E}}[X]$.

We also have Fatou’s lemma (Lemma 2.11 in Bai-Lin [2]) in the $G$-framework:

######  Lemma 2.5.

Assume that $\\{X^{n}\\}_{n\in\mathbb{N}}$ is a sequence in $L^{0}(\Omega)$
and for a $Y\in L^{0}(\Omega)$ satisfying $\bar{\mathbb{E}}[|Y|]<\infty$ and
all $n\in\mathbb{N}$, $X^{n}\geq Y$, q.s., then

$\bar{\mathbb{E}}[\liminf_{n\rightarrow\infty}X^{n}]\leq\liminf_{n\rightarrow\infty}\bar{\mathbb{E}}[X^{n}].\\\
$

In order to introduce stopping time in $G$-framework, Li-Peng [8] first
consider the process space generalized by $B_{b}(\Omega)$ and define the
related Itô’s integral. Let $L^{p}_{*}(\Omega)$ be the completion of
$B_{b}(\Omega)$, and consider the following type of simple processes:

$\displaystyle M^{0}_{b}([0,T])$
$\displaystyle=\\{\eta:\eta_{t}(\omega)=\sum_{i=0}^{N-1}\xi_{i}(\omega)\mathbf{1}_{[t_{i},t_{i+1})}(t),$
$\displaystyle\forall N\in\mathbb{N},\ 0=t_{0}<\cdots<t_{N}=T,\ \xi_{i}\in
B_{b}(\Omega_{t_{i}}),\ i=0,\cdots,N-1\\}.$

######  Definition 2.6.

For $p\geq 1$, we denote by $M_{*}^{p}([0,T])$ the completion of
$M_{b}^{0}([0,T])$ under the following norm:

$\|\eta\|_{p}:=\bigg{(}\bar{\mathbb{E}}\bigg{[}\frac{1}{T}\int_{0}^{T}|\eta_{t}|^{p}dt\bigg{]}\bigg{)}^{1/p}.$

Here below is the definition of the $G$-Itô integral. In the sequel, $B^{\bf
a}$ denotes the inner product of ${\bf a}\in\mathbb{R}^{d}$ and $B$, which is
still a $G$-Brownian motion, and $\sigma_{\bf aa^{\rm
T}}:=\bar{\mathbb{E}}[({\bf a},B_{1})^{2}]$.

######  Definition 2.7.

For each $\eta\in M^{0}_{b}([0,T])$, we define the Itô type integral

$\mathcal{I}_{[0,T]}(\eta)=\int^{T}_{0}\eta_{t}dB^{\bf
a}_{t}:=\sum^{N-1}_{k=0}\xi_{k}(B^{\bf a}_{t_{k+1}}-B^{\bf a}_{t_{k}}).$

Then, the linear mapping $\mathcal{I}_{[0,T]}$ on $M^{0}_{b}([0,T])$ can be
continuously extended to $\mathcal{I}_{[0,T]}:M^{2}_{*}([0,T])\rightarrow
L^{2}_{*}(\Omega_{T})$ and for each $\eta\in M^{2}_{*}([0,T])$, we define
$\int^{T}_{0}\eta_{t}dB^{\bf a}_{t}:=\mathcal{I}_{[0,T]}(\eta)$.

We notice that Gao [5] proves the $G$-Itô integral
$X_{\cdot}=\int_{0}^{\cdot}\eta_{s}dB^{\bf a}_{s}$ has a continuous
$\bar{C}$-modification, for any $\eta\in M_{G}^{2}(0,T)\subset
M_{*}^{2}([0,T])$, where $M_{G}^{2}([0,T])$ is defined in [12] by the
completion of simple process formed by random variables in
$C_{b,lip}(\Omega)$. In fact, this result still holds for $\eta\in
M_{*}^{2}(0,T)$, that is to say, there exists $\bar{X}$ which is continuous in
$t$ pathwisely and $\bar{C}(|X_{t}-\bar{X}_{t}|\neq 0)=0$ for all $t\in[0,T]$.
Furthermore, we can deduce the following proposition, so that we do not care
the choice of the $t$-continuous $\bar{C}$-modification for the $G$-Itô
integral $(\int^{t}_{0}\eta_{s}dB_{s})_{0\leq t\leq T}$.

######  Proposition 2.8.

Suppose that $X$ and $X^{\prime}$ are two processes that have $t$-continuous
paths and for each $t\in[0,T]$, $X_{t}=X^{\prime}_{t}$, q.s.. Then, $X$ and
$X^{\prime}$ are indistinguishable in the q.s. (indistinguishable, for short
in the rest part of this paper).

##  3\. Localisation techniques and extension of the $G$-Itô type integral

In this section, we follow Li-Peng [8] to define the $G$-Itô type integral for
a certain type of locally integrable processes and generalize some results in
[8]. The notion of stopping times always refer to the ones with respect to
$\mathcal{F}$.

######  Definition 3.1.

For fixed $p\geq 1$, a stochastic process $\eta$ is said to be in
$M^{p}_{w}([0,T])$, if it is associated with a sequence
$\\{\sigma_{m}\\}_{m\in\mathbb{N}}$ of stopping times incresing to infinity
such that

$\eta{\bf 1}_{[0,\sigma_{m}\wedge T]}\in M^{p}_{*}([0,T]),\ \ \forall
m\in\mathbb{N}.$

######  Remark 3.2.

One can easily deduce from this definition that

(3.1) $\int^{T}_{0}|\eta_{t}|^{p}dt<\infty,\ q.s..$

This definition is a slight modification of original one in [8]. We can define
the related Itô’s integral in the similar way.

For a given $\eta\in M^{2}_{w}([0,T])$ associated with
$\\{\sigma_{m}\\}_{m\in\mathbb{N}}$, we note $\tau_{m}:=\sigma_{m}\wedge T$
and consider the $t$-continuous $\bar{C}$-modification of
$(\int^{t}_{0}\eta_{s}{\bf 1}_{[0,\tau_{m}]}(s)dB^{\bf a}_{s})_{0\leq t\leq
T}$. For each $m$, $n\in\mathbb{N}$, $n>m$, by Lemma 4.2 in [8] and the
continuity of the paths of the $G$-Itô integral, we can find a polar set
$\hat{A}^{m,n}$, such that for all $\omega\in(\hat{A}^{m,n})^{c}$, the
following equality holds:

$\displaystyle\int^{t\wedge\tau_{m}}_{0}\eta_{s}dB^{\bf a}_{s}(\omega)$
$\displaystyle=\int^{t}_{0}\eta_{s}{\bf 1}_{[0,\tau_{m}]}(s)dB^{\bf
a}_{s}(\omega)$ (3.2) $\displaystyle=\int^{t}_{0}\eta_{s}{\bf
1}_{[0,\tau_{m}]}(s){\bf 1}_{[0,\tau_{n}]}(s)dB^{\bf a}_{s}(\omega)$
$\displaystyle=\int^{t\wedge\tau_{m}}_{0}\eta_{s}{\bf
1}_{[0,\tau_{n}]}(s)dB_{s}^{\bf a}(\omega),\ 0\leq t\leq T.$

Define a polar set

$\hat{A}:=\bigcup_{m=1}^{\infty}\bigcup_{n=m+1}^{\infty}\hat{A}^{m,n}.$

For each $m\in\mathbb{N}$ and $(\omega,t)\in\Omega\times[0,T]$, we set

$X^{m}_{t}(\omega):=\left\\{\begin{aligned} \int^{t}_{0}\eta_{s}{\bf
1}_{[0,\tau_{m}]}(s)dB^{\bf a}_{s}(\omega)\ &,\
\omega\in\hat{A}^{c}\cap\bar{\Omega};\\\ 0\ &,\ {\rm
otherwise}.\end{aligned}\right.$

For each $\omega\in\hat{A}^{c}$ and $m$, $n\in\mathbb{N}$, $n>m$,
$X^{n}(\omega)\equiv X^{m}(\omega)$ on $[0,\tau_{m}(\omega)]$. Therefore, we
can define unambiguously a process by stipulating that it equal to $X^{m}$ on
$[0,\tau_{m}(\omega)]$.

######  Definition 3.3.

Giving $\eta\in M^{2}_{w}([0,T])$, for each $(\omega,t)\in\Omega\times[0,T]$,
we define

$\int^{t}_{0}\eta_{s}dB^{\bf
a}_{s}(\omega):=\lim_{m\rightarrow\infty}X^{m}_{t}(\omega).$

######  Remark 3.4.

For $\eta\in M^{p}_{w}([0,T])$, where $p\geq 2$, associated with
$\\{\sigma_{m}\\}_{m\in\mathbb{N}}$, define
$X_{t}:=\int^{t}_{0}\eta_{s}dB_{s}$, then $X$ stays in $M^{p}_{w}([0,T])$,
where the sequence of stopping times can be chosen as
$\\{\sigma_{m}\\}_{m\in\mathbb{N}}$.

At the end of this section, we generalize the BDG type inequality in [8] for
$\eta\in M^{2}_{w}([0,T])$.

######  Lemma 3.5.

Let $\eta\in M^{p}_{w}([0,T])$, the following inequality holds: for $p>0$,
there exists $C_{p}>0$ depends only on $p$, such that

$\bar{\mathbb{E}}\bigg{[}\sup_{0\leq s\leq
t}\bigg{|}\int^{s}_{0}\eta_{u}dB^{\bf a}_{u}\bigg{|}^{p}\bigg{]}\leq
C_{p}\sigma^{p/2}_{\bf aa^{\rm
T}}\bar{\mathbb{E}}\bigg{[}\bigg{(}\int^{t}_{0}|\eta_{s}|^{2}ds\bigg{)}^{p/2}\bigg{]}.$

Proof: Suppose $\eta$ is associated with stopping times
$\\{\sigma_{m}\\}_{m\in\mathbb{N}}$, then we have

$\bar{\mathbb{E}}\bigg{[}\sup_{0\leq s\leq t}\bigg{|}\int^{s}_{0}\eta_{u}{\bf
1}_{[0,\sigma_{m}\wedge T]}dB^{\bf a}_{u}\bigg{|}^{p}\bigg{]}\leq
C_{p}\sigma^{p/2}_{\bf aa^{\rm
T}}\bar{\mathbb{E}}\bigg{[}\bigg{(}\int^{t}_{0}|\eta_{s}|^{2}{\bf
1}_{[0,\sigma_{m}\wedge T]}ds\bigg{)}^{p/2}\bigg{]}\leq C_{p}\sigma^{p/2}_{\bf
aa^{\rm
T}}\bar{\mathbb{E}}\bigg{[}\bigg{(}\int^{t}_{0}|\eta_{s}|^{2}ds\bigg{)}^{p/2}\bigg{]}.$

Since $\int^{\cdot}_{0}\eta_{t}dB^{\bf a}_{t}$ has continuous paths,

$\sup_{0\leq s\leq t}\bigg{|}\int^{s}_{0}\eta_{u}{\bf 1}_{[0,\sigma_{m}\wedge
T]}dB^{\bf a}_{u}\bigg{|}^{p}=\sup_{0\leq s\leq
t}\bigg{|}\int^{s\wedge\sigma_{m}\wedge T}_{0}\eta_{u}dB^{\bf
a}_{u}\bigg{|}^{p}\rightarrow\sup_{0\leq s\leq
t}\bigg{|}\int^{s}_{0}\eta_{u}dB^{\bf a}_{u}\bigg{|}^{p},\ q.s..$

Finally, Fatou’s lemma (Lemma 2.5) gives desired result. $\square$

##  4\. Locally Lipschtiz $G$-stochastic differential equations

In this section, we study the solvability of the following $n$-dimensional
stochastic differential equation driven by $d$-dimensional $G$-Brownian
motion:

(4.1) $X_{t}=x+\int^{t}_{0}f(s,X_{s})ds+\int^{t}_{0}h(s,X_{s})d\langle
B,B\rangle_{s}+\int^{t}_{0}g(s,X_{s})dB_{s},\ 0\leq t\leq T,\ q.s.,$

where $x\in\mathbb{R}^{n}$ is the initial value and $\langle
B,B\rangle=(\langle B^{i},B^{j}\rangle)_{i,j=1,\ldots,d}$ is the mutual
variation matrix of $B$.
In a first stage, we consider this equation under the so-called uniformly
Lipschitz conditions:
(H1) For some $p\geq 2$ and each $x\in\mathbb{R}^{n}$, $f(\cdot,x)$,
$h^{ij}(\cdot,x)$, $g^{j}(\cdot,x)\in M^{p}_{*}([0,T];\mathbb{R}^{n})$,
$i,j=1,\ldots,d$;
(H2) The coefficients $f$, $h$ and $g$ are uniformly Lipschitz in $x$, i.e.,
for each $t\in[0,T]$ and $x,x^{\prime}\in\mathbb{R}^{n}$,

$\displaystyle|f(t,x)-f(t,x^{\prime})|+||h(t,x)-h(t,x^{\prime})||+||g(t,x)-g(t,x^{\prime})||\leq
C_{L}|x-x^{\prime}|,$

where $||\cdot||$ is the Hilbert-Schmidt norm of a matrix.
We notice that the domain of coefficients here is a little larger than the
ones in Gao [5] and Peng [12], however, the same method as in [5] can be
applied to prove the uniqueness and existence theorem as follows:

######  Theorem 4.1.

Let (H1) and (H2) hold. Then, for any $p\geq 2$, there exists a unique process
$X\in M^{p}_{*}([0,T];\mathbb{R}^{n})$ that has $t$-continuous paths and
satisfies the GSDE (4.1). Denote by $X^{x}$ and $X^{y}$ the solutions starting
with $x$, $y\in\mathbb{R}^{n}$, then there exists $C>0$ that depends only on
$p$, $T$ and $C_{L}$, such that

$\bar{E}[\sup_{t\in[0,T]}|X^{x}_{t}-X^{y}_{t}|^{p}]\leq C|x-y|^{p}.$

######  Remark 4.2.

Similarly to Lemma 5.1 in Bai-Lin [2], we have the following assertion:
suppose that for some $p\geq 1$, $\zeta$ is a function such that
$\zeta(\cdot,x)\in M^{p}_{*}([0,T];\mathbb{R}^{n})$, for each
$x\in\mathbb{R}^{n}$ and that $\zeta(\cdot,\cdot)$ is uniformly Lipschitz in
$x$, then for each $X\in M^{p}_{*}([0,T];\mathbb{R}^{n})$,
$\zeta(\cdot,X_{\cdot})$ is an element in $M^{p}_{*}([0,T];\mathbb{R}^{n})$.
This result ensures that (4.1) is well defined.

In what follows, we consider such a GSDE, whose coefficients satisfy both a
locally Lipschitz condition and a Lyapunov’s type condition. Here below are
two assumptions that we concern: from now on, we adopt the Einstein notation.
(H2’) The coefficients $f(\cdot,\cdot)$, $h^{ij}(\cdot,\cdot)$,
$g^{j}(\cdot,\cdot):[0,T]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ are
deterministic functions continuous in $t$ and locally Lipschitz in $x$, i.e.,
for each $x$, $x^{\prime}\in\\{a:|a|\leq{R}\\}$, there exists a positive
constant $C_{R}$ that depends only on $R$, such that for each $t\in[0,T]$,

$\displaystyle|f(t,x)-f(t,x^{\prime})|+||h(t,x)-h(t,x^{\prime})||+||g(t,x)-g(t,x^{\prime})||\leq
C_{R}|x-x^{\prime}|.$

(H3’) There exist a deterministic nonnegative Lyapunov function
$V\in\mathcal{C}^{1,2}([0,T]\times\mathbb{R}^{n})$, such that

$\inf_{|x|\geq R}\inf_{t\in[0,T]}V(t,x)\rightarrow\infty,\ {\rm as}\
R\rightarrow\infty,$

and for some constant $C_{LY}>0$ and all $(t,x)\in[0,T]\times\mathbb{R}^{n}$,

$\mathcal{L}V(t,x)\leq C_{LY}V(t,x),$

where $\mathcal{L}$ is a differential operator defined by

$\displaystyle\mathcal{L}V=\partial_{t}V+\partial_{x^{\nu}}Vf^{\nu}+G\bigg{(}\left(\partial_{x^{\nu}}V\cdot(h^{\nu
ij}+h^{\nu ji})+\partial_{x^{\mu}x^{\nu}}^{2}V\cdot g^{\mu i}g^{\nu
j}\right)^{d}_{i,j=1}\bigg{)}.$

######  Remark 4.3.

The coefficients $f$, $h^{ij}$, $g^{j}$, $i,j=1,\ldots,d$, are continuous in
$t$ and thus uniformly continuous on $[0,T]$, so for each
$x\in\mathbb{R}^{n}$, $f(\cdot,x)$, $h^{ij}(\cdot,x)$, $g^{j}(\cdot,x)\in
M^{p}_{G}([0,T];\mathbb{R}^{n})\subset M^{p}_{*}([0,T];\mathbb{R}^{n})$, for
any $p\geq 1$.

######  Lemma 4.4.

For each $X\in M^{1}_{w}([0,T];\mathbb{R}^{n})$ that has continuous paths and
for each function $V\in\mathcal{C}^{1,2}([0,T]\times\mathbb{R}^{n})$, we have
$\eta(V,X)\in M^{p}_{w}([0,T];\mathbb{S}^{d})$, for any $p\geq 1$, where

$\eta^{ij}_{\cdot}(V,X):=\partial_{x^{\nu}}V(\cdot,X_{\cdot})(h^{\nu
ij}(\cdot,X_{\cdot})+h^{\nu
ji}(\cdot,X_{\cdot}))+\partial_{x^{\mu}x^{\nu}}^{2}V(\cdot,X_{\cdot})g^{\mu
i}(\cdot,X_{\cdot})g^{\nu j}(\cdot,X_{\cdot}).$

Proof: Similarly to the proof of Theorem 5.4 in Li-Peng [8], we can find a
sequence of stopping times $\\{\sigma_{m}\\}_{m\in\mathbb{N}}$ satisfying
Definition 3.1, such that on $[0,\sigma_{m}]$, $|X|$ is bounded by $m$. It is
obvious that $X{\bf 1}_{[0,\sigma_{m}]}\in M^{p}_{*}([0,T];\mathbb{R}^{n})$,
for any $p\geq 1$.
Noticing that $\partial_{x^{\mu}}V$, $\partial^{2}_{x^{\mu}x^{\nu}}V$,
$h^{ij}$ and $g^{j}$, $\mu$, $\nu=1,\ldots n$, $i,j=1,\ldots,d$, are bounded
and uniformly continuous on compact sets, we can approximate these functions
uniformly on $[0,T]\times B(0,m)$ by $C_{b,lip}([0,T];\mathbb{R}^{n})$ ones
and thus, we have $\eta^{ij}_{\cdot\wedge\sigma_{m}}(V,X)\in
M^{p}_{*}([0,T])$, for any $p\geq 1$, which implies the desired result.
$\square$

######  Theorem 4.5.

Let (H2’) and (H3’) hold. Then, for any $p\geq 2$, there exists a unique
solution of the GSDE (4.1) in $M^{p}_{w}([0,T];\mathbb{R}^{n})$ that has
$t$-continuous paths on $[0,T]$, and the following estimate holds:

$\bar{\mathbb{E}}[V(t,X^{x}_{t})]\leq e^{C_{LY}T}V(0,x).$

Proof: Existence. For each $N\in\mathbb{N}$, we first consider the following
truncated GSDE:

(4.2)
$X^{N}_{t}=x+\int^{t}_{0}f^{N}(s,X^{N}_{s})ds+\int^{t}_{0}h^{N}(s,X^{N}_{s})d\langle
B,B\rangle_{s}+\int^{t}_{0}g^{N}(s,X^{N}_{s})dB_{s},\ 0\leq t\leq T,\ q.s.,$

where $f^{N}$, $(h^{ij})^{N}$ and $(g^{j})^{N}$, $i$, $j=1,\ldots,d$, are
defined in the following way:

$\displaystyle\zeta^{N}(t,x)=\left\\{\begin{array}[]{c@{,}l}\zeta(t,x)&\
{\rm{if}}\ |x|\leq N;\\\\[3.0pt] \zeta({t,Nx}/{|x|})&\ {\rm{if}}\
|x|>N.\end{array}\right.$

It is easy to verify that $f^{N}$, $h^{N}$ and $g^{N}$ are all bounded
functions and uniformly Lipschitz in $x$. Then, by the result of Lipschitz
GSDEs with coefficients in $M^{p}_{G}([0,T];\mathbb{R}^{n})$, for example in
[5] and [P1], the truncated GSDE (4.2) admits a unique solution $X^{N}\in
M^{p}_{G}([0,T];\mathbb{R}^{n})$, for any $p\geq 2$, whose paths are
$t$-continuous.
Define a sequence of stopping times by

$\tau_{N}:=\inf\\{t:|X^{N}_{t}|\geq N\\}\wedge T,$

which satisfies $\\{\tau_{N}\leq t\\}\in\mathcal{F}_{t}$. Thanks to Lemma
LABEL:ere, we can deduce from (4.2) that

$\displaystyle X^{N}_{t\wedge\tau_{N}}$
$\displaystyle=x+\int^{t}_{0}f^{N}(s,X^{N}_{s}){\bf 1}_{[0,\tau_{N}]}(s)ds$
$\displaystyle+\int^{t}_{0}h^{N}(s,X^{N}_{s}){\bf 1}_{[0,\tau_{N}]}(s)d\langle
B,B\rangle_{s}+\int^{t}_{0}g^{N}(s,X^{N}_{s}){\bf 1}_{[0,\tau_{N}]}(s)dB_{s}$
$\displaystyle=x+\int^{t}_{0}f^{N+1}(s,X^{N}_{s}){\bf 1}_{[0,\tau_{N}]}(s)ds$
$\displaystyle+\int^{t}_{0}h^{N+1}(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)d\langle
B,B\rangle_{s}+\int^{t}_{0}g^{N+1}(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)dB_{s}$
$\displaystyle=x+\int^{t\wedge\tau_{N}}_{0}f^{N+1}(s,X^{N}_{s})ds$
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}h^{N+1}(s,X^{N}_{s})d\langle
B,B\rangle_{s}+\int^{t\wedge\tau_{N}}_{0}g^{N+1}(s,X^{N}_{s})dB_{s},\ 0\leq
t\leq T,\ q.s..$

On the other hand, by the definition of $X^{N+1}$, we have

$\displaystyle
X^{N+1}_{t\wedge\tau_{N}}=x+\int^{t\wedge\tau_{N}}_{0}f^{N+1}(s,X^{N+1}_{s})ds$
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}h^{N+1}(s,X^{N+1}_{s})d\langle
B,B\rangle_{s}$ (4.3)
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}g^{N+1}(s,X^{N+1}_{s})dB_{s},\ 0\leq
t\leq T,\ q.s..$

By the uniqueness of the solution to the truncated GSDE (4), for each
$N\in\mathbb{N}$, $X^{N}$ and $X^{N+1}$ are distinguishable on $[0,\tau_{N}]$.
This also implies that the sequence $\\{\tau_{N}\\}_{N\in\mathbb{N}}$ are q.s.
increasing.
Now we aim to show that

(4.4)
$\bar{C}\bigg{(}\bigcup_{N=1}^{\infty}\\{\omega:\tau_{N}(\omega)=T\\}\bigg{)}=1.$

Because $|X^{N}_{\cdot}|$ never exceeds $N$ on $[0,\tau_{N}]$, we have

$\displaystyle f(t,X^{N}_{t}){\bf 1}_{[0,\tau_{N}]}(t)=f^{N}(t,X^{N}_{t}){\bf
1}_{[0,\tau_{N}]}(t);\ $ $\displaystyle h(t,X^{N}_{t}){\bf
1}_{[0,\tau_{N}]}(t)=h^{N}(t,X^{N}_{t}){\bf 1}_{[0,\tau_{N}]}(t);$ (4.5)
$\displaystyle g(t,X^{N}_{t}){\bf 1}_{[0,\tau_{N}]}(t)=g^{N}(t,X^{N}_{t}){\bf
1}_{[0,\tau_{N}]}(t),\ 0\leq t\leq T,$

where the right-hand side are $M^{p}_{*}([0,T];\mathbb{R}^{n})$ processes (cf.
Lemma 4.2 in [8]), for any $p\geq 2$. As a result,

$\displaystyle X^{N}_{t\wedge\tau_{N}}$
$\displaystyle=x+\int^{t\wedge\tau_{N}}_{0}f^{N}(s,X^{N}_{s})ds+\int^{t\wedge\tau_{N}}_{0}h^{N}(s,X^{N}_{s})d\langle
B,B\rangle_{s}+\int^{t\wedge\tau_{N}}_{0}g^{N}(s,X^{N}_{s})dB_{s}$
$\displaystyle=x+\int^{t\wedge\tau_{N}}_{0}f^{N}(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)ds$
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}h^{N}(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)d\langle
B,B\rangle_{s}+\int^{t\wedge\tau_{N}}_{0}g^{N}(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)dB_{s}$ $\displaystyle=x+\int^{t}_{0}f(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)ds$ $\displaystyle+\int^{t}_{0}h(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)d\langle B,B\rangle_{s}+\int^{t}_{0}g(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)dB_{s},\ 0\leq t\leq T,\ q.s..$

Then, applying $G$-Itô’s formula (cf. Theorem 5.4 in [8]) to
$\Phi(t\wedge\tau_{N},X^{N}_{t\wedge\tau_{N}}):=\exp(-C_{LY}(t\wedge\tau_{N}))V(t\wedge\tau_{N},X^{N}_{t\wedge\tau_{N}})$,
we have

$\displaystyle\Phi(t\wedge\tau_{N},X^{N}_{t\wedge\tau_{N}})$
$\displaystyle-\Phi(0,x)$
$\displaystyle=\int^{t\wedge\tau_{N}}_{0}(\partial_{t}\Phi(s,X^{N}_{s})+\partial_{x^{\nu}}\Phi(s,X^{N}_{s})f^{\nu}_{s}(s,X^{N}_{s}))ds$
(4.6)
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}\partial_{x^{\nu}}\Phi(s,X^{N}_{s})g^{\nu
j}_{s}(s,X^{N}_{s})dB^{j}_{s}$
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}\partial_{x^{\nu}}\Phi(s,X^{N}_{s})h^{\nu
ij}_{s}(s,X^{N}_{s})$
$\displaystyle+\frac{1}{2}\partial^{2}_{x^{\mu}x^{\nu}}\Phi(s,X^{N}_{s})g^{\mu
i}_{s}(s,X^{N}_{s})g^{\nu j}_{s}(s,X^{N}_{s})d\langle B^{i},B^{j}\rangle_{s},\
q.s..$

Borrowing the notation in Lemma 4.4, the above equality can be written into

$\displaystyle Phi(t\wedge\tau_{N},X^{N}_{t\wedge\tau_{N}})$
$\displaystyle-\Phi(0,x)$
$\displaystyle=\int^{t\wedge\tau_{N}}_{0}\partial_{t}\Phi(s,X^{N}_{s})+\partial_{x^{\nu}}\Phi(s,X^{N}_{s})f^{\nu}_{s}(s,X^{N}_{s})+G\big{(}\eta_{s}(\Phi,X^{N})\big{)}ds$
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}\partial_{x^{\nu}}\Phi(s,X^{N}_{s})g^{\nu
j}_{s}(s,X^{N}_{s})dB^{j}_{s}$
$\displaystyle+\int^{t\wedge\tau_{N}}_{0}\eta^{ij}_{s}(\Phi,X^{N})d\langle
B^{i},B^{j}\rangle_{s}-\int^{t\wedge\tau_{N}}_{0}G(\eta_{s}(\Phi,X^{N}))ds$
$\displaystyle=\int^{t}_{0}\mathcal{L}\Phi(s,X^{N}_{s}){\bf
1}_{[0,\tau_{N}]}(s)ds+\int^{t}_{0}\partial_{x^{\nu}}\Phi(s,X^{N}_{s})g^{\nu
j}_{s}(s,X^{N}_{s}){\bf 1}_{[0,\tau_{N}]}(s)dB^{j}_{s}$
$\displaystyle+\int^{t}_{0}\eta^{ij}_{s}(\Phi,X^{N}){\bf
1}_{[0,\tau_{N}]}(s)d\langle
B^{i},B^{j}\rangle_{s}-\int^{t}_{0}G(\eta_{s}(\Phi,X^{N}){\bf
1}_{[0,\tau_{N}]}(s))ds,\ q.s..$

From (4) and the fact that $\partial_{x}V(t,x)$ is uniformly continuous in $t$
and uniformly Lipschitz in $x$ on $[0,T]\times B(0,N)$, it is readily observed
that $\partial_{\nu}\Phi(\cdot,X^{N}_{\cdot})g^{\nu j}(X^{N}_{\cdot}){\bf
1}_{[0,\tau_{N}]}(\cdot)\in M^{p}_{*}([0,T])$, for any $p\geq 2$ (cf.
Proposition 3.11 in [8]). Then, we obtain

$\bar{\mathbb{E}}\bigg{[}\int^{t}_{0}\partial_{x^{\nu}}\Phi(s,X^{N}_{s})g^{\nu
j}_{s}(s,X^{N}_{s}){\bf 1}_{[0,\tau_{N}]}(s)dB^{j}_{s}\bigg{]}=0.$

On the other hand, $\eta(\Phi,X)\in M^{p}_{w}([0,T];\mathbb{S}^{d})$, for any
$p\geq 1$, which implies $\eta(\Phi,X){\bf 1}_{[0,\tau_{N}]}\in
M^{p}_{w}([0,T];\mathbb{S}^{d})$. Here, we claim that

$\bar{\mathbb{E}}\bigg{[}\int^{t}_{0}\eta^{ij}_{s}(\Phi,X^{N}){\bf
1}_{[0,\tau_{N}]}(s)d\langle
B^{i},B^{j}\rangle_{s}-\int^{t}_{0}G\big{(}\eta_{s}(\Phi,X^{N}){\bf
1}_{[0,\tau_{N}]}(s)\big{)}ds\bigg{]}\leq 0,$

whose proof is postponed to the appendix (see Lemma 6.3). Hence,

$\bar{\mathbb{E}}[\Phi(T\wedge\tau_{N},X^{N}_{T\wedge\tau_{N}})]-\Phi(0,x)\leq\bar{\mathbb{E}}\bigg{[}\int^{T\wedge\tau_{N}}_{0}\mathcal{L}\Phi(t,X^{N}_{t})dt\bigg{]}.$

Because $\mathcal{L}V\leq C_{LY}V$ implies $\mathcal{L}\Phi\leq 0$,

$\bar{\mathbb{E}}[V(T\wedge\tau_{N},X^{N}_{T\wedge\tau_{N}})]\leq
V(0,x)\exp(C_{LY}T).$

In particular, we have

$\bar{\mathbb{E}}[{\bf
1}_{\\{\tau_{N}<T\\}}V(T\wedge\tau_{N},X^{N}_{T\wedge\tau_{N}})]\leq
V(0,x)\exp(C_{LY}T).$

Since $X^{N}$ has $t$-continuous paths, $\tau_{N}<T$ implies
$|X^{N}_{T\wedge\tau_{N}}|=N$, q.s., from which we deduce

$\bar{C}(\\{\omega:\tau_{N}(\omega)<T\\})\inf_{|x|\geq
N}\inf_{t\in[0,T]}V(t,x)\leq V(0,x)\exp(C_{LY}T).$

As $N\rightarrow\infty$, by (H3’), we obtain

$1\geq\lim_{N\rightarrow\infty}\bar{C}(\\{\omega:\tau_{N}(\omega)=T\\})\geq
1-\lim_{N\rightarrow\infty}\bar{C}(\\{\omega:\tau_{N}(\omega)<T\\})=1.$

Since $\\{\omega:\tau_{N}(\omega)=T\\}$ is increasing, the upwards convergence
theorem yields (4.4). Therefore, there exists a polar set $A$, such that for
all $\omega\in A^{c}$, the following assertion holds: one can find an
$N_{0}(\omega)$ that depends on $\omega$, such that for all $N\geq
N_{0}(\omega)$, $N\in\mathbb{N}$, $\tau_{N}(\omega)=T$. Then, we define

(4.7) $X_{t}(\omega)=\left\\{\begin{aligned} X^{N_{0}(\omega)}_{t}(\omega),\
0\leq t\leq T\ &,\ \omega\in A^{c};\\\ 0\ &,\ \omega\in
A.\end{aligned}\right.$

From the argument above, we have for each $\tau_{N}$, $X{\bf
1}_{[0,\tau_{N}]}=X^{N}{\bf 1}_{[0,\tau_{N}]}\in
M^{p}_{*}([0,T];\mathbb{R}^{n})$ and thus, $X\in
M^{p}_{w}([0,T];\mathbb{R}^{n})$, for any $p\geq 2$. Moreover,

$\displaystyle X_{t\wedge\tau_{N}}=X^{N}_{t\wedge\tau_{N}}$
$\displaystyle=x+\int^{t\wedge\tau_{N}}_{0}f^{N}(s,X^{N}_{s})ds+\int^{t\wedge\tau_{N}}_{0}h^{N}(s,X^{N}_{s})d\langle
B,B\rangle_{s}+\int^{t\wedge\tau_{N}}_{0}g^{N}(s,X^{N}_{s})dB_{s}$
$\displaystyle=x+\int^{t\wedge\tau_{N}}_{0}f(s,X_{s})ds+\int^{t\wedge\tau_{N}}_{0}h(s,X_{s})d\langle
B,B\rangle_{s}$ $\displaystyle+\int^{t\wedge\tau_{N}}_{0}g(s,X_{s})dB_{s},\
0\leq t\leq T,\ q.s.,$

which implies that $X$ satisfies (4.1).
Thanks to the positivity of $C_{LY}$, we have for a fixed $t\in[0,T]$,

$\bar{\mathbb{E}}[V(t\wedge\tau_{N},X^{N}_{t\wedge\tau_{N}})]\leq
V(0,x)\exp(C_{LY}T).$

Letting $N\rightarrow\infty$, (4.4) and (4.7) yield
$V(t\wedge\tau_{N},X^{N}_{t\wedge\tau_{N}})\rightarrow V(t,X_{t})$, q.s., then
by Fatou’s Lemma we can deduce

$\bar{\mathbb{E}}[V(t,X_{t})]\leq V(0,x)\exp(C_{LY}T).$

Uniqueness. Suppose that $X$ and $X^{\prime}\in
M^{p}_{w}([0,T];\mathbb{R}^{n})$ are two solutions of the GSDE (4.1)
associated with $\\{\mu_{m}\\}_{m\in\mathbb{N}}$, and
$\\{\mu^{\prime}_{m}\\}_{m\in\mathbb{N}}$. Then, by the continuity of
solutions, we can define a new sequence of stopping times by
$\tau_{m}:=\sigma_{m}\wedge\sigma^{\prime}_{m}\wedge\mu_{m}\wedge\mu^{\prime}_{m}$,
where

$\sigma_{m}:=\inf\\{t:|X_{t}|\geq m\\}\wedge T,\ \mbox{and}\
\sigma^{\prime}_{m}:=\inf\\{t:|X^{\prime}_{t}|\geq m\\}\wedge T,\
m\in\mathbb{N}.$

From the pathwise uniqueness of Lipschitz GSDEs, we know that $X$ and
$X^{\prime}$ are indistinguishable on $[0,\tau_{m}]$, for each
$m\in\mathbb{N}$. Thus, the uniqueness of the GSDE (4.1) can be deduced from
the fact $\bar{C}(\lim_{m\rightarrow\infty}\\{\tau_{m}=T\\})=1$. $\square$

######  Remark 4.6.

The equality (4) can be obtained in the following way: first applying
$G$-Itô’s formula to $\Phi(t,X^{N}_{t})$, one can see that (4) holds q.s. for
each $t\in[0,T]$. Since the processes on both sides of (4) have continuous
paths, they are indistinguishable. Thus, (4) holds q.s. for any bounded
stopping time, such as $t\wedge\tau$.

######  Example 4.7.

In particular, locally Lipschitz GSDEs with a linear growth condition can be
regarded as a special case satisfying (H2’) and (H3’). For example, letting
$n=d=1$, suppose all the coefficients are continuous in $t$, locally Lipschtz
in $x$ and for some $K>0$,

$xf(t,x)+xh(t,x)+|g(t,x)|^{2}\leq K(1+|x|^{2}).$

In this case, $V$ can be defined by $V(x)=C_{K}(1+|x|^{2})$, where $C_{K}$
depends on $\Sigma$ and $K$.

######  Example 4.8.

In fact, (H2’) allows us to consider some GSDEs with polynomial growth
coefficients. Here is an example, which is similar to Duffing and van der Pol
oscillators in random mechanics (see Arnold [1] for more examples): letting
$n=2$ and $d=1$, consider the following GSDE:

$d\left(\begin{array}[]{c}X_{t}\\\ Y_{t}\\\
\end{array}\right)=\left(\begin{array}[]{c}Y_{t}\\\ -\alpha X_{t}-\beta
X^{3}_{t}-\gamma Y_{t}\\\ \end{array}\right)d\langle
B\rangle_{t}+\left(\begin{array}[]{c}0\\\ \sigma\\\ \end{array}\right)dB_{t},$

where $\alpha$, $\beta$, $\gamma$ and $\sigma$ are positive constants. In this
case, the Lyapunov function could be

$V(x,y)=1+\frac{1}{2}y^{2}+\frac{\alpha}{2}x^{2}+\frac{\beta}{4}x^{4}.$

##  5\. Stability of $G$-stochastic differential equations

In this section, we study the moment stability of the G-stochastic system
(4.1) under perturbations of the initial value. Moreover, we consider the
relationship between the moment stability and the q.s. stability under the
linear growth condition and obtain a property of the sample paths of the
$G$-Brownian motion.
In order to study the behavior of the solution to the GSDE (4.1) over an
infinite horizon, we first give the following definition:

######  Definition 5.1.

We call $X$ a global solution of the GSDE (4.1) if $X$ has $t$-continuous
paths on $[0,\infty)$ and for each $T>0$, $X_{\cdot\wedge T}\in
M^{2}_{w}([0,T];\mathbb{R}^{n})$ is a solution of the GSDE (4.1) on $[0,T]$.

######  Remark 5.2.

Under (H2’) and (H3’), such a global solution exists, since we can first
consider the solution of the GSDE (4.1) on $[0,N]$, for each $N\in\mathbb{N}$,
and then concatenate all these solutions. The uniqueness can be deduced in the
light of the continuity of paths.

Throughout this section, we always consider the GSDE (4.1) with the locally
Lipschitz condition (H2’). Meanwhile, we shall limit ourselves to conditions
for stability of the trivial solution $X\equiv 0$. Accordingly, we assume that
(H4’)

$f(t,0)=h(t,0)=g(t,0)\equiv 0.$

######  Definition 5.3.

Denote by $X^{s,x}$ the solution of the GSDE (4.1) starting with $X_{s}=x$,
$x\in\mathbb{R}^{n}$. The trivial solution of this $G$-stochastic system in
$\mathbb{R}^{n}$ is said to be

  1. (i)

$p$-stable, for some $p>0$, if for each $\varepsilon>0$, there exists a
$\delta>0$, such that

$\sup_{|x|\leq\delta}\sup_{t\geq
s}\bar{\mathbb{E}}[|X^{s,x}_{t}|^{p}]<\varepsilon;$

  2. (ii)

asymptotically $p$-stable, if it is $p$-stable and moreover

$\bar{\mathbb{E}}[|X^{s,x}_{t}|^{p}]\rightarrow 0,{\rm as}\
t\rightarrow\infty;$

  3. (iii)

exponentially $p$-stable, if for some positive constants $C$ and $\lambda$

$\bar{\mathbb{E}}[|X^{s,x}_{t}|^{p}]\leq C|x|^{p}e^{-\lambda(t-s)}.$

In particular, when $p=2$, we call this system is (asymptotically,
exponentially) stable in mean square.

######  Example 5.4.

Letting $n=1$ and $d=1$, we consider the “population growth model” in the
$G$-framework, i.e. the following GSDE:

(5.1) $X^{x}_{t}=x+\int^{t}_{0}\alpha X_{s}ds+\int^{t}_{0}\beta X_{s}d\langle
B\rangle_{s}+\int^{t}_{0}\gamma X_{s}dB_{s},$

where $\alpha$, $\beta$ and $\gamma$ are constants. The explicit solution of
the above equation is given by Theorem 1.2 in §V-1 of [12]:

$X^{x}_{t}=x\exp\bigg{(}\alpha
t+\big{(}\beta-\frac{1}{2}\gamma^{2}\big{)}\langle B\rangle_{t}+\gamma
B_{t}\bigg{)}.$

Thus, for any $p>0$,

$|X^{x}_{t}|^{p}=|x|^{p}\exp\bigg{(}\alpha
pt+\frac{1}{2}p\big{(}2\beta+\gamma^{2}(p-1)\big{)}\langle
B\rangle_{t}\bigg{)}\mathcal{E}(\gamma pB).$

It is obviously that $\gamma pB$ satisfies Assumption 2.1 in Xu-Shang-Zhang
[13], which implies that $\mathcal{E}(\gamma pB):=\exp(\gamma
pB-\frac{1}{2}(\gamma p)^{2}\langle B\rangle)$ is a $G$-martingale. Therefore,
(a) if $2\beta+\gamma^{2}(p-1)\geq 0$ and
$\alpha+\frac{1}{2}\overline{\sigma}^{2}\big{(}2\beta+\gamma^{2}(p-1)\big{)}<0$,
or (b) if $2\beta+\gamma^{2}(p-1)<0$ and
$\alpha+\frac{1}{2}\underline{\sigma}^{2}\big{(}2\beta+\gamma^{2}(p-1)\big{)}<0$,
then the trivial solution of the GSDE (5.4) is exponentially $p$-stable.

Similar to Theorem 4.5, the sufficient condition for the stability of the
$G$-stochastic system (4.1) will be given in terms of Lyapunov functions.
Without loss of generality, we consider only the case $s=0$.

######  Theorem 5.5.

Consider the $G$-stochastic system (4.1) with (H2’) and (H4’). Suppose that
there exists a function
$V(t,x)\in\mathcal{C}^{1,2}([0,\infty)\times\mathbb{R}^{n})$ such that for all
$t\geq 0$, some positive constant $c_{1}$ and $c_{2}$, and some $p>0$,

(5.2) $c_{1}|x|^{p}\leq V(t,x)\leq c_{2}|x|^{p}.$

Then,

  1. (a)

the trivial solution is $p$-stable, if

(5.3) $\mathcal{L}V\leq 0,$

  2. (b)

the trivial solution is exponentially $p$-stable, if there exists a
$\lambda>0$ such that for all $(t,x)\in([0,\infty)\times\mathbb{R}^{n})$,

(5.4) $\mathcal{L}V(t,x)\leq-\lambda V(t,x).$

Proof: Obviously, (H3’) is satisfied in both cases, so that fixing $t>0$, we
can follow the procedure in the proof of Theorem 4.5 to construct the solution
of the GSDE (4.1) on $[0,T]$ with the initial value $x$, for some $T>t$.
Meanwhile, the following inequality can be deduced:

$\bar{\mathbb{E}}[V(t,X^{x}_{t})]\leq V(0,x)\exp(-\lambda t).$

Finanlly, (5.2) implies

$\bar{\mathbb{E}}[|X^{x}_{t}|^{p}]\leq\frac{c_{2}}{c_{1}}|x|^{p}\exp(-\lambda
t),$

from which we can deduce both (a) and (b). $\square$

######  Corollary 5.6.

Let $p>0$. Consider the stochastic system (4.1) with (H2’) and (H4’). Assume
that there exists a $P\in\mathbb{S}^{n}_{+}$, and constants
$\alpha_{1}\in\mathbb{R}$, $0\leq\alpha_{2}<\alpha_{3}$, such that for all
$(t,x)\in[0,\infty)\times\mathbb{R}^{n}$,

$x^{\rm T}Pf(x,t)+G\big{(}(x^{\rm
T}P(h^{ij}(x,t)+h^{ji}(x,t)))^{d}_{i,j=1}+g^{\rm
T}(x,t)Pg(x,t)\big{)}\leq\alpha_{1}x^{\rm T}Px,$

and

$\alpha^{2}_{2}(x^{\rm T}Px)^{2}\leq 2G\big{(}g^{\rm T}(x,t)P^{\rm T}xx^{\rm
T}Pg(x,t)\big{)}\leq\alpha^{2}_{3}(x^{\rm T}Px)^{2}.$

Then, the trivial solution is exponentially $p$-stable if (a) $\alpha_{1}<0$
and $p<2+|\alpha_{1}|/\alpha_{3}^{2}$; or if (b)
$0\leq\alpha_{1}<\alpha^{2}_{2}$ and $p<2-2\alpha_{1}/\alpha_{2}^{2}.$

Proof: Set $V(x,t):=(x^{\rm T}Px)^{\frac{p}{2}}$. The fact
$P\in\mathbb{S}^{n}_{+}$ implies

$\lambda_{min}^{\frac{p}{2}}(P)|x|^{p}\leq
V(x)\leq\lambda_{max}^{\frac{p}{2}}(P)|x|^{p}.$

By the definition of the differential operator $\mathcal{L}$ and the sub-
additivity of the function $G$, we have

$\displaystyle\mathcal{L}V(x,t)$ $\displaystyle\leq p(x^{\rm
T}Px)^{\frac{p}{2}-1}\bigg{(}x^{\rm T}Pf(x,t)+G\big{(}(x^{\rm
T}P(h^{ij}(x,t)+h^{ji}(x,t)))^{d}_{i,j=1}+g^{\rm
T}(x,t)Pg(x,t)\big{)}\bigg{)}$
$\displaystyle+2p\bigg{(}\frac{p}{2}-1\bigg{)}(x^{\rm
T}Px)^{\frac{p}{2}-2}G\big{(}g^{\rm T}(x,t)P^{\rm T}xx^{\rm T}Pg(x,t)\big{)}.$

Then, if (a) holds true (without loss of generality, assuming $p\geq 2$),

$\mathcal{L}V(x,t)\leq-p\bigg{(}|\alpha_{1}|-\big{(}\frac{p}{2}-1\big{)}\alpha^{2}_{3}\bigg{)}V(x,t);$

if (b) holds true,

$\mathcal{L}V(x,t)\leq-p\bigg{(}\big{(}\frac{p}{2}-1\big{)}\alpha^{2}_{2}-\alpha_{1}\bigg{)}V(x,t).$

In both cases, the exponentially $p$-stability follows from the previous
theorem. $\square$
Now we have the following examples to show the algebraic criteria for the
exponential stability in mean square of a linear $G$-stochastic system:

######  Example 5.7.

Consider the following linear GSDE driven by a scalar $G$-Brownian motion:

(5.5) $X_{t}=x+\int^{t}_{0}FX_{s}ds+\int^{t}_{0}HX_{s}d\langle
B\rangle_{s}+\int^{t}_{0}CX_{s}dB_{s},$

where $F$, $H$ and $C\in\mathbb{R}^{n\times n}$. If there exists a matrix
$P\in\mathbb{S}^{n}_{+}$, such that for any $x\in\mathbb{R}^{n}$,

(5.6) $x^{\rm T}\big{(}PF+I_{n}\big{)}x+G\big{(}x^{\rm T}(2PH+C^{\rm
T}PC)x\big{)}\leq 0.$

Then, one can verify that the Lyapunov function $V(x):=x^{\rm T}Px$ satisfies
(5.2) and (5.3) and thus, the trivial solution is exponential stable in mean
square.
Indeed,

$D_{x}V(x)=2Px,\ D^{2}_{xx}V(x)\equiv 2P,$

thus,

$\mathcal{L}V(x)=2\langle Px,Fx\rangle+G\big{(}\langle 2Px,2Hx\rangle+\langle
2PCx,Cx\rangle\big{)}.$

Obviously, if $P$ satisfies (5.6), then

$\mathcal{L}V(x)\leq=-2|x|^{2}.$

In fact, (5.6) is the Riccati type inequality related to the stability problem
of (5.5) in the $G$-framework, but it is no longer a linear matrix inequality
(LMI) because of the non-linearity of the function $G$. However, a sufficient
condition to ensure that $P\in\mathbb{S}^{d}_{+}$ satisfies (5.6) can be given
by the following system of LMIs: for some $\alpha\in\mathbb{R}$,

$\displaystyle\left\\{\begin{array}[]{l}2PH+C^{\rm T}PC\leq\alpha
I_{n};\\\\[3.0pt] 2PF+I_{n}\leq-G(\alpha)I_{n}.\end{array}\right.$

In what follows, we shall consider a GSDE whose coefficients satisfy a linear
growth condition. Similar to the result in the classical framework, the
exponentially $p$-stability of the trivial solution to such a GSDE implies the
q.s. exponential stability.

######  Theorem 5.8.

Consider the $G$-stochastic system (4.1) with (H2’), (H4’) and the following
conditions: for some $K>0$ and all $(t,x)\in[0,\infty)\times\mathbb{R}^{n}$,

(5.7) $x^{\rm T}f(t,x)\vee||x^{\rm T}h(t,x)||\vee||g(t,x)||^{2}\leq K|x|^{2}.$

If the trivial solution is exponential $p$-stable, for some $p>0$, namely, for
all $x\in\mathbb{R}^{n}$,

(5.8) $\bar{\mathbb{E}}[|X^{s,x}_{t}|^{p}]\leq|x|^{p}\exp(-\lambda(t-s)),$

where $\lambda>0$. Then,

$\limsup_{t\rightarrow\infty}\frac{1}{t}\log(|X^{s,x}_{t}|)\leq-\frac{\lambda}{p}<0,\
q.s.,$

which implies that the trivial solution is q.s. exponentially stable.

Proof: Without loss of generality, we consider only the case $s=0$. Applying
$G$-Itô’s formula to $(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}$, we have for
each $m\in\mathbb{N}$,

$\displaystyle(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}=(|X^{x}_{m-1}|^{2}+\delta)^{\frac{p}{2}}$
$\displaystyle+\int^{t}_{m-1}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}f(u,X^{x}_{u})du$
$\displaystyle+\int^{t}_{m-1}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}g(u,X^{x}_{u})dB_{u}$
$\displaystyle+\int^{t}_{m-1}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}h^{ij}(u,X^{x}_{u})$
$\displaystyle+\frac{1}{2}\big{(}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}g^{i}(u,X^{x}_{u})g^{j}(u,X^{x}_{u})$
$\displaystyle+p(p-2)(|X^{x}_{u}|^{2}+\delta)^{\frac{p-4}{2}}(X^{x}_{u})^{\mu}(X^{x}_{u})^{\nu}g^{\mu
i}(u,X^{x}_{u})g^{\nu j}(u,X^{x}_{u})\big{)}d\langle B^{i}B^{j}\rangle_{u},\
q.s..$

From (5.7), we have

$\displaystyle(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}\leq(|X^{x}_{m-1}|^{2}+\delta)^{\frac{p}{2}}$
$\displaystyle+\int^{t}_{m-1}pK(|X^{x}_{u}|^{2}+\delta)^{\frac{p}{2}}du+\int^{t}_{m-1}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}g(u,X^{x}_{u})dB_{u}$
$\displaystyle+c(\Sigma)\int^{t}_{m-1}pK(|X^{x}_{u}|^{2}+\delta)^{\frac{p}{2}}+\frac{pK}{2}\big{(}c(d)+c(n,d)|p-2|\big{)}(|X^{x}_{u}|^{2}+\delta)^{\frac{p}{2}}du$
$\displaystyle\leq(|X^{x}_{m-1}|^{2}+\delta)^{\frac{p}{2}}+c_{1}\int^{t}_{m-1}(|X^{x}_{u}|^{2}+\delta)^{\frac{p}{2}}du$
(5.9)
$\displaystyle+\int^{t}_{m-1}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}g(u,X^{x}_{u})dB_{u},$

where $c_{1}>0$ is a constant that depends only on $n$, $d$, $p$, $K$ and
$\Sigma$.
Then, we deduce by Lemma 3.5,

$\displaystyle\bar{\mathbb{E}}\bigg{[}\sup_{m-1\leq t\leq
m}\bigg{|}\int^{t}_{m-1}p(|X^{x}_{u}|^{2}+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}g(u,X^{x}_{u})dB_{u}\bigg{|}\bigg{]}$ $\displaystyle\leq\ $ $\displaystyle
C\bar{\mathbb{E}}\bigg{[}\bigg{(}\int^{m}_{m-1}p^{2}(|X^{x}_{u}|^{2}+\delta)^{(p-2)}|(X^{x}_{u})^{\rm
T}g(t,X^{x}_{u})|^{2}du\bigg{)}^{\frac{1}{2}}\bigg{]}$ $\displaystyle\leq\ $
$\displaystyle C\bar{\mathbb{E}}\bigg{[}\bigg{(}\big{(}\sup_{m-1\leq t\leq
m}(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}\big{)}\int^{m}_{m-1}p^{2}K(|X^{x}_{u}|^{2}+\delta)^{\frac{p}{2}}du\bigg{)}^{\frac{1}{2}}\bigg{]}.$

On the other hand, by Young’s inequality, we can always find a constant
$c_{2}$ that depends on $c_{1}$ such that

$\displaystyle\bar{\mathbb{E}}\bigg{[}\sup_{m-1\leq t\leq
m}\int^{t}_{m-1}p(|X^{x}_{u}|^{2}$
$\displaystyle+\delta)^{\frac{p-2}{2}}(X^{x}_{u})^{\rm
T}g(u,X^{x}_{u})dB_{u}\bigg{]}$ (5.10)
$\displaystyle\leq\frac{1}{2}\bar{\mathbb{E}}\big{[}\sup_{m-1\leq t\leq
m}(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}\big{]}+c_{2}p^{2}K\bar{\mathbb{E}}\bigg{[}\int^{m}_{m-1}(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}dt\bigg{]}.$

Thanks to the positivity of $(|X^{x}|^{2}+\delta)^{\frac{p}{2}}$, we apply
Fubini’s theorem under each $\mathbb{P}\in\mathcal{P}_{G}$ to
$\mathbb{E}^{\mathbb{P}}[\int^{m}_{m-1}(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}dt]$
and obtain

(5.11)
$\bar{\mathbb{E}}\bigg{[}\int^{m}_{m-1}(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}dt\bigg{]}\leq\int^{m}_{m-1}\bar{\mathbb{E}}[(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}]dt\leq\max\\{1,2^{\frac{p}{2}-1}\\}\int^{m}_{m-1}(\bar{\mathbb{E}}[|X^{x}_{t}|^{p}]+\delta^{\frac{p}{2}})dt.$

We conclude from (5.8)-(5.11) that

$\displaystyle\bar{\mathbb{E}}[\sup_{m-1\leq t\leq m}|X^{x}_{t}|^{p}]$
$\displaystyle\leq\bar{\mathbb{E}}[\sup_{m-1\leq t\leq
m}(|X^{x}_{t}|^{2}+\delta)^{\frac{p}{2}}]$ $\displaystyle\leq
2\max\\{1,2^{\frac{p}{2}-1}\\}\bar{\mathbb{E}}[|X^{x}_{m-1}|^{p}]+c_{3}\int^{m}_{m-1}\bar{\mathbb{E}}[|X^{x}_{t}|^{p}]dt+c^{\prime}_{3}\delta^{\frac{p}{2}}$

where $c_{3}$, $c^{\prime}_{3}>0$ depends only on $c_{1}$, $c_{2}$ and $p$.
Letting $\delta\rightarrow 0$, there exists a constant $c_{4}$ depends only on
$c_{3}$, $c^{\prime}_{3}$ and $x$, such that

$\bar{\mathbb{E}}[\sup_{m-1\leq t\leq m}|X^{x}_{t}|^{p}]\leq
c_{4}\exp(-\lambda(m-1)).$

Applying Markov’s inequality (Lemma 2.2) gives, for arbitrary
$\varepsilon\in(0,\lambda)$,

$\bar{C}\bigg{\\{}\sup_{m-1\leq t\leq
m}|X^{x}_{t}|^{p}>e^{-(\lambda-\varepsilon)(m-1)}\bigg{\\}}\leq
c_{4}\exp(-\varepsilon(m-1)),$

which implies

$\sum^{\infty}_{n=1}\bar{C}\bigg{\\{}\sup_{m-1\leq t\leq
m}|X^{x}_{t}|^{p}>e^{-(\lambda-\varepsilon)(m-1)}\bigg{\\}}<\infty.$

From Borel-Cantelli lemma (Lemma 2.3), we see that for q.s. every $\omega$,
there exists $N_{0}:=N_{0}(\omega)$, such that for $m\geq N_{0}(\omega)$ and
$t\in[m-1,m]$,

$\frac{1}{t}\log(|X^{x}_{t}|)=\frac{1}{pt}\log(|X^{x}_{t}|^{p})<-\frac{(\lambda-\varepsilon)(m-1)}{pm},$

Passing to limit as $\varepsilon\rightarrow 0$ and $m\rightarrow\infty$, we
obtain the desired result. $\square$
The next property of the paths of the $G$-Brownian motion follows from the
theorem above.

######  Corollary 5.9.

$\frac{|B_{t}|}{t}\rightarrow 0,\ {\rm{as}}\ t\rightarrow\infty,\ q.s..$

Proof. Without loss of generality, we only prove this property for scalar
$G$-Brownian motion. Consider a particular case of Example 5.4, i.e. in (5.1),
$\alpha=-1$, $\beta=\frac{1}{2}$ and $\gamma=1$. Then, we calculate for some
$0<p<\frac{2}{\overline{\sigma}^{2}}$,

$\bar{\mathbb{E}}[|X^{x}_{t}|^{p}]\leq|x|^{p}\exp\big{(}-p(1-\frac{p\overline{\sigma}^{2}}{2})t\big{)}.$

By Theorem 5.8, we obtain

$\limsup_{t\rightarrow\infty}\frac{1}{t}\log(|X^{x}_{t}|)=\limsup_{t\rightarrow\infty}\frac{1}{t}\log(|\exp(-t+B_{t})|)=-1+\limsup_{t\rightarrow\infty}\frac{B_{t}}{t}\leq-{(1-\frac{p\overline{\sigma}^{2}}{2})},$

which implies
$\limsup_{t\rightarrow\infty}\frac{B_{t}}{t}\leq\frac{p\overline{\sigma}^{2}}{2}$.
On the other hand, we consider $\alpha=-1$, $\beta=\frac{1}{2}$ and
$\gamma=-1$ in (5.1) and obtain
$\liminf_{t\rightarrow\infty}\frac{B_{t}}{t}\geq-\frac{p\overline{\sigma}^{2}}{2}$.
Letting $p\rightarrow 0$, we complete the proof. $\square$ Now we turn to
discuss the sufficient condition for the moment instability. In order to
ensure the solvability of (4.1) in the following theorem, we always assume
Lipschitz conditions on its coefficients.

######  Theorem 5.10.

Consider the $G$-stochastic system (4.1), whose coefficients are deterministic
functions continuous in $t$ and Lipschitz in $x$. If there exists a function
$V(t,x)\in\mathcal{C}^{1,2}([0,\infty)\times\mathbb{R}^{n})$ such that for all
$t\geq 0$, some positive constant $c_{1}$, $c_{2}$, and some $q>0$,

(5.12) $c_{1}|x|^{q}\leq V(t,x)\leq c_{2}|x|^{q}$

and

(5.13) $\mathcal{L}V(t,x)\geq\lambda V(t,x).$

Then, the trivial solution is exponentially $q$-instable, namely,

$\bar{\mathbb{E}}[|X^{s,x}_{t}|]\geq C|x|^{q}\exp(\lambda(t-s)).$

Proof: Without loss of generality, we prove only the case $s=0$. In the light
of the property of the solution of a Lipschitz GSDE, one can define a sequence
of stopping times by $\tau_{N}:=\inf\\{t:X^{x}_{t}\geq N\\},$ such that
$X^{x}{\bf 1}_{[0,\tau_{N}]}\in M^{p}_{*}([0,T];\mathbb{R}^{n})$, for any
$p\geq 1$. Applying $G$-Itô’s formula to
$\Phi(t\wedge\tau_{N},X_{t\wedge\tau_{N}}):=-\exp(-\lambda(t\wedge\tau_{N}))V(t\wedge\tau_{N},X_{t\wedge\tau_{N}})$,
one can follow the proof of Theorem 4.5 and obtain

$\bar{\mathbb{E}}[\Phi(t,X^{x}_{t})]\leq\liminf_{N\rightarrow\infty}\bar{\mathbb{E}}[\Phi(t\wedge\tau_{N},X^{x}_{t\wedge\tau_{N}})]\leq\Phi(0,x).$

Thus,

$\bar{\mathbb{E}}[V(t,X^{x}_{t})]\geq V(0,x)\exp(\lambda t),$

which implies

$\bar{\mathbb{E}}[|X^{x}_{t}|^{q}]\geq\frac{c_{1}}{c_{2}}|X^{x}_{t}|^{q}\exp(\lambda
t).$

$\square$

######  Example 5.11.

Consider the linear GSDE (5.5), a sufficient condition to ensure that the
trivial solution is $q$-unstable can be given by the following system of LMIs:
for some $\alpha\in\mathbb{R}$,

$\displaystyle\left\\{\begin{array}[]{l}2PH+C^{\rm T}PC\geq\alpha
I_{n};\\\\[3.0pt] 2PF-I_{n}\geq-G(\alpha)I_{n}.\end{array}\right.$

##  6\. Appendix

In the Appendix, we study the property of stochastic integrals with respect to
the quadratic variation process $\langle B,B\rangle$, and we obtain a
generalized result of Proposition 1.4 in §IV-1 of Peng [12], for a $\eta\in
M_{w}^{1}([0,T],\mathbb{S}^{d})$.

######  Lemma 6.1.

Let $\eta=(\eta^{ij})^{d}_{i,j=1}\in M^{1}_{*}([0,T];\mathbb{S}^{d})$ and
$M_{t}:=\int^{t}_{0}\eta^{ij}_{s}d\langle
B^{i}B^{j}\rangle_{s}-\int^{t}_{0}2G(\eta_{s})ds$. Then, for each $t\in[0,T]$,
$M_{t}\leq 0$, q.s., thus $\bar{\mathbb{E}}[M_{t}]\leq 0$.

Proof: Without loss of generality, we prove only the case $d=1$. By the
definition, we know that $\langle B\rangle$ has continuous and increasing
paths on $[0,T]$, q.s.. From Corollary 5.4 in §III-5 of Peng [12], we have,
for any $0\leq t_{1}\leq t_{2}\leq T$, in the sense of $L^{1}_{G}(\Omega_{T})$
(consequently in the sense of q.s.),

(6.1) $\underline{\sigma}^{2}(t_{2}-t_{1})\leq\langle B\rangle_{t_{2}}-\langle
B\rangle_{t_{1}}\leq\overline{\sigma}^{2}(t_{2}-t_{1}).$

Thus, one can pick up a $\bar{C}$-null set $A$, such that for each $\omega\in
A^{c}$, for each $t_{1}$, $t_{2}\in\mathbb{Q}\cap[0,T]$, (6.1) holds true,
furthermore, by the continuity of the path, (6.1) indeed holds true for all
$t_{1}$, $t_{2}\in[0,T]$. Therefore, $\langle B\rangle_{\cdot}(\omega)$ induce
a finite measure on $[0,T]$, which is absolutely continuous w.r.t. the
Lebesgue measure. We denote by $\hat{a}_{\cdot}(\omega)$ the R-N derivative
$\frac{d\langle B\rangle_{t}(\omega)}{dt}$, which is in
$[\underline{\sigma}^{2},\overline{\sigma}^{2}]$, $\lambda$-a.s.. As a result,

$M_{t}(\omega):=\int^{t}_{0}\eta_{s}(\omega)d\langle
B\rangle_{s}(\omega)-\int^{t}_{0}2G(\eta_{s}(\omega))ds\leq 0,\ q.s..$

Taking $\bar{\mathbb{E}}[\cdot]$ on both sides, we obtain the desired result.
$\square$

######  Remark 6.2.

In §2.4.1 of Lin [10], the following fact is proved: for $\eta\in
M^{1}_{*}([0,T])$, the stochastic integral of $\eta$ w.r.t. $\langle B\rangle$
on $[0,T]$ can be defined in both the Lebesgue-Stieltjes sense and the sense
of a contracting mapping $M^{1}_{*}([0,T])\rightarrow L^{1}_{*}([0,T])$; these
two definitions are indistinguishable in the $M^{1}_{*}(\Omega_{T})$ sense.

Furthermore, we can generalize this lemma to the case that
$\eta=(\eta^{ij})^{d}_{i,j=1}\in M^{1}_{w}([0,T];\mathbb{S}^{d})$.

######  Lemma 6.3.

Let $\eta=(\eta^{ij})^{d}_{i,j=1}\in M^{1}_{w}([0,T];\mathbb{S}^{d})$ and
$M_{t}:=\int^{t}_{0}\eta^{ij}_{s}d\langle
B^{i}B^{j}\rangle_{s}-\int^{t}_{0}2G(\eta_{s})ds$. Then, for each $t\in[0,T]$,
$\bar{\mathbb{E}}[M_{t}]\leq 0$.

Proof: The definition of $M$ is well posed because of (3.1) and the Lipschtiz
continuity of the function $G$. Suppose $\eta$ is associated with the sequence
of stopping time $\\{\sigma_{m}\\}_{m\in\mathbb{N}}$. Then,

$\displaystyle M_{t\wedge\sigma_{m}\wedge T}$
$\displaystyle=\int^{t\wedge\sigma_{m}\wedge T}_{0}\eta^{ij}_{s}d\langle
B^{i}B^{j}\rangle_{s}-\int^{t\wedge\sigma_{m}\wedge T}_{0}2G(\eta^{ij}_{s})ds$
$\displaystyle=\int^{t}_{0}\eta^{ij}_{s}{\bf 1}_{[0,\sigma_{m}\wedge
T]}(s)d\langle B^{i}B^{j}\rangle_{s}-\int^{t}_{0}2G(\eta_{s}{\bf
1}_{[0,\sigma_{m}\wedge T]}(s))ds\leq 0,\ q.s..$

Fatou’s Lemma implies

$\bar{\mathbb{E}}[M_{t}]=\bar{\mathbb{E}}[\lim_{m\rightarrow\infty}M_{t\wedge\sigma_{m}\wedge
T}]\leq\liminf_{m\rightarrow\infty}\bar{\mathbb{E}}[M_{t\wedge\sigma_{m}\wedge
T}]\leq 0.$

$\square$
Acknowledgement The authors expresses special thanks to Prof. Ying HU for his
useful suggestions and also to Prof. Shige PENG for his constructive advice to
improve this paper.

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