# Discovery of merging twin quasars at $z$ = 6.05

Yoshiki Matsuoka Research Center for Space and Cosmic Evolution, Ehime
University, Matsuyama, Ehime 790-8577, Japan.  Takuma Izumi National
Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588, Japan.  Masafusa
Onoue Kavli Institute for the Physics and Mathematics of the Universe, WPI,
The University of Tokyo, Kashiwa, Chiba 277-8583, Japan.  Center for Data-
Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa,
Chiba 277-8583, Japan.  Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, P.R.China.  Michael A. Strauss Department of
Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ
08544, USA.  Kazushi Iwasawa ICREA and Institut de Ciències del Cosmos,
Universitat de Barcelona, IEEC-UB, Martí i Franquès, 1, 08028 Barcelona,
Spain.  Nobunari Kashikawa Department of Astronomy, School of Science, The
University of Tokyo, Tokyo 113-0033, Japan.  Masayuki Akiyama Astronomical
Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan.  Kentaro Aoki
Subaru Telescope, National Astronomical Observatory of Japan, Hilo, HI 96720,
USA.  Junya Arita Department of Astronomy, School of Science, The University
of Tokyo, Tokyo 113-0033, Japan.  Masatoshi Imanishi National Astronomical
Observatory of Japan, Mitaka, Tokyo 181-8588, Japan.  Department of
Astronomical Science, Graduate University for Advanced Studies (SOKENDAI),
Mitaka, Tokyo 181-8588, Japan.  Rikako Ishimoto Department of Astronomy,
School of Science, The University of Tokyo, Tokyo 113-0033, Japan.  Toshihiro
Kawaguchi Department of Economics, Management and Information Science,
Onomichi City University, Onomichi, Hiroshima 722-8506, Japan.  Kotaro Kohno
Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 181-0015,
Japan.  Research Center for the Early Universe, University of Tokyo, Tokyo
113-0033, Japan.  Chien-Hsiu Lee W. M. Keck Observatory, Kamuela, HI 96743,
USA  Tohru Nagao Research Center for Space and Cosmic Evolution, Ehime
University, Matsuyama, Ehime 790-8577, Japan.  John D. Silverman Kavli
Institute for the Physics and Mathematics of the Universe, WPI, The University
of Tokyo, Kashiwa, Chiba 277-8583, Japan.  Yoshiki Toba National Astronomical
Observatory of Japan, Mitaka, Tokyo 181-8588, Japan.  Academia Sinica
Institute of Astronomy and Astrophysics, Taipei 10617, Taiwan.  Research
Center for Space and Cosmic Evolution, Ehime University, Matsuyama, Ehime
790-8577, Japan.

###### Abstract

We report the discovery of two quasars at a redshift of $z$ = 6.05, in the
process of merging. They were serendipitously discovered from the deep multi-
band imaging data collected by the Hyper Suprime-Cam (HSC) Subaru Strategic
Program survey. The quasars, HSC $J$121503.42$-$014858.7 (C1) and HSC
$J$121503.55$-$014859.3 (C2), both have luminous ($>$1043 erg s-1) Ly$\alpha$
emission with a clear broad component (full width at half maximum $>$1000 km
s-1). The rest-frame ultraviolet (UV) absolute magnitudes are
$M_{1450}=-23.106\pm 0.017$ (C1) and $-22.662\pm 0.024$ (C2). Our crude
estimates of the black hole masses provide $\log(M_{\rm BH}/M_{\odot})=8.1\pm
0.3$ in both sources. The two quasars are separated by 12 kpc in projected
proper distance, bridged by a structure in the rest-UV light suggesting that
they are undergoing a merger. This pair is one of the most distant merging
quasars reported to date, providing crucial insight into galaxy and black hole
build-up in the hierarchical structure formation scenario. A companion paper
will present the gas and dust properties captured by Atacama Large
Millimeter/submillimeter Array observations, which provide additional evidence
for and detailed measurements of the merger and also demonstrate that the two
sources are not gravitationally-lensed images of a single quasar.

dark ages, reionization, first stars — galaxies: active — galaxies: high-
redshift — intergalactic medium — quasars: general — quasars: supermassive
black holes

##  1 Introduction

Quasars at high redshifts are an important and unique probe of the epoch of
reionization (EoR, referring to $z\geq 6$ in this letter), a critical epoch
for understanding the seeding and initial growth of supermassive black holes
(SMBHs), the evolution of the host galaxies at an early stage of hierarchical
structure formation, and the spatial and temporal progress of the
reionization. A significant number of EoR quasars have been discovered in the
past few decades, exploiting wide-field ($>$100-deg2 class) optical and near-
infrared (IR) imaging surveys (e.g., Fan et al., 2023, and references
therein). The ultraviolet (UV) quasar luminosity function (QLF) has now been
established at $z=6$ and $7$ (e.g., Matsuoka et al., 2018c, 2023; Schindler et
al., 2023), demonstrating that UV emission from quasars makes only a minor
contribution to cosmic reionization, if the flat QLF slope below the
characteristic luminosity continues to the unobserved faint end.

In the meantime, the James Webb Space Telescope (JWST) has produced
groundbreaking results in the past two years. Its extremely high IR
sensitivity has revealed a weak broad component in Balmer emission lines of
many high-$z$ galaxies, signaling the presence of low-luminosity active
galactic nuclei (AGNs) out to $z\sim 11$ (e.g., Greene et al., 2023; Maiolino
et al., 2023). The number density of such AGNs exceeds the extrapolation of
the classical QLF by several orders of magnitude (e.g., Harikane et al., 2023;
Matthee et al., 2023), changing the paradigm of SMBH activity happening in the
EoR.

On the other hand, there are still missing pieces in the AGN demographics in
the EoR, one of which is pairs of quasars or AGNs in mergers. Hierarchical
structure formation within the Lambda Cold Dark Matter model suggests that
galaxies grow via frequent mergers. If a significant fraction of those
galaxies contain a SMBH at the center, as implied from the measurements in the
local Universe (e.g., Kormendy & Ho, 2013), then one would naturally expect
SMBH pairs in the merging galaxies. If the merger induces gas inflow toward
the SMBHs (e.g., Hopkins et al., 2006), then such systems would be observed as
pairs of quasars or AGNs. The observed frequency of such pairs constrains many
key factors, such as the relative importance of mergers for galaxy and SMBH
evolution, the timescales associated with SMBH interaction and coalescence,
and the number density of possible gravitational wave sources. Quasar pairs
can also serve as a signpost of galaxy overdense regions (e.g., Onoue et al.,
2018) and as a probe of the small-scale distribution of the foreground
intergalactic medium (Rorai et al., 2017).

Searches for pairs of quasars or AGNs have used various techniques, typically
based on wide-field surveys (e.g., De Rosa et al., 2019, and references
therein). The most recent efforts include those reported by Silverman et al.
(2020) and Tang et al. (2021), who used Subaru Hyper Suprime-Cam (HSC; see
below) high-resolution images and identified pairs at $z\leq 3$ among the
known Sloan Digital Sky Survey (SDSS; York et al., 2000) quasars. Shen et al.
(2021) exploited astrometry information from the Gaia satellite mission (Gaia
Collaboration et al., 2016) to find two pairs among the SDSS quasars at
$z=2-3$. On the other hand, no pairs were found by Sandoval et al. (2023) in
their search from a large X-ray catalog at $z\sim 3$. There are also projects
to search for quasar pairs motivated by investigation of gravitational lensing
(e.g., Richards et al., 2006; Inada et al., 2012; Yue et al., 2023). From such
a project based on the Dark Energy Survey (Abbott et al., 2018), Yue et al.
(2021) found a candidate of quasar pair at $z=5.66$, with a separation of 7.3
kpc. If confirmed, this would be the first quasar pair reported in the EoR. In
addition, quasars with merging galaxy companions have been reported in the EoR
(e.g., Decarli et al., 2017, 2019b), in which the companion galaxies are
frequently invisible in the rest-UV and are only identified by submillimeter
observations. Most recently, JWST observations are finding signatures of dual
AGNs in individual EoR galaxies, via double components or off-nucleus emission
of broad Balmer lines (Maiolino et al., 2023; Übler et al., 2023).

This letter presents the discovery of a pair of merging quasars at $z$ = 6.05,
HSC $J$121503.42$-$014858.7 and HSC $J$121503.55$-$014859.3 (C1 and C2,
hereafter). The two quasars are separated by 12 kpc, forming one of the most
distant pairs of quasars or AGNs reported to date. We describe the target
selection and spectroscopic observations in §2. The nature of the two sources
is discussed in §3, based on their imaging and spectroscopic properties. A
summary appears in §4. We adopt the cosmological parameters $H_{0}$ = 70 km
s-1 Mpc-1, $\Omega_{\rm M}$ = 0.3, and $\Omega_{\rm\Lambda}$ = 0.7. All
magnitudes refer to CModel magnitudes from the HSC data reduction pipeline,
which are measured by fitting galaxy models convolved with the point spread
function (PSF) to the observed source profile (Bosch et al., 2018). The
magnitudes have been corrected for Galactic extinction (Schlegel et al.,
1998), and are reported in the AB system (Oke & Gunn, 1983). A companion paper
(T. Izumi et al., in prep.) will present the gas and dust properties of these
quasars captured by Atacama Large Millimeter/submillimeter Array (ALMA)
observations, as well as their kinematic modeling.

##  2 Observations

Figure 1: Top: three-color (HSC $r$, $i$, and $z$-band) composite image around
C1 and C2, the two reddest sources at the center. North is up and East to the
left, and the image size is approximately 90″$\times$ 25″. The limiting
magnitude for point sources is $\sim$26\. The inset shows an expanded view of
C1 and C2, with the thin dotted lines representing a 1″.0 slitlet used for
FOCAS spectroscopy. Bottom: two-dimensional FOCAS spectrum of C1 (upper trace
of light) and C2 (lower trace), created by stacking all available data.

Figure 1 presents a three-color (HSC $r$, $i$, and $z$-band) composite image
around the two quasars, C1 (West) and C2 (East). Their observed properties are
summarized in Table 1. Here $\mu_{z/y}$ represents the second-order moment of
the source on the $z/y$-band image, normalized to those of field stars as a
model of PSF (i.e., an unresolved source has $\mu_{z/y}=1$). C1 was originally
selected from the HSC Subaru Strategic Program (SSP; Aihara et al., 2018)
imaging survey. Its red $i-z$ and relatively blue $z-y$ colors as well as the
fact that it is not (or only marginally) spatially resolved made it an EoR
quasar candidate in our “Subaru High-$z$ Exploration of Low-Luminosity Quasars
(SHELLQs)” project (Matsuoka et al., 2016, 2018a, 2018b, 2018c, 2019a, 2019b,
2022, 2023).111 We clarify that the present two quasars were not included in
the previous SHELLQs publications, and are reported here for the first time.
The initial follow-up spectroscopy was carried out with Subaru Telescope on
April 24, 2018, as a part of the Subaru intensive program S16B-011I. We used
the Faint Object Camera and Spectrograph (FOCAS; Kashikawa et al., 2002) in
the multi-object spectroscopy mode. The combination of the VPH900 grism, SO58
order-sorting filter, and 1″.0 slitlets yielded spectral coverage from 0.75 to
1.05 $\mu$m with resolution $R\sim 1200$. The slit angle222Slit angle is
measured from North to East, such that 90∘ refers to a slit aligned to the
East – West direction.  was set to 90∘. We took seven 10-min exposures under
the clear sky, with the seeing conditions of 0″.8 – 1″.0. The data reduction
was performed with the Image Reduction and Analysis Facility (IRAF) using the
dedicated FOCASRED package in a standard manner. The wavelength scale was
calibrated with reference to sky emission lines, and the flux calibration was
tied to Feige 34, a white dwarf standard star, observed on the same night.
Slit losses were corrected for by scaling the spectrum to match the HSC
$z$-band magnitude.

Table 1: Imaging and spectroscopic measurements

Object | R. A. | Decl. | $g_{\rm AB}$ | $r_{\rm AB}$ | $i_{\rm AB}$ | $z_{\rm AB}$ | $y_{\rm AB}$
---|---|---|---|---|---|---|---
C1 | 12:15:03.42 |  $-$01:48:58.7 | $26.25\pm 0.28$ | $<26.09$ | $25.73\pm 0.22$ | $23.78\pm 0.11$ | $23.14\pm 0.12$
C2 | 12:15:03.55 |  $-$01:48:59.3 | $<26.75$ | $<26.32$ | $<26.50$ | $24.40\pm 0.15$ | $23.75\pm 0.18$
|  $\mu_{z}$ (HSC) |  $\mu_{y}$ (HSC) |  $\mu_{z}$ (FOCAS) |  $\mu_{y}$ (FOCAS) |  |  |
C1 | 1.35 $\pm$ 0.16 | 1.27 $\pm$ 0.15 | $1.29\pm 0.16$ | $1.34\pm 0.15$ |  |  |
C2 | 1.60 $\pm$ 0.20 | 0.99 $\pm$ 0.23 | $1.92\pm 0.25$ | $1.63\pm 0.21$ |  |  |
| $z_{\rm Ly\alpha}$ | $M_{1450}$ |  $L_{\rm bol}$ (erg s-1) |  $v_{\rm FWHM}$ (km s-1) | EWrest (Å) |  $L_{\rm line}$ (erg s-1) | Comment
C1 | 6.053 | $-23.106\pm 0.017$ | $(6.2\pm 0.1)\times 10^{45}$ |  |  |  |
|  |  |  | 1450 $\pm$ 170 | 15 $\pm$ 2 | $(1.34\pm 0.13)\times 10^{43}$ | Ly$\alpha$ (broad)
|  |  |  | 360 $\pm$ 30 | 12 $\pm$ 2 | $(1.02\pm 0.12)\times 10^{43}$ | Ly$\alpha$ (narrow)
C2 | 6.053 | $-22.662\pm 0.024$ | $(4.1\pm 0.1)\times 10^{45}$ |  |  |  |
|  |  |  | 1290 $\pm$ 60 | 27 $\pm$ 2 | $(1.84\pm 0.07)\times 10^{43}$ | Ly$\alpha$ (broad)

Note. — The magnitude lower limits are given at 3$\sigma$ confidence level.
The line FWHMs ($v_{\rm FWHM}$) have been corrected for line broadening due to
the finite instrumental resolution. The equivalent widths (EWrest) are
reported in the rest frame.

The initial spectroscopy revealed strong and asymmetric Ly$\alpha$ emission at
the observed wavelength of $\lambda_{\rm obs}=8576$ Å, indicating that C1
exists at $z_{\rm Ly\alpha}=6.053$. Soon after the spectroscopic
identification, we noticed that C1 is accompanied by a fuzzy source with
similar $i-z$ and $z-y$ colors (see Figure 1). This fuzz, named C2, is
separated by 2″.0 from C1 toward the East. We carried out another set of
spectroscopy with FOCAS on April 25, 26, and May 10, 2019, as a part of the
Subaru intensive program S18B-071I. This time we oriented the slit angle to
106∘, so that C1 and C2 were observed simultaneously. The total exposure time
in this run was 270 min. The sky condition was mostly clear, with the seeing
of 0″.4 – 0″.7. All the other instrument configurations and data reduction
were identical to those in the initial spectroscopy. We further obtained
additional exposures totaling 100 min on March 2, 2021, using the same
observational settings as in the 2019 run. The sky condition was clear with
the seeing of 0″.6.

We also acquired near-IR spectra of the two sources with the Fast Turnaround
program (ID: GN-2020A-FT-106) at the Gemini North telescope. We used the
Gemini Near-InfraRed Spectrograph (GNIRS; Elias et al., 2006) in the cross-
dispersed mode, with the 32 l/mm grating and the central wavelength set to
1.65 $\mu$m. The slit width was 1″.0, giving spectral coverage from 0.85 to
2.5 $\mu$m and resolution $R\sim 500$. We oriented the slit angle to 106∘ and
took 63 $\times$ 5-min exposures in total, spread over a month (June 3, 4, 14,
July 6, and 7, 2020). The observations were carried out in the queue mode,
with the requested sky conditions of 50-percentile cloud coverage and
70-percentile image quality. The data reduction was performed with IRAF using
the Gemini GNIRS package in a standard manner. The wavelength scale was
calibrated with reference to Argon lamp spectra. The flux calibration and
telluric absorption correction were tied to standard stars HIP 54849 and HIP
61637, observed immediately before or after the target observations at similar
airmass. We scaled the GNIRS spectrum to match the FOCAS spectrum where they
overlap in wavelength.

##  3 Results and discussion

###  3.1 Nature of the two sources

Figure 1 presents the two-dimensional FOCAS spectra of C1 (West) and C2
(East), co-added across all exposures. The extracted one-dimensional spectra
are shown in Figure 2 (upper panels). We detected a strong emission line also
from C2, whose peak wavelength is consistent with that measured in C1. The
asymmetric profiles and the presence of the Gunn & Peterson (1965) trough at
the shorter wavelengths confirm the identification of the line as Ly$\alpha$,
redshifted to $z_{\rm Ly\alpha}=6.053$. Since Ly$\alpha$ redshifts of EoR
objects are relatively uncertain (see also below), we report a formal redshift
of $z=6.05$ in this paper. The full widths at half maximum (FWHM) of
Ly$\alpha$, uncorrected for intergalactic medium (IGM) absorption, are $v_{\rm
FWHM}$ = 320 $\pm$ 20 km s-1 and 810 $\pm$ 180 km s-1 for C1 and C2,
respectively. We also detected flat continuum emission redwards of the line.
The rest-UV absolute magnitudes of the two sources are $M_{1450}=-23.106\pm
0.017$ (C1) and $-22.662\pm 0.024$ (C2) at the rest-frame wavelength
$\lambda_{\rm rest}$ = 1450 Å. These values were obtained by extrapolating the
continuum flux density at $\lambda_{\rm obs}=9000-9300$ Å, where the sky
emission is relatively weak, with a power-law model with a slope $\alpha=-1.5$
($F_{\lambda}\propto\lambda^{-1.5}$; e.g., Vanden Berk et al., 2001). Assuming
a quasar bolometric correction of BC1350 = 3.81 (Shen et al., 2011), we get
the bolometric luminosity of $L_{\rm bol}$ = (6.2 $\pm$ 0.1) $\times$ 1045 erg
s-1 and (4.1 $\pm$ 0.1) $\times$ 1045 erg s-1 for C1 and C2, respectively.

Figure 2:  Upper panels: FOCAS spectra of C1 (top) and C2 (middle) created by
stacking all available data, along with a sky spectrum as a guide to the
expected noise (bottom). The dotted lines represent the expected positions of
Ly$\alpha$ and N V $\lambda$1240 emission lines, as well as interstellar
absorption lines of Si II $\lambda$1260, Si II $\lambda$1304, and C II
$\lambda$1335, given the redshift of $z_{\rm Ly\alpha}=6.053$. An unidentified
line at $\lambda_{\rm obs}$ = 9082 Å in C1 (see text) is marked by an arrow.
Lower panels: Continuum subtracted spectra of C1 (left) and C2 (right) around
Ly$\alpha$. The thick red lines represent the best-fit models, while the thin
red lines (only in C1) represent their broad and narrow components. The
spectral window used for the fitting ($\lambda_{\rm obs}=8576-8615$ Å) is
shown by the vertical dashed lines. The gray shaded area marks the wavelength
range affected by strong sky emission. The spectra in both upper and lower
panels were smoothed using inverse-variance-weighted means over 3 pixels, for
display purposes.

It is clear from Figure 2 (upper panels) that the Ly$\alpha$ profile has a
relatively broad component in both sources, with a narrow core component seen
only in C1. We fit two Gaussians and one Gaussian to the C1 and C2 spectra
redward of the line peak, respectively, as displayed in Figure 2 (lower
panels). The local continuum emission was estimated at $\lambda_{\rm
obs}=8694-8738$ Å, where strong lines from the targets and the sky are absent,
and was subtracted before the model fitting. We found that the broad
Ly$\alpha$ components of the two sources have similar widths, $v_{\rm FWHM}$ =
1450 $\pm$ 170 km s-1 (C1) and 1290 $\pm$ 60 km s-1 (C2). Luminosity and other
line properties from the best fit models are reported in Table 1.

While the redshift was fixed to $z_{\rm Ly\alpha}=6.053$ during the model
fitting, adopting alternative values doesn’t change our conclusion that a
broad line component is present. Due to severe absorption from the IGM, the
intrinsic peak of Ly$\alpha$ is often located blueward of the observed peak,
which would indicate an intrinsically broader line width than estimated above.
This is likely the case for C2, whose ALMA observations of the [C II] 158
$\mu$m line indicate $z_{\rm[CII]}=6.044$ (T. Izumi et al., in prep.). On the
other hand, C1 has $z_{\rm[CII]}=6.057$, i.e., the observed Ly$\alpha$ peak is
blueshifted relative to [C II]. When we fix the Ly$\alpha$ redshift of the
broad component to $z_{\rm[CII]}=6.057$, we get its width of $v_{\rm FWHM}$ =
1100 $\pm$ 100 km s-1 in C1 and 1100 $\pm$ 60 km s-1 in C2. We also note that
the FWHM estimate remains almost unchanged when we fit only the nuclear part
of the spatially resolved C2 spectrum.

The spectral properties mentioned above suggest the presence of quasars in
both sources. The widths of the Ly$\alpha$ broad components exceed the common
threshold of quasar classification, $v_{\rm FWHM}$ = 500 – 1000 km s-1 (e.g.,
Schneider et al., 2010; Pâris et al., 2012), and are similar to values found
in faint AGNs revealed by JWST spectroscopy of EoR galaxies (e.g., Greene et
al., 2023; Harikane et al., 2023; Maiolino et al., 2023). These values are
found at the lower end of the FWHM distribution of low-$z$ Seyfert 1 galaxies
in SDSS (e.g., Hao et al., 2005) and of high-$z$ low-luminosity quasars found
in SHELLQs. On the other hand, star-forming galaxies cannot produce line
components that are significantly broader than $\sim$500 km s-1, even with
outflows (e.g., Newman et al., 2012; Swinbank et al., 2019). We note that
Ly$\alpha$ is spatially resolved in C2, and the line component extending to
$>$1000 km s-1 belongs to the nuclear part of the two-dimensional spectrum
(see also below). The observed Ly$\alpha$ luminosities of $>$1043 erg s-1 are
also very high for non-AGN galaxies, and overlap with the lower end of the
distribution of other SHELLQs quasars (e.g., Onoue et al., 2021). At lower
redshifts ($z\sim 2-3$), Ly$\alpha$ emitters with such high luminosities
almost always harbor AGNs, identified via characteristic X-ray, UV, radio
continuum emission and/or high-ionization optical lines (e.g., Konno et al.,
2016; Sobral et al., 2018; Spinoso et al., 2020). The continuum luminosities
of C1 and C2 ($M_{1450}\sim-23$ mag) are roughly 10 times higher than the
characteristic luminosity of the galaxy luminosity function at $z=6$ (Harikane
et al., 2022), and it would be unexpected (though not impossible) if a close
pair of such luminous high-$z$ galaxies were found.

Other than Ly$\alpha$, no strong emission lines are detected from C1 or C2. We
found a small spectral bump at the expected wavelength of N V $\lambda$1240 in
both C1 and C2 (see Figure 2), but the adjacent bright sky emission hampers
robust identification of this feature. The 3$\sigma$ upper limit of the N
V/Ly$\alpha$ (broad) ratio is $\sim$0.2 in both sources, which is consistent
with the ratio measured in low-$z$ SDSS quasars ($\sim$0.02; Vanden Berk et
al., 2001). The GNIRS spectra of the two targets are very noisy (see Figure 3)
even with $>$5-hour on-source exposure, only allowing us to identify continuum
emission from C1. There is a spectral bump at the expected position of C IV
$\lambda$1549 in C1, but the detection is marginal at most. On the other hand,
the optical spectrum of C1 (Figure 2) exhibits a weak but clear emission line
at $\lambda_{\rm obs}=9082$ Å, which is also apparent in the two-dimensional
spectrum in Figure 1. This line corresponds to $\lambda_{\rm rest}=1288$ Å at
$z_{\rm Ly\alpha}=6.053$, where no emission line is known. It could be due to
an overlapping foreground source, whose faint blue emission extends Northward
of C1 (see the HSC image of Figure 1), but the present data cannot provide any
robust identification.

Figure 3: GNIRS spectra of C1 (top) and C2 (middle) created by stacking all
available data, along with the error spectrum dominated by the sky background
(bottom). The vertical lines represent the expected positions of Ly$\alpha$, N
V $\lambda$1240, C IV $\lambda$1549, C III] $\lambda$1906, and Mg II
$\lambda$2800, given the redshift of $z_{\rm Ly\alpha}=6.053$. The spectra
were smoothed using inverse-variance-weighted means over 9 pixels, for display
purposes.

It is well known that quasar emission line properties, in particular those of
C IV $\lambda$1549, Mg II $\lambda$2800, and H$\beta$, are sensitive to SMBH
masses ($M_{\rm BH}$). Correlation in the form of $M_{\rm BH}\propto v_{\rm
FWHM}^{2}L_{\rm line}^{\gamma}\equiv\mathscr{M}_{\rm line}$ is observed for
the above three lines, where $L_{\rm line}$ is the line luminosity and
$\gamma$ is a constant close to 0.5 (e.g., Vestergaard & Peterson, 2006). Here
we obtain crude mass estimates of the two quasars via the broad Ly$\alpha$
component, which is also sensitive to $M_{\rm BH}$ (e.g., Takahashi et al.,
2024). As is clear from Table 1, C1 and C2 have similar Ly$\alpha$ properties
in the broad components, suggesting similar $M_{\rm BH}$. We looked into the
spectroscopic properties of SDSS quasars measured by Rakshit et al. (2020),
and found 678 / 579 quasars whose $\mathscr{M}_{\rm Ly\alpha}$ ($\gamma=0.5$
is assumed) values lie within $\pm$0.1 dex of the Ly$\alpha$ broad component
of C1 / C2. Both of these matched samples have median masses $\log(M_{\rm
BH}/M_{\odot})=8.1$, with a relatively small scatter of 0.3 dex. We thus
estimate that both C1 and C2 have $\log(M_{\rm BH}/M_{\odot})=8.1\pm 0.3$. The
corresponding Eddington ratios are $\sim$0.4 and $\sim$0.3 for C1 and C2,
respectively. These estimates are approximate at most, and need to be updated
with future measurements of, e.g., Balmer lines in the rest-frame optical with
JWST.

Similar objects with luminous ($>$1043 erg s-1) and relatively narrow (total
FWHM of $<500$ km s-1, uncorrected for IGM absorption) Ly$\alpha$ have been
identified at $z\geq 6$ in our SHELLQs project (e.g., Matsuoka et al., 2022).
Though most of them do not exhibit a broad Ly$\alpha$ component, we consider
them be candidate (possibly obscured) quasars, based on their high Ly$\alpha$
luminosities. We have carried out follow-up IR spectroscopy of seven such
objects with $L_{\rm Ly\alpha}=10^{43.3}-10^{44.3}$ erg s-1, and found
positive evidence for the presence of AGN overall. JWST observations revealed
broad components in H$\beta$ and H$\alpha$ from two objects, providing clear
signatures of AGNs. Two other objects observed with Keck/MOSFIRE show strong
high-ionization lines, C IV $\lambda\lambda$1548, 1550 in one object (Onoue et
al., 2021) and N IV $\lambda\lambda$1483, 1487 in the other (M. Onoue et al.,
in prep.).333 The ionization potentials of C IV and N IV are 64 eV and 77 eV,
respectively. Their large rest-frame EWs are difficult to explain with star-
forming activity alone, pointing to the presence of hard AGN radiation. The
remaining three objects were observed with X-Shooter on the Very Large
Telescope, but no emission lines other than Ly$\alpha$ were detected. However,
the sensitivity of the observations is somewhat lower than those mentioned
above. Overall, accumulating evidence suggests that at least some of the
SHELLQs objects with luminous and narrow Ly$\alpha$ host AGNs.

###  3.2 Extended emission and merging signatures

Figure 4: Close-up view of the two-dimensional Ly$\alpha$ spectra of C1 (upper
trace, at $\sim$22 kpc) and C2 (lower trace, at $\sim$10 kpc). The left panel
displays the stacked data of all available exposures, totaling 370 min (i.e.,
an enlarged view of Figure 1). The white boxes represent the regions used to
calculate the S/N of the bridging emission (signal in the “bridge” box and
noise in the “blank” boxes). The right panel displays a single 20-min exposure
taken under the best seeing condition ($\sim$0″4), which reveals a spatially
resolved Ly$\alpha$ profile in C2. The angular distance of 1″ corresponds to
5.7 kpc at the source redshift.

The two objects are likely in physical association with each other, as
indicated by their close separation in both the transverse and line-of-sight
directions. The angular separation of 2″.0 corresponds to a projected distance
of 12 kpc (proper) or 82 kpc (comoving). Moreover, we see extended Ly$\alpha$
emission bridging the two objects, as is clear from Figure 4 (left). This
emission component is detected with the signal-to-noise ratio (S/N) of
$\sim$8, when the signal and noise are measured in the “bridge” and “blank”
boxes indicated in the figure, respectively. More spectacular bridging, tails,
and other extended structures have been identified around C1 and C2 in the [C
II] 158$\mu$m line emission with our ALMA observations, whose analysis will be
presented in a companion paper (T. Izumi et al., in prep.).

The rest-UV emission connecting the two sources is also visible in our deep
FOCAS images presented in Figure 5. The bridging emission was detected only in
the $z$ band containing Ly$\alpha$, with S/N $\sim$ 5 when the signal and
noise are measured in the “bridge” and “blank” boxes indicated in the figure.
The $z$\- and $y$-band images were taken on May 10 – 11, 2019, under the
seeing condition of 0″.4–0″.6. The total exposure time is 30 min in each
filter.

Similar extended rest-UV emission is seen around other high-$z$ quasars
(Farina et al., 2019), in some cases accompanied by a merging galaxy (Decarli
et al., 2019a). While such emission is sometimes observed around an isolated
quasar, the fact that it is visible only in the area connecting C1 and C2 in
the present case provides a strong indication of a merger in progress. If the
nearly 1:1 ratio of black hole masses indicates similar stellar masses in the
merging two quasars, then it agrees with the results from hydrodynamical
simulations, which predict that dual SMBH activity appears most frequently in
merging galaxies with a mass ratio close to one, in close separations ($<$10
kpc; e.g., Capelo et al., 2015, 2017).

The above FOCAS images also confirmed the spatial extendedness of C2, which
has the normalized second-order moments of $\mu_{z}=1.92\pm 0.25$ and
$\mu_{y}=1.63\pm 0.21$. The large $\mu_{z}$ is most likely due to the
spatially resolved Ly$\alpha$ of C2 (see Figure 4), while the measurement of
$\mu_{y}>1$ may suggest a contribution from the host galaxy to the continuum
emission. C1 has a more compact shape, with $\mu_{z}=1.29\pm 0.16$ and
$\mu_{y}=1.34\pm 0.15$. On the other hand, the FOCAS spectra in Figure 2 show
no evidence of interstellar absorption lines such as Si II $\lambda$1260, Si
II $\lambda$1304, and C II $\lambda$1335, indicating that the host galaxy
makes a subdominant contribution at most to the continuum spectrum, in either
C1 or C2. We note that our SMBH mass estimates are not affected by host galaxy
contamination, since they are derived from the luminosity and width of the
broad Ly$\alpha$ component only. The bolometric luminosity and Eddington
ratios reported above would be upper limits, if there were significant host
galaxy contamination.

We have ruled out the possibility that these two sources are gravitationally-
lensed images of a single quasar. The ALMA observations mentioned above
revealed significantly brighter ($>$5 times) far-IR continuum emission from C1
than from C2, in contrast to their similar brightness in the rest-UV
($\sim$0.6 mag difference; see Table 1). A continuous velocity gradient is
observed throughout C1, C2, and the surrounding extended [C II] 158 $\mu$m
emission (blueshift to redshift from East to West, overall). In addition, only
C2 has a spatially resolved Ly$\alpha$ profile, in which the West and East
edges have a relative velocity offset of $\sim$300 km s-1 and the nucleus has
a velocity component extending to $\sim$1000 km s-1. We will present detailed
analysis of gas kinematics, combining both the Ly$\alpha$ and [C II]
measurements, in the companion paper.

Figure 5: FOCAS $z$-band (left) and $y$-band (right) images around the two
quasars, obtained under seeing conditions of 0″.4–0″.6. North is up and East
to the left. The scale bars represent 1″. The boxes represent the regions used
to calculate the S/N of the bridging emission (signal in the “bridge” box and
noise in the “blank” boxes).

##  4 Summary

This letter is the twentieth in a series of publications from the SHELLQs
project, a high-$z$ quasar survey based on HSC-SSP imaging. We report the
serendipitous discovery of two merging quasars at $z$ = 6.05, one of the most
distant pairs of quasars or AGNs known to date. The quasars, HSC
$J$121503.42$-$014858.7 (C1) and HSC $J$121503.55$-$014859.3 (C2), have
similar rest-UV properties overall, with $M_{1450}=-23.106\pm 0.017$ (C1) and
$-22.662\pm 0.024$ (C2). Both sources have a broad Ly$\alpha$ component with
FWHM $>$ 1000 km s-1, giving crude estimates of SMBH masses of $\log(M_{\rm
BH}/M_{\odot})=8.1\pm 0.3$. The close separation (2″.0, corresponding to a
projected proper distance of 12 kpc) and the bridging emission structure
indicate that the two objects are undergoing a merger, which may have caused
the observed quasar activity. Indeed, ALMA observations have revealed
spectacular extended structure surrounding the two quasars, whose detailed
analysis will be presented in our companion paper (T. Izumi et al., in prep.).

This research is based on data collected at Subaru Telescope, which is
operated by the National Astronomical Observatory of Japan. We are honored and
grateful for the opportunity of observing the Universe from Maunakea, which
has the cultural, historical and natural significance in Hawaii. We appreciate
the staff members of the telescope for their support during our FOCAS
observations. This research is based in part on data obtained at the
international Gemini Observatory, a program of NSF’s NOIRLab, via the time
exchange program between Gemini and the Subaru Telescope. The international
Gemini Observatory at NOIRLab is managed by the Association of Universities
for Research in Astronomy (AURA) under a cooperative agreement with the
National Science Foundation on behalf of the Gemini partnership: the National
Science Foundation (United States), the National Research Council (Canada),
Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia,
Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia,
Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science
Institute (Republic of Korea). Y. M. was supported by the Japan Society for
the Promotion of Science (JSPS) KAKENHI Grant No. 21H04494. K. I. acknowledges
the support under the grant PID2022-136827NB-C44 provided by MCIN/AEI
/10.13039/501100011033 / FEDER, EU. The HSC collaboration includes the
astronomical communities of Japan and Taiwan, and Princeton University. The
HSC instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University. This letter is based on data collected at the
Subaru Telescope and retrieved from the HSC data archive system, which is
operated by Subaru Telescope and Astronomy Data Center (ADC) at NAOJ. Data
analysis was in part carried out with the cooperation of Center for
Computational Astrophysics (CfCA) at NAOJ. This letter makes use of software
developed for Vera C. Rubin Observatory. We thank the Rubin Observatory for
making their code available as free software at http://pipelines.lsst.io/. The
Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made
possible through contributions by the Institute for Astronomy, the University
of Hawaii, the Pan-STARRS Project Office, the Max Planck Society and its
participating institutes, the Max Planck Institute for Astronomy, Heidelberg,
and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns
Hopkins University, Durham University, the University of Edinburgh, the
Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics,
the Las Cumbres Observatory Global Telescope Network Incorporated, the
National Central University of Taiwan, the Space Telescope Science Institute,
the National Aeronautics and Space Administration under grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation grant No. AST-1238877, the
University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos
National Laboratory, and the Gordon and Betty Moore Foundation.

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# The weak rate of convergence for the Euler-Maruyama approximation of one-
dimensional stochastic differential equations involving the local times of the
unknown process

M.Benabdallah M.Benabdallah : e-mail<EMAIL_ADDRESS>Department of
Mathematics, faculty of science, University of Ibn Tofail, Kenitra, Morocco
K.Hiderah K.Hiderah: e-mail<EMAIL_ADDRESS>Department of Mathematics,
faculty of science, University of Ibn Tofail, Kenitra, Morocco

###### Abstract

In this paper, we consider the weak convergence of the Euler-Maruyama
approximation for one dimensional stochastic differential equations involving
the local times of the unknown process. We use a transformation in order to
remove the local time $L_{t}^{a}$ from the stochastic differential equations
of type:

$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})dB_{s}+\int_{\mathds{R}}\nu(da)L_{t}^{a}$

where $B$ is a one-dimensional Brownian motion,
$\varphi:\mathds{R}\rightarrow\mathds{R}$ is a bounded measurable function,
and $\nu$ is a bounded measure on $\mathds{R}$ and we provide the
approximation of Euler-maruyama for the stochastic differential equations
without local time. After that, we conclude the approximation of Euler-
maruyama $X_{t}^{n}$ of the above mentioned equation, and we provide the rate
of weak convergence
$\text{Error}=\mathds{E}\left|G(X_{T})-G(X_{T}^{n})\right|$, for any function
$G$ in a certain class.
Keywords: Euler-Maruyama approximation, weak convergence, stochastic
differential equation, local time, bounded variation.
MSC 2010: 60H35, 41A25, 60H10, 60J55, 65C30

##  1 Introduction

Let $X=\\{X_{t}:t\geq 0\\}$ be a process stochastic involving the local time
defined by the stochastic differential equations:

$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})dB_{s}+\int_{\mathds{R}}\nu(da)L_{t}^{a}$
(1)

where $B$ is a one-dimensional Brownian motion,
$\varphi:\mathds{R}\rightarrow\mathds{R}$ is a bounded measurable function,
and $\nu$ is a bounded measure on $\mathds{R},L_{t}^{a}$ denotes the local
time at $a$ for the time $t$ of the semimartingale X.
In [References], if $\nu$ is absolutely continuous with respect to the
Lebesgue measure on $\mathds{R}$ (i.e.$\nu(da)=g(a)da)$ then (1) becomes the
usual Itô equation :

$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})dB_{s}+\int_{0}^{t}(g\varphi^{2})ds$
(2)

In general, we consider the following stochastic differential equation(SDE)
with coefficients $b$ and $\sigma$, driven by a Brownian motion $B$ in
$\mathds{R}$:

$X_{t}=X_{0}+\int_{0}^{t}\sigma(X_{s})dB_{s}+\int_{0}^{t}b(X_{s})ds$ (3)

where the drift coefficient $b$ and the diffusion coefficient $\sigma$ are
Borel-measurable functions from $\mathds{R}$ into $\mathds{R}$ and $X_{0}$ is
an $\mathds{R}$-valued random variable, which is independent of $B$ and $b$.
The continuous Euler scheme $\\{X^{n}_{t},0\leq t\leq T\\}$ for the SDE (3) on
the time interval $[0,T]$ is defined as follows: $X^{n}_{0}=X_{0}$, and

$X_{t}^{n}=X_{0}^{n}+\int_{0}^{t}\sigma(X^{n}_{\eta_{k}(s)})dB_{s}+\int_{0}^{t}b(X^{n}_{\eta(k)})ds,$
(4)

For $\eta_{k}\leq t\leq\eta_{k+1},k=0,1,2,...,n-1$, where
$0=\eta_{0}\leq\eta_{1}\leq...\leq\eta_{n}=T$ is a sequence of random
partitions of $[0,T]$. The weak convergence of the stochastic differential
equations has been studied by Avikainen [References], Bally and
Talay[References], Mikulevičius and Platen[References], D.Talay and
L.Tubaro[References], and the weak convergence of SDEs with discontinuous
coefficients has been studied by Chan and Stramer [References], Yan
[References], Arturo Kohatsu-Higa [References].

We use the same criteria that mentioned in [References], that the definition
of weak convergence with order $\gamma>0$ is that for all functions $f$ in a
certain class, there exists a positive constant $C$, which does not depend on
$\Delta t$, such that:

$\left|\mathds{E}[f(X_{T})]-\mathds{E}[f(X^{n}_{T})]\right|\leq C(\Delta
t)^{\gamma}$ (5)

The stochastic differential equations of the type (1) have been studied by
Stroock and Yor [References], Portenko [References], Le Gall
[References,References], Blei and Engelbert [References], Bass and
Chen[References] and the approximation of Euler-Maruyama for SDE of type (1)
has been studied by,e.g., Benabdallah, Elkettani and Hiderah[References].
In [References], when $\nu=\beta\delta_{(0)}$ ($\delta{(0)}$ denotes the Dirac
measure at 0) and $\varphi=1$, we get

$X_{t}=X_{0}+B_{t}+\beta L_{t}^{0}(X),|\beta|\leq 1$ (6)

The solution of equation (6) is the well-known process called the skew
Brownian motion which has been studied by Harrison and Shepp [References],
Ouknine [References,References], Bouhadou and Ouknine[References],
Lejay[References], Barlow [References], Étoré and Martinez [References],
Walsh[References].

Our goal of this paper is that under the assumption that the SDE (1) has a
weak solution and that it is unique, we study the conditions under which the
Euler scheme $\\{X^{n}_{t}:0\leq t\leq T\\}$ converges weakly to the exact
solution $\\{X_{t}:0\leq t\leq T\\}$ of the SDE (1).

We face two major problems: the presence of local time in equation (1) and the
inability to provide a simple discretization scheme. and the presence of a
discontinuity for the diffusion coefficient $\varphi$.
Our paper is divided as follows: In section (2), we present the important
propositions, assumptions and stochastic differential equations involving
local time. Our main results on the weak convergence of Euler scheme of one-
dimensional stochastic differential equations involving the local time are
given in section (3). Some numerical examples are given in section (4). All
proofs of the theorems are given in section (5).

##  2 Preliminaries and approximation

In this section, we provide the definitions, important propositions,
assumptions and stochastic differential equations involving local time.

######  Definition 2.1.

Let

$T_{f}(x):=\sup\sum\limits_{j=1}^{N}|f(x_{j})-f(x_{j-1})|,$

where the supremum is taken over $N$ and all partitions
$-\infty<x_{0}<x_{1}<...<x_{N}=x$ be the total variation function of $f$. Then
we say that $f$ is a function of bounded variation, if
$V(f):=\lim\limits_{x\rightarrow\infty}T_{f}(x)$ is finite, and call $V(f)$
the (total) variation of $f$.

######  Definition 2.2.

We say that a function $f$ has at most polynomial growth in $\mathds{R}$ is
there exist an integer $k$ and a constant $C>0$ such that $|f(x)|\leq
C(1+|x|^{k})$ for any $x\in\mathds{R}$

######  Remark 2.3.

Let $C^{k}_{p}(\mathds{R})$ denote the space of all $C^{k}$-functions of
polynomial growth (together with their derivatives).

######  Note 2.4.

BV($\mathds{R}$) will denote the space of all functions
$\varphi:\mathds{R}\rightarrow\mathds{R}$ of bounded variation on $\mathds{R}$
such that:

  1. 1.

$\varphi$ is right continuous.

  2. 2.

There exists an $\varepsilon>0$ such that : $\varphi(x)\geq\varepsilon$ for
all $x$.

If $\varphi$ is in $BV(\mathds{R})$, $\varphi(x^{-})$ will denote the left-
limit of $\varphi$ at point $x$ and $\varphi^{{}^{\prime}}(dx)$ will be the
bounded measure associated with $\varphi$.

######  Note 2.5.

M($\mathds{R}$) will denote the space of all bounded measures $\nu$ on
$\mathds{R}$ such that:

$|\nu(\\{x\\})|<1,\forall x\in\mathds{R}$

We have the stochastic differential equation (1), if $\varphi$ is in
$BV(\mathds{R})$ and $\nu$ is in $M(\mathds{R})$, then the stochastic
differential equation (1) has a unique strong solution as soon as
$|\nu(da)|<1$.

######  Theorem 2.6.

Let $\varphi$ be in $BV(\mathds{R})$ and $\nu$ be in $M(\mathds{R})$. Then
existence and pathwise uniqueness of solution hold for (1).

######  Proposition 2.7.

Let $\nu$ be in M($\mathds{R}$). There exists a function $f$ in
BV($\mathds{R}$), unique up to a multiplicative constant, such that:

$f^{{}^{\prime}}(dx)+(f(x)+f(x^{-}))\nu(dx)=0$ (7)

If we require that $f(x)\xrightarrow[x\rightarrow-\infty]{}1$, then $f$ is
unique and is given by:

$f(x)=f_{\nu}(x)=\exp(-2\nu^{c}(]-\infty;x]))\prod\limits_{y\leq
x}\left(\frac{1-\nu(\\{y\\})}{1+\nu(\\{y\\})}\right)$ (8)

Where $\nu^{c}$ denotes the continuous part of $\nu$.

######  Proposition 2.8.

Let $\varphi$ be in $BV(\mathds{R})$ and $\nu$ be in $M(\mathds{R})$ and
$f_{\nu}$ be define by(8) and set:

$F_{\nu}(x)=\int_{0}^{x}f_{\nu}(y)dy$ (9)

Then $X$ is a solution of equation (1), if and only if $Y:=F_{\nu}(X)$ is a
solution of:

$dY_{t}=(\varphi f_{\nu})\circ F^{-1}(Y_{t})dB_{t}$ (10)

The Euler scheme ${Y^{n}_{t}:0\leq t\leq T}$ for the SDE (10) on the time
interval $[0,T]$ is defined as follows: $Y^{n}_{0}=Y_{0}$, and

$Y^{n}_{t}=Y^{n}_{\eta_{k}}+(\varphi f_{\nu})\circ
F^{-1}(Y^{n}_{\eta_{k}})(B_{t}-B_{\eta_{k}})$ (11)

for $\eta_{k}<t\leq\eta_{k+1},k=0,1,2,...,n$ where
$0=\eta_{0}\leq\eta_{1}\leq...\leq\eta_{n}=T$. This Euler scheme can be
written as

$Y^{n}_{t}=Y^{n}_{0}+\int_{0}^{t}(\varphi f_{\nu})\circ
F^{-1}(Y^{n}_{\eta_{k}})dB_{s}$ (12)

In [References], the SDE(1) has existence of strong solutions and pathwise
uniqueness when $\varphi$ is a function on $\mathds{R}$ that is bounded above
and below by positive constants and such that there is a strictly increasing
function $f$ on $\mathds{R}$ such that

$\left|\varphi(x)-\varphi(y)\right|^{2}\leq\left|f(x)-f(y)\right|,x,y\in\mathds{R}$

and $\nu$ is a finite measure with $|\nu(\\{a\\})|\leq 1$ for every
$a\in\mathds{R}$.

######  Theorem 2.9.

Suppose $\varphi$ is a measurable function on $\mathds{R}$ that is bounded
above and below by positive constants and suppose that there is a strictly
increasing function f on R such that

$\left|\varphi(x)-\varphi(y)\right|^{2}\leq\left|f(x)-f(y)\right|,x,y\in\mathds{R}$
(13)

For any finite signed measure $\nu$ on $\mathds{R}$ such that
$\nu(\\{x\\})<\frac{1}{2}$ for each $x\in\mathds{R}$ and
every$x_{0}\in\mathds{R}$, the SDE

$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})dB_{s}+\int_{\mathds{R}}\nu(da)L_{t}^{a},t\geq
0$ (14)

has a continuous strong solution and the continuous solution is pathwise
unique.

Define

Let

$\pi(x):=\begin{cases}-\frac{\log(1-2x)}{2x},&\mbox{if
}x\in(-\infty,0)\cup(0,\frac{1}{2})\\\ 1,&\mbox{if }x=0.\end{cases}$

Let $\mu(dx):=\pi(\nu(\\{x\\})\nu(dx)$ which is a finite signed measure.
Define

$S(x):=\int_{0}^{x}e^{-2\mu(-\infty,y]}dy$

Since $\mu$ is a finite measure, $S^{{}^{\prime}}$ is right continuous and
strictly positive. Hence $S$ is increasing and one-to-one. Let $S^{-1}$ denote
the inverse of $S$, let $S^{{}^{\prime}}_{\ell}$ denote the left continuous
version of $S^{{}^{\prime}}$, i.e., the left hand derivative of s, and let
$\sigma^{{}^{\prime}}_{\ell}$ denote the left hand derivative of $S^{-1}$.
Since $\mu$ is a finite signed measure, $S^{{}^{\prime}}$ is of bounded
variation. Let $\\{Y_{t},t\leq 0\\}$ solve

$dY_{t}=(S^{{}^{\prime}}_{\ell}\varphi)\circ
S^{-1}(Y_{t})dB_{t},Y_{0}=S(X_{0})$ (15)

Let $X=S^{-1}(Y)$, and $Y_{t}$ define by (15), in [References], they must show
that $X$ is a solution to (14).

##  3 The main results

In this section, we provide the main theorems. we study two cases: the
continuous of function $\left(\varphi.f_{\nu}\right)\circ F^{-1}(.)$ and the
discontinuous of function $\left(\varphi.f_{\nu}\right)\circ F^{-1}(.)$.

###  3.1 Main Theorems

In this section, we present the following results on the rates of the Euler-
Maruyama approximation.
Let $X_{t}$ be defined as in equation (1), $X_{t}^{n}$ be the Euler scheme for
equation (1), let $F$ is defined by equation (9), and let $f_{\nu}$ be from
the proposition(2.7). Here, we suppose that the assumptions of
proposition(2.8) satisfied, and the constant $C$ may change from line to line
and from theorem to theorem.

######  Theorem 3.1.

For any function $g:\mathds{R}\rightarrow\mathds{R}$, if for each $G:=g\circ
F^{-1}\in C_{P}^{2(\gamma+1)}(\mathds{R})$, and
$\psi(.):=\left(\varphi.f_{\nu}\right)\circ F^{-1}(.)$ is continuous, then,
there exists a constant $C>0$, which does not depend on $n$, such that

$\mathds{E}\left|g(X_{T}^{n})-g(X_{T})\right|\leq\frac{C}{n^{\gamma}}$

holds.

Next we give a sufficient condition under which the Euler scheme converges
weakly to the weak solution of SDE (2) case: discontinuous of function
$\left(\varphi.f_{\nu}\right)\circ F^{-1}(.)$. Let
$\psi_{1}(z):=\liminf\limits_{y\rightarrow z}\psi^{2}(y)>0$

######  Theorem 3.2.

Suppose that $\psi:=(\varphi.f)\circ F^{-1}(.)$ has at most linear growth with
$D_{\psi}$ of Lebesgue measure zero and $\mathds{E}(Y_{0})^{4}<\infty$. If
$\psi_{1}(z)>0$ for $z\in D_{\psi}$,and $G:=g\circ F^{-1}\in
C_{P}^{2(\gamma+1)}(\mathds{R})$, then the Euler scheme of SDE(1) converges
weakly to the unique weak solution of SDE(1), where $D_{\psi}$ is the set of
discontinuous points of $\psi(y)$.

If we suppose that the conditions of theorem (2.9) satisfied, we show the
following note.

######  Note 3.3.

The same results in the theorems (3.1,3.2) stay valid, if the conditions of
theorem (2.9) are hold.

##  4 Some examples

######  Example 4.1.

In equation (1), let $\nu(dx)=\alpha\delta_{0}(dx),|\alpha|<1$ where
($\delta_{0}$ is the Dirac measure at 0 ) and

$\varphi=\begin{cases}\frac{1+\alpha}{1-\alpha}\exp\left(-\frac{1-\alpha}{1+\alpha}x\right)&\mbox{
if }x\geq 0\\\ \exp\left(x\right)&\mbox{ if }x<0\end{cases}$

Using proposition (2.7), we have

$f_{\nu}(x)=\begin{cases}\frac{1-\alpha}{1+\alpha}&\mbox{if }x\geq 0\\\
1&\mbox{if }x<0\end{cases}$

Using proposition (2.8), we have

$F(x)=\begin{cases}\frac{1-\alpha}{1+\alpha}x&\mbox{if }x\geq 0\\\ x&\mbox{if
}x<0\end{cases}$

Then

$\psi(y):=(\varphi.f)\circ F^{-1}(y)=\begin{cases}\exp\left(-y\right)&\mbox{if
}y\geq 0\\\ \exp\left(y\right)&\mbox{if }y<0\end{cases}$

We note $\psi$ is continuous, then for any function
$g:\mathds{R}\rightarrow\mathds{R}$, such that $g\circ F^{-1}\in
C^{2(\gamma+1)}_{p}(\mathds{R})$, for any $\gamma>0$, for example

$g(x)=\frac{1}{1+\left(\frac{1-\alpha}{1+\alpha}x\right)^{2}}.1_{x\geq
0}+\frac{1}{1+x^{2}}1_{x<0}$

there exist a constant $C>0$, which does not depend on $n$, such that

$\left|\mathds{E}\left[g(X_{T}^{n})\right]-\mathds{E}\left[g(X_{T})\right]\right|\leq\frac{C}{n^{\gamma}}$

######  Example 4.2.

In equation (1), let $\nu(x)=\alpha\delta_{0}(x),|\alpha|<1$, where
$\delta_{0}$ is the Dirac measure at 0, and
$\mathds{E}(X_{0})^{4}=\mathds{E}(Y_{0})^{4}<\infty$ we define $\varphi=1$,
such that
Using proposition (2.7), we have

$\displaystyle f(x)=f_{\nu}(x)=\begin{cases}\frac{1-\alpha}{1+\alpha}&\mbox{if
}x\geq 0\\\ 1&\mbox{if }x<0\end{cases}$

And by using proposition (2.8), we have
$F_{\nu}(x)=\begin{cases}\frac{1-\alpha}{1+\alpha}x&\mbox{if }x\geq 0\\\
x&\mbox{if }x<0\end{cases}$
$F^{-1}_{\nu}(y)=\begin{cases}\frac{1+\alpha}{1-\alpha}y&\mbox{if }y\geq 0\\\
y&\mbox{if }y<0\end{cases}$ Then, we obtain

$\psi(y)=\left(\varphi.f_{\nu}\right)\circ
F^{-1}(y)=\begin{cases}\frac{1-\alpha}{1+\alpha}&\mbox{if }y\geq 0\\\
1&\mbox{if }y<0\end{cases}$ (16)

And

$\displaystyle Y_{t}$ $\displaystyle=Y_{0}+\int_{0}^{t}\psi(Y_{s})dB_{s}$ (17)

The Euler scheme $\left\\{Y^{n}_{t}:0\leq t\leq T\right\\}$ for the SDE (17),
on the time interval $[0,T]$ is defined as follows: $Y^{n}_{0}=Y_{0}$, and

$\displaystyle Y^{n}_{t}$
$\displaystyle=Y^{n}_{\eta_{k}}+\psi(Y^{n}_{\eta_{k}})(B_{t}-B_{\eta_{k}})$
(18)

For $\eta_{k}<t\leq\eta_{k+1},k=0,1,2,...,n$, where
$0=\eta_{0}\leq\eta_{1}\leq...\leq\eta_{n}=T$, the coefficient of equation
(17) is discontinuous at point 0, and, we have:

  * •

$D_{\psi}=\\{0\\}$

  * •

$\psi_{1}(z):=\liminf\limits_{z\rightarrow y}\psi^{2}(y)>0$

  * •

$\psi(\cdot)$ is at most linear growth with $D_{\psi}$

We have the Euler scheme of SDE(18) converges weakly to the unique weak
solution of SDE((17)), then the Euler scheme $X_{t}^{n}$ converges weakly to
the unique weak solution$X_{t}$ of SDE((1)).

##  5 Proofs of theorems

Before proving the theorems below, we introduce some notations.

######  Note 5.1.

The constant $C$ may change from line to line and from theorem to theorem.

Let $B$ be a one-dimensional brownian motion $(B_{t})_{t\in[0,T]}$, and
$\varphi$ is in $BV(\mathds{R})$ and $\nu$ is in $M(\mathds{R})$ and

$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})dB_{s}+\int_{\mathds{R}}\nu(da)L_{t}^{a}.$
(19)

and $Y_{t}$ is solution of the equation

$dY_{t}=\left(\varphi.f_{\nu}\right)\circ F^{-1}(Y_{t})dB_{t}.$ (20)

for which, as we get by the proposition(2.8), where $F$ be defined by the
equation (9), $f_{\nu}$ get by the proposition(2.7), and let $Y_{t}^{n}$ is a
solution by Euler scheme for equation(10) and defined by equation(12).

###  5.1 Proof of theorem (3.1)

######  Definition 5.2.

In [References], a time discrete approximation $X^{n}$ converges weakly with
order $\gamma>0$ to $X$ at time $T$ as $n\rightarrow\infty$, if for each $G\in
C_{P}^{2(\gamma+1)}(\mathds{R})$, there exists a constant $C>0$, which does
not depend on $n$, such that

$\left|\mathds{E}(G(X_{T}^{n}))-\mathds{E}(G(X_{T}))\right|\leq\frac{C}{n^{\gamma}}\text{holds.}$

Using the above definition, we can prove theorem (3.1).

  1. 1.

The proposition (2.8) is satisfied , then $X_{t}=F^{-1}(Y_{t})$ is solution
uniqueness of the SDE (1).

  2. 2.

$G:=g\circ F^{-1}\in C^{2(\gamma+1)}_{p}$

  3. 3.

By using definition (5.2), we have

$\displaystyle\left|\mathds{E}[g(X_{T})]-\mathds{E}[g(X^{n}_{T})]\right|$
$\displaystyle=\left|\mathds{E}[g\circ F^{-1}(Y_{T})]-\mathds{E}[g\circ
F^{-1}(Y^{n}_{T})]\right|$
$\displaystyle=\left|\mathds{E}[G(Y_{T})]-\mathds{E}[G(Y^{n}_{T})]\right|\leq\frac{C}{n^{\gamma}}$

###  5.2 Proof of theorem (3.2)

If we have SDE of type

$Y_{t}=Y_{0}+\int_{0}^{t}\sigma(Y_{s})dB_{s}$ (21)

and the Euler scheme of equation (21) give by

$Y_{t}=Y_{0}+\int_{0}^{t}\sigma(Y_{\eta_{n}(s)})dB_{s}$ (22)

Here, we present the following theorem(see [References]):

######  Theorem 5.3.

Suppose that $\sigma(y)$ has at most linear growth with $D_{\sigma}$ of
Lebesgue measure zero and $\mathds{E}(Y_{0})^{4}<\infty$. If
$\sigma_{1}(z):=\liminf\limits_{z\rightarrow y}\sigma^{2}(y)>0$ for $z\in
D_{\psi}$, then the Euler scheme of SDE(21) converges weakly to the unique
weak solution of SDE(21), where $D_{\sigma}$ is the set of discontinuous
points of $\sigma(y)$.

Using the above theorem, we can prove theorem (3.2).

  1. 1.

The proposition (2.8) is satisfied , then $X_{t}=F^{-1}(Y_{t})$ is solution
uniqueness of the SDE (1).

  2. 2.

$G:=g\circ F^{-1}\in C^{2(\gamma+1)}_{p}$

  3. 3.

$(\varphi.f_{\nu})\circ F^{-1}(\cdot)$ has at most linear growth with
$D_{\sigma}$ of Lebesgue measure zero.

  4. 4.

$\mathds{E}(Y_{0})^{4}<\infty$

  5. 5.

By using definition (5.2) and theorem (5.3) , we have

$\displaystyle\left|\mathds{E}[g(X_{T})]-\mathds{E}[g(X^{n}_{T})]\right|$
$\displaystyle=\left|\mathds{E}[g\circ F^{-1}(Y_{T})]-\mathds{E}[g\circ
F^{-1}(Y^{n}_{T})]\right|$
$\displaystyle=\left|\mathds{E}[G(Y_{T})]-\mathds{E}[G(Y^{n}_{T})]\right|\leq\frac{C}{n^{\gamma}}$

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