# Toward Wireless System and Circuit Co-Design
for the Internet of Self-Adaptive Things

Diptashree Das, Mohammad Abdi, Minghan Liu, Marvin Onabajo and Francesco
Restuccia
Department of Electrical and Computer Engineering, Northeastern University,
United States

###### Abstract

The deployment of a growing number of devices in Internet of Things (IoT)
networks implies that uninterrupted and seamless adaptation of wireless
communication parameters (e.g., carrier frequency, bandwidth and modulation)
will become essential. To utilize wireless devices capable of switching
several communication parameters requires real-time self-optimizations at the
radio frequency integrated circuit (RFIC) level based on system level
performance metrics during the processing of complex modulated signals. This
article introduces a novel design verification approach for reconfigurable
RFICs based on end-to-end wireless system-level performance metrics while
operating in a dynamically changing communication environment. In contrast to
prior work, this framework includes two modules that simulate a wireless
channel and decode waveforms. These are connected to circuit-level modules
that capture device- and circuit-level non-idealities of RFICs for design
validation and optimization, such as transistor noises,
intermodulation/harmonic distortions, and memory effects from parasitic
capacitances. We demonstrate this framework with a receiver (RX) consisting of
a reconfigurable complementary metal-oxide semiconductor (CMOS) low-noise
amplifier (LNA) designed at the transistor level, a behavioral model of a
mixer, and an ideal filter model. The seamless integration between system-
level wireless models with circuit-level and behavioral models (such as
VerilogA-based models) for RFIC blocks enables to preemptively evaluate
circuit and system designs, and to optimize for different communication
scenarios with adaptive circuits having extensive tuning ranges. An exemplary
case study is presented, in which simulation results reveal that the LNA power
consumption can be reduced up to 16x depending on system-level requirements.

###### Index Terms:

System-level validation, adaptive wireless systems and circuits,
hardware/software co-design, energy-aware optimization, simulation-based
system testing.

Figure 1: Overview of the framework for co-design and optimization of
reconfigurable RF circuits and wireless communication systems.

##  I Introduction

By 2030, over 50 billion devices will be absorbed into the Internet of Things
(IoT) [1]. The sheer number of IoT devices implies that continuous and
seamless adaptation of wireless communication parameters (e.g., carrier
frequency, bandwidth and modulation) will become essential. One of the
wireless system design goals is to provide sensing services for a plethora of
applications. However, the implementations are constrained because sensing and
communication circuit parameters have to be optimized for different frequency
bands, modulation schemes and channel conditions [2]. For this reason,
reconfigurability and self-optimization will be a cornerstone of IoT
networking paradigms [3, 4, 5]. Furthermore, it becomes increasingly important
to jointly simulate circuit and system level components for design
optimization and validation prior to the fabrication of chips. On the other
hand, conventional radio frequency integrated circuits (RFICs) are still
statically optimized, which does not allow for real-time self-optimization at
the intersection of hardware and software. Low-power RFICs are typically
designed and optimized specifically for the worst-case scenario of a given
communication standard, which leads to performance limitations and excessive
power consumption. Furthermore, circuit-level tuning usually optimizes block-
level performance. Conversely, in real systems, the circuit-level linearity
and dynamic range requirements strongly depend on the presence of nearby
interference signals and bias conditions [6, 7, 8, 9, 10], while slow-varying
aspects such as temperature sensitivity or device-level aging effects can only
be computed and compensated throughout the device lifetime [11]. Digitally-
controlled calibration is a popular design approach to improve the performance
and testability of mixed-signal integrated circuits [12, 13]. However, RFIC
calibrations of multiple interconnected circuit blocks typically do not
address the interdependence of the circuit-specific parameters during tuning,
which creates limitations during simulations for design validations. They
normally also do not account for system-level parameters such as symbol error
rate (SER) or throughput, particularly when relying on single-tone/two-tone
signals or other alternative test signals instead of the actual modulated
signals [14]. A comprehensive survey about the integration of machine learning
(ML) into integrated circuit design has been provided in [15, 16]. The use of
ML during design and optimization can be another potential method to ensure
functionality under consideration of interdependence between circuit and
system parameters.

Considering the above-mentioned challenges and opportunities, we propose a new
RFIC co-design and validation paradigm at the intersection of hardware and
software, which is summarized in Fig. 1. Our joint simulation framework
considers system-level performance metrics to facilitate the design of
adaptive RFIC circuits. The developed interoperability between different
design tools allows to (i) generate arbitrary modulated waveforms while
modeling different wireless channel conditions, (ii) utilize foundry-supplied
models that capture transistor-level non-idealities, (iii) simulate both
behavioral analog RFIC blocks and circuit level designs, and (iv) extract
circuit-level and system-level performance metrics. Furthermore, we introduce
simulations with modulated signal packets and modeled channel impairments to
extract system-level parameters such as bit error rate (BER) and error vector
magnitude (EVM); and accordingly optimize several circuit conditions (e.g.,
gain, noise figure and linearity characteristics) for optimization of
energy/performance tradeoffs. As a use-case scenario, we leverage our
framework to optimize a receiver (RX) composed of a reconfigurable
complementary metal-oxide semiconductor (CMOS) low-noise amplifier (LNA)
designed at the transistor-level, a behavioral model of a mixer and an ideal
filter model. The simulation results show that our framework can reduce the
LNA power consumption by up to 16x under varying BER requirements. In general,
the simulation framework can be used as a tool during the design and
validation of adaptive wireless RXs that operate with dynamically changing
requirements.

##  II Existing Work and Current Challenges

The fast-changing IoT ecosystem leads to a very dynamic nature of the wireless
channel that calls for complex hardware and software systems, including
adaptive and tunable transceiver designs. It has been explained in [17] how
tunable and reconfigurable radio frequency (RF) technologies provide potential
solutions for efficient spectrum sharing. In [18], a simulation-based approach
develops a multi-user IoT communication system by taking into account carrier
synchronization and data broadcasts on multiple channels for several of
low‐power devices present in the network. The work in [19] realized a
learning-based RF signal classifier on a field-programmable gate array (FPGA)
to reduce latency and power consumption, which requires prior knowledge of
signals and spectrum.

Adaptability in the RX front-end opens up the opportunity to collect and
process data from a dynamic wireless channel. CMOS RFIC prototypes have been
designed to enhance linearity and power handling requirements for cellular
applications [20]. However, RFICs can still be complemented with real-time
adaptation algorithms to optimize transceiver operation. [21] introduces a
real-time two-dimensional real-time adaptation method to configure a RX for
optimum NF and linearity with a certain power budget and desired signal level.
The design of adaptive wireless RXs with single or multi-parameter
optimization is highly relevant for specific incoming signals, and requires
validating performance for different wireless standards, channel conditions,
and chip-level performance variations (e.g., CMOS fabrication process
variations). A key consideration during the design of adaptive RFICs is that
reconfigurability is tightly coupled with power consumption. In addition, the
need for wide tuning range and energy/power scalable designs calls for the
seamless combination of circuit and system level adjustments. An evaluation of
performance versus power trajectory for RF front-end functional blocks has
been explored in [22]. In addition to determining the inter-dependencies of
circuit parameters for each block in the RF front-end, the optimization of
analog RF circuits based on feedback control with digitally-controlled
features has been demonstrated [23, 24, 25, 26, 27, 13], which requires to
design complex control strategies for different conditions in the presence of
channel and device level variations.

The problem of energy efficiency and channel conditions is conventionally
controlled by adaptive modulation and coding [28], making it harder for the RF
front-end to adapt to any changes in the channel. Integrating tunable RFICs
into spectrum-agile wireless networks can allow to self-optimize RF circuit
parameters to produce a desired output signal within the optimum power budget
based on existing channel conditions. [29] describes a channel-adaptive RX
design with process variation tolerance. Furthermore, a neural network based
self-learning RF system has been demonstrated in [30], which is able to reduce
power consumption of wireless transceiver systems by dynamically tuning the
circuit components while monitoring the effects of real-time wireless channel
conditions and the fabrication process variations to produce a desired BER and
threshold EVM. This is achieved with an on-chip look-up table that requires to
be updated based on expected channel conditions.

The difficulty of simulating the entire system is a major impediment to verify
the merits of the integrated hardware-software based wireless system. This
work aims to design and validate a joint simulation platform that can address
the inter-dependencies of circuit parameters in the RF front-end to develop
self-optimized wide-range reconfigurable RX architecture together with
wireless network that is capable of changing communication parameters.
Furthermore, this approach to incorporate and verify system-level performance-
driven tuning features using reconfigurable RFIC blocks is especially
compelling to enhance resilience to sudden changes in the environment and
accordingly optimize to achieve performance targets with optimum power
consumption. Hence, the presented simulation framework is expected to ease the
adaptive design and optimization of closed-loop self-supervised Internet of
Self-adaptive Things with different modulated signals, wireless channel models
and adaptive RFIC designs together with circuit level non-idealities.

##  III Optimization Framework Overview

We consider a scenario as depicted in Fig. 1, where we model multipath effects
and dynamic fading along with arbitrary modulated signal packets during the
design of robust RFIC adaptability to receive and process information. Here,
the waveform generation allows to change the modeled Physical-layer (PHY)
parameters (such as modulation scheme, power carried by the spectrum
components, RF sampling frequency, channel coding and bandwidth). Currently,
only the Signal-to-Noise-Ratio (SNR) is used as a primary indicator of
variable channel conditions. The variable SNR-based model encompasses several
channel impairments such as additive white Gaussian noise (AWGN), multi-path
fading, variable distance between transmitter (TX) and RX, and path-loss among
other interference as indicated in Fig. 1. The sensitivity requirement of the
RF front-end strongly depends on the SNR. [31] includes a description of
typical SNR values for channels, where the comparative results with different
modulation schemes provide insights into the expected SNR values for various
channel conditions. It is also envisioned that future spectrum-agile TXs will
lead to more variations of interference levels, SNR values, and modulation
schemes.

As depicted in Fig. 1, the baseband waveform is processed to infer parameters
associated with system-level performance such as BER, SER, EVM, modulation
error ratio (MER) and packet error rate (PER). This approach is based on the
goal to develop algorithms for accurate data-driven optimization of RFICs
based on system performance. Next, we present an architecture with a
reconfigurable RF front-end circuit that is evaluated through the simulation
framework from this work.

Figure 2: Co-design simulation platform for wireless systems with dynamically
reconfigurable RF front-end circuits.

##  IV Design and Modeling of RF Front-end Circuits with Dynamic
Reconfigurability

###  IV-A Flexible RF Front-end Architecture

Analog RF front-end reconfigurability enhances co-existence and spectrum
sharing in crowded environments [32], as well as allows to vary parameters
such as data rates on demand [33]. As shown in Fig. 2, the reconfigurable RF
RX front-end in this work consists of a digitally programmable LNA circuit
designed at the transistor-level with tunable bias current, a behavioral model
of a direct down-conversion in-phase/quadrature (I/Q) mixer stage, and two
ideal low-pass filter (LPF) models in the I and Q paths. Circuit-level
simulations with device-level non-idealities can capture impacts of frequency
response limitations, inter-modulation products, thermal and flicker noises,
parasitic capacitances/resistances, and higher-order non-linearities of the
transistors, that allows block-level specifications assessment such as gain,
noise figure (NF), input third-order inter-modulation intercept point (IIP3)
and impedance matching conditions. At the same time, the ability to include
some behavioral models of circuit blocks aids the early design and system-
level verification phase. Most importantly, the accurate transient simulations
allow to account for the impacts of circuit-level imperfections to assess
system-level metrics with changing environment.

To overcome existing inflexible wireless standards, inefficient spectrum use
and potential security threats in the wireless network, the flexible
adaptation of PHY parameters of the signal proves an effective and long-
standing solution. The work in [34] demonstrated if TXs were allowed to
dynamically switch PHY parameters such as carrier frequency and symbol
modulation, the TXs would become less jamming-prone and achieve more efficient
spectrum occupation. To give an example, Fig. 2 shows the selected test signal
with a format that corresponds to the Zigbee PHY packet structure. This signal
with any modeled channel impairments has been applied as input signal during
circuit and behavioral simulations of the RF front-end. The frame starts with
a known preamble for synchronization, which exhibits high auto-correlation and
low cross-correlation features. The preamble is followed by a start-of-frame
delimiter that marks the beginning of the header. The header consists of the
frame length in bytes and the modulation code associated with the modulation
scheme used. A data checksum (CSC) is attached to the header and data parts
respectively, such that erroneous frames can be detected and discarded. The
data part of the frame can support a MAC Protocol Data Unit (MPDU) with a size
of up to $2^{8}-1$ bytes.

As depicted in Fig. 2, the emulated transmitted packets with added channel
noise and imperfections are transferred to the signal source of a circuit
simulator (Cadence Spectre) for the RF front-end simulation. The model-based
design simulator (Matlab) is capable of saving the raw data in the comma-
separated values (CVS) file format, which has been incorporated into the
circuit simulator by the virtue of a piece-wise linear signal source. Thus,
the reconfigurability settings and biasing conditions of the RF front-end
blocks can be evaluated during the circuit design phase to optimize
characteristics such as noise levels, linearity, RX sensitivity and impedance
matching conditions. As mentioned earlier, the mathematical analysis of the
circuit simulator output provides system-level performance metrics (such as
BER, SER, MER, and EVM) for both circuit and system-level optimization with
modulated signals. Here, the BER is considered as one of the most significant
PHY performance indicators. Since not all inaccuracies lead to bit flips, EVM
is another appropriate measure to quantify the quality of the received signal
after processing in the RF front-end, which allows to capture important
channel and RX non-idealities [35, 36]. Various imperfections such as changing
channel conditions have impacts on the EVM since they can cause the received
constellation points to deviate from their original ideal locations.

###  IV-B Reconfigurable LNA

A 2.4 GHz single-ended cascode common-source LNA with inductive source
degeneration has been selected as an exemplary reconfigurable narrowband LNA
design, of which the bias current is tuned to control performance/power
tradeoffs as shown in Fig. 2. Since the LNA is the first block of the RF
front-end, its performance is particularly crucial when receiving noisy
packets at low power levels. The standalone LNA was designed in a standard
65nm CMOS technology, and simulated for bias currents ranging from 31.25
$\mu$A to 500 $\mu$A. By changing the bias current +/-20% to +/-50% from its
design point (125 $\mu$A), the circuit characteristics such as gain, IIP3, and
NF can be adjusted with a corresponding change of the power consumption that
is directly proportional to the bias current. Fig. 3 summarizes the gain, NF
and IIP3 of the LNA from transistor-level simulations.

Figure 3: Simulated LNA performance parameters vs. bias current.

###  IV-C Mixer and Baseband Signal Processing

To portray the capability of combining transistor-level schematic simulations
(i.e., the LNA) with behavioral blocks during transient circuit simulations to
assess system-level metrics under specified/changeable wireless channel
conditions, a direct down-conversion I/Q mixer has been modeled in VerilogA,
followed by two ideal LPFs (one in each RX path). The mixer model includes
variable gain, IIP3 and NF based on typical reconfigurable circuit parameters
of down-conversion mixers. In this proof-of-concept, the performance
assessment of the RF front-end was with QPSK-modulated signals and channel
impairments according to the signal generation in Fig. 2.

The channel model accounts for AWGN, path-loss and generic frequency-selective
multi-path fading that can introduce different path attenuation, delay, and
Doppler shift. The baseband demodulator after the LPF is also implemented
mathematically, where according to the decision boundaries defined by the
associated constellation, the I/Q samples are detected and compared against
the ground-truth I/Q samples to compute the desired system-level parameters.
Future work will be devoted to automatic parameter tuning parameters in the RF
front-end circuits based on the extracted system-level performance metrics to
optimize with varying spectrum conditions under specified power consumption
targets.

##  V Case Study:
Reconfigurable RF Front-end Simulation

Figure 4: Results from the simulation with generated QPSK packets: transient
RF front-end characteristics and reconstructed constellations. TABLE I:
System-level performance summary for the re-configurable RF front-end with LNA
bias tuning from simulations with a QPSK burst (4272 bits) having a power of
-100 dBm.

This section summarizes results from the use of our co-simulation framework
for the example RF front-end configuration described in the previous section.
The simulations were primarily carried out to evaluate the performance vs.
power trade-offs associated with the reconfigurable LNA design. As mentioned
in the previous section, the complete testbench used for the simulations
includes a VerilogA based mixer and ideal LPF. The mixer has been modeled with
flexible circuit parameters in which the gain, IIP3 and NF can be changed as
part of the design exploration. In this work, we have used an ideal behavioral
mixer model with a gain of 10 dB, IIP3 of 5 dBm and NF of 10dB for the proof-
of-concept simulations [37, 38, 39]. The upper left image in Fig. 4 displays
the generated QPSK-modulated signal with the packet structure defined in Fig.
2. A binary ground-truth message is randomly generated and up-converted to
produce the QPSK modulation with a center frequency of 2.4 GHz. The SNR of the
received signal was selected as 20 dB to emulate typical wireless network
conditions with channel impairments and distortion [40, 41]. The corresponding
waveform of the generated QPSK signal in Fig. 4 is corrupted by the channel
imperfections and noise, and fed to the RF front-end. The simulated I/Q
signals at the LPF outputs are shown on the right side of Fig. 4. These I/Q
signals are transferred for mathematical baseband processing using an envelope
detector and moving average filter (as displayed in Fig. 2) during model-based
simulations.

The demodulated I/Q samples are then compared and verified with the ground-
truth data to extract BER, SER, EVM and MER. The lower values of the BER and
SER relust from high accuracy of demodulation process. In this case, the BER
and SER are in the range of $10^{-5}$-$10^{-3}$ for all LNA bias current
conditions. On the other hand, the MER values are 18.9 dB and 11.2 dB for LNA
bias currents of 500 $\mu$A and 31.25 $\mu$A respectively. The proximity of
the MER value to the specified channel SNR (20 dB) is an indication of a noise
resilient system, which results from the LNA bias with high current (i.e.,
high power consumption) to achieve a low NF. Fig. 4 includes the extracted
constellations of the QPSK signal after the processing by the RF front-end
with the highest and lowest LNA bias currents, which also show the power vs.
performance tradeoff. Table I includes an overview of the system-level
performance for different LNA bias current settings. We have simulated two
QPSK packets (4272 bits) with randomly generated ground-truth data to evaluate
the high-level system performance with the co-design platform. It can be seen
from Table I that the BER, SER and EVM are considerably lower for LNA bias
currents in the 125 $\mu$A to 500 $\mu$A range. In addition, no packet errors
(PEs) occurred in the 125 $\mu$A to 500 $\mu$A bias current range. The
simulation results show robust adaptability, which will be realized with an
application-specific feedback control loop as depicted in Fig. 1. Depending on
the application-specific BER requirement and channel conditions, energy
consumption can be significantly reduced through the 62.5 $\mu$A and 31.25
$\mu$A bias current settings.

##  VI Next Steps: Beyond Traditional Wireless System and RFIC Integration

We foresee that the research presented in this paper will be the foundation
for the development of reconfigurable RXs with real-time self-optimization
capabilities. The work described in this article is the first step associated
with the co-design and verification of wireless systems and circuits that can
tolerate and adapt to interference conditions using novel RFIC optimization
methods with unprecedented design flexibility based on specified system-level
performance metrics. As depicted in Fig. 5, we anticipate that the presented
co-simulation and design verification framework will contribute to the
development of several novel features: (i) ML-based self-decisive CMOS RF
front-ends to adapt changing network conditions, (ii) development of hardware-
software prototypes based on joint integrated circuit simulations and wireless
data collection for enhanced modeling, (iii) several digitally-controlled
tuning knobs in each analog block as indicated in Fig. 1, and (iv) collection
of waveform datasets through the experimental testbenches to train ML
algorithms. (v) Once the ML algorithms are developed, the deep reinforcement
learning (DRL) agent will be trained to collect data, both experimentally and
synthetically utilizing the proposed framework. During the data collection,
one can deploy the optimal policy on FPGA-based platforms such as software-
defined radios (SDRs) [42, 43] to meet the challenging time constraints
involved, and to reduce the overall power consumption. The presented
simulation framework features tools to jointly optimize the power-efficiency
of digitally-controlled analog circuits and the computation resources to
implement adaptive ML-based control. Once the envisioned ML algorithm is
developed and the DRL agent is trained experimentally and synthetically
collected data using the proposed framework, one could deploy the optimal
policy on FPGA-based platforms such as SDRs [42] so as to meet the challenging
time constraints involved. To further minimize the power consumption
introduced by running the ML method, one can change the operational frequency
or even limit it to the times when the system-level performance drops below a
certain threshold or experiences a sudden change. The simulation results in
Section V show that the RF front-end in this case study has a sensitivity of
-100 dBm. We have presented a case study with a QPSK packet, but other
modulation schemes (e.g. ASK, BPSK) with different SNR values, data rate,
bandwidth, center frequencies can be employed. The reconfigurable RX design
and simulation approach is intended to facilitate the use of flexible wireless
system parameters by reconfiguring circuit parameters for the optimum
performance and spectrum sharing in real-time.

Figure 5: Envisioned ML-based real-time RF front-end optimization.

##  VII Conclusion

This article presented a new co-design simulation platform to evaluate trade-
offs between performance and power consumption by integrating the design and
modeling of wireless systems and reconfigurable RF front-end circuits
together. A main contribution of the integrated framework is to provide an
effective and long-lasting tool for the design of spectrum-agile RXs with
adaptive RFICs for dynamic optimizations under changing environmental
conditions. The execution of the joint summation taking into account combining
dynamic wireless channel model and reconfigurable RF front-end with circuit
level non-idealities accounts for incorporation of several modeling and design
software to validate the performance. The reconfigurability of the RF front-
end aims to reduce power consumption significantly based on application-
specific system-level performance targets. However, the primary goal is to
execute the seamless adaptation of the RF front-end by optimizing its circuit
parameters during real-time execution with spectrum-agile transmission to
produce desired end-to-end system level performance. The integration of
digital tuning capabilities in each of the analog blocks within RFICs will be
particularly useful for realizing future wireless system paradigms with an
unprecedented degree-of-freedom.

##  VIII Acknowledgement

This material is in part based upon work supported by the NSF under grant no.
ECCS-2146754, CCF-2218845, CNS-2134973, CNS-2120447, ECCS-2229472, and is
supported in part by funds from OUSD R&E, NIST, and industry partners as
specified in the Resilient & Intelligent NextG Systems (RINGS) program, as
well as by the Air Force Office of Scientific Research under contract number
FA9550-23-1-0261, and by the Office of Naval Research under award number
N00014-23-1-2221. The U.S. Government is authorized to reproduce and
distribute reprints for Governmental purposes notwithstanding any copyright
notation thereon. The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official
policies or endorsements of the NSF and the U.S. Government.

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# Martingale Decomposition and BSDE on Time Scales

Guofeng Tang  Zhongtai Securities Institute for Financial Studies
Shandong University
<EMAIL_ADDRESS>
###### Abstract

In this paper, we present martingale decomposition on time scales. We
establish the related backward stochastic dynamic equations on time scales
(this paper BS$\nabla$E for short, concerning $\nabla$-integral on time
scales) which unify backward stochastic differential equations and backward
stochastic difference equations. We prove the existence and uniqueness theorem
of BS$\nabla$E. This work can be considered as a unification and a
generalization of similar results in backward stochastic difference equations
and backward stochastic differential equations.

keywords: Time Scales, Martingale Decomposition on time scales, BSDE on time
scales

##  1 Introduction

In 1988, Stefan Hilger [21] introduced the calculus of measure chains to unify
continuous and discrete analysis. Since then, this topic has received much
attention, especially on the deterministic analysis [19], [20].

For the stochastic integral on time scales, Suman [29] tried to define
stochastic integral on time scales , but he just dealt with isolated time
scales. David Grow and Suman Sanyal [30] extended the stochastic integral for
Brownian motion to a general time scale. The existence and uniqueness of the
strong solutions of a certain class of stochastic dynamic equations
(S$\Delta$E) was proved. Wenqing Hu [22] provided an Itô’s formula with
respect to (w.r.t.) the Brownian Motion on time scales and demonstrated a
Girsanov’s change of measure formula in the case of general time scales. Jocad
and Shiryaev[23] showed that the discrete case actually reduced to a
particular case in the general continuous one. To that effect, they considered
a discrete stochastic basis
$\mathcal{B}=(\Omega,\mathcal{F},\mathbf{F}=(\mathcal{F}_{n})_{n\in\mathds{N}},\mathbf{P})$.
They associated to $\mathcal{B}$ a ”continuous ” stochastic basis
$\mathcal{B}^{\prime}$ as follows:
$\mathcal{B}^{\prime}=(\Omega,\mathcal{F}^{\prime}_{t},\mathbf{F}^{\prime}),~{}\text{with}~{}\mathcal{F}_{t}^{\prime}=\mathcal{F}_{n},~{}\text{for}~{}t\in[n,n+1)$,
and the discrete process $X_{n}$ on $\mathcal{B}$ by
$X_{t}^{\prime}=X_{n},t\in\left[n,n+1\right)$. Bohner [9] constructed a
stochastic integral and stochastic differential equations on general time
scales with the same method. These integrals are $\Delta$-integration on the
semi-open intervals of the form $\left[t_{i},t_{i+1}\right)$. Meanwhile, by
the requirement of the predictable integrand, it is easy to consider semi-open
intervals $\left(t_{i},t_{i+1}\right]$. Nguyen Huu Du and Nguyen Thanh Dieu
[26] defined a stochastic calculus on time scales (for the $\nabla$ case), in
which they presented the Doob-Meyer decomposition theorem for a submartingale
indexed by a time scale. They also defined the $\nabla$-stochastic
integration, Itô’s formula with respect to the square integral martingales. In
[27], they considered the $\nabla$-stochastic dynamic equation on time scales
(S$\nabla$E), in which they gave the conditions for the existence and
uniqueness of solutions, Markovian property of the solutions. For the Backward
Stochastic calculus on time scales, there are few results.

The theory of BSDE on continuous-time is a mature field, while the discrete
counterpart BSDE was rarely studied. For BSDE on continuous-time:

$\left\\{\begin{array}[]{ll}Y_{t}=Y_{T}+\int_{t}^{T}g(u,Y_{u},Z_{u})du-\int_{t}^{T}Z_{u}dW_{u},\quad
t\in\mathbf{R}_{+};\\\ Y_{T}=\xi.\\\ \end{array}\right.$ (1)

A solution of this equation, associated with the terminal value $\xi$ and
generator $g(\omega,t,y,z)$, is a couple of stochastic processes
$(Y_{t},Z_{t})_{t\in\mathbf{R}_{+}}$ which satisfies the equation. A general
nonlinear BSDE was first introduced by Pardoux and Peng [28].

Concerning Backward Stochastic Difference Equations, the formal work focus on
the order of convergence as a numerical scheme, rarely the discrete scheme
itself. Many algorithms for BSDE are based on random walk framework. This
equation has a unique solution since the driven martingale has the Predictable
Representation Property (PRP). Cohen [11] studied the Backward Stochastic
Difference Equations on spaces related to discrete-time, which is driven by
finite-state processes. The finite-state driven process has the predictable
representation property. This kind of difference equation:

$\left\\{\begin{array}[]{ll}Y_{t+1}-Y_{t}=-g(t,Y_{t},Z_{t})+Z_{t}(M_{t+1}-M_{t}),\quad
t\in\mathds{N};\\\ Y_{T}=\xi,\quad T\in\mathds{N}.\\\ \end{array}\right.$ (2)

has a unique solution $(Y_{t},Z_{t})_{t\in\mathds{N}}$. The typical example of
a model based on a finite probability space is the binomial model, also known
as the Cox-Ross-Rubinstein model in finance.

Another kind of Backward Stochastic Difference Equation is driven by increment
independent processes, such as i.i.d normal distributions. By switching from
the Brownian motion on continuous-time to discrete-time, we lose the
predictable representation property. It is well known that we need to include
in the formulation of the BSDE on discrete-time an additional orthogonal
martingale term.

$\left\\{\begin{array}[]{ll}Y_{t+1}-Y_{t}=-g(t,Y_{t},Z_{t})+Z_{t}(W_{t+1}-W_{t})+N_{T}-N_{t},\quad
t\in\mathds{N};\\\ Y_{T}=\xi.\\\ \end{array}\right.$ (3)

By the Galtchouk-Kunita-Watanabe Theorem, the solution of this kind equation
is a triple tuple $(Y,Z,N)$, where $N_{t}$ is an orthogonal martingale to the
integrals w.r.t the driven process $W_{t}$. Bielecki [2] first studied the
existence and uniqueness of the solutions by the Galtchouk-Kunita-Watanabe
decomposition of discrete BSDE. In [12], Cohen considered BSDE in discrete
time with infinitely many states, the driven process was a martingale
difference process.

For the general difference equations, there are many types: implicit or
explicit.

$\left\\{\begin{array}[]{ll}Y_{k+1}-Y_{k}=-g(\omega,k,Y_{k},Z_{k})+Z_{k}(W_{k+1}-W_{k})+N_{k+1}-N_{k}\\\
Y_{T}=\xi\\\ \end{array}\right.$ (4)
$\left\\{\begin{array}[]{ll}Y_{k+1}-Y_{k}=-g(\omega,k+1,Y_{k+1},Z_{k})+Z_{k}(W_{k+1}-W_{k})+N_{k+1}-N_{k}\\\
Y_{T}=\xi\\\ \end{array}\right.$ (5)
$\left\\{\begin{array}[]{ll}Y_{T}-Y_{T-1}=-h(\omega,Y_{T})+Z_{T-1}(W_{T}-W_{T-1})+(N_{T}-N_{T-1})\\\
Y_{k+1}-Y_{k}=-g(\omega,k+1,Y_{k+1},Z_{k+1})+Z_{k}(W_{k+1}-W_{k})+(N_{k+1}-N_{k})\\\
Y_{T}=\xi\\\ \end{array}\right.$ (6)

This paper aims to establish the existence and uniqueness of solutions to
BSDEs on time scales (BS$\nabla$E for our case). The backward stochastic
dynamic equations w.r.t. Brownian motion on time scales are similar to
traditional BSDEs on continuous-time driven by general
c$\grave{a}$dl$\grave{a}$g martingales [10]. We consider the following
BS$\nabla$E:

$\left\\{\begin{array}[]{ll}Y_{t}=Y_{T}+\int_{t}^{T}g(u,Y(u-),Z(u))\nabla
u-\int_{t}^{T}Z(u)\nabla W_{u}+N_{T}-N_{t};\quad t\in\mathds{T},\\\
Y_{T}=\xi.\\\ \end{array}\right.$ (7)

The solution is given by $(Y,Z,N)$, where $N$ can not be constructed by the
integrals w.r.t. the Brownian motion on time scales. We assume the usual
conditions on BSDE: $\xi\in
L^{2}(\Omega,\mathcal{F},\mathbf{P};\mathbf{R}^{n})$, and $g$ satisfies the
Lipschitz condition. Under the conditions, we provide existence and uniqueness
results of Backward Stochastic Dynamic Equations on time scales.

The paper is organized as follows. In section 2, we provide basic definitions
regarding time scales. In section 3, we give some stochastic notations and
results on time scales, we state and prove the martingale decomposition
theorem on time scales. And finally, in section 4, we give the main results:
the existence and uniqueness solution of backward stochastic dynamic equations
on time scales.

##  2 Preliminaries

For the theory of calculus on time scales, we refer to the original work by
Higer [21]. In this section, we give basic definitions concerning the time
scales and present some applications. For more additional information on the
theory of time scales, we refer the reader to the monograph by Bohner and
Peterson [5].

A _time scale_ $\mathds{T}$ is a nonempty closed subset of the real numbers
$\mathbf{R}$. The distance between the points $t,s\in\mathds{T}$ is defined as
the normal distance on $\mathbf{R}:|t-s|$. For t$\in\mathds{T}$, we define the
_forward jump operator_ $\sigma$: $\mathds{T}\rightarrow\mathds{T}$ by
$\sigma(t)=\inf\\{s\in\mathds{T}:s>t\\}$, while the _backward jump operator_
$\rho$: $\mathds{T}\rightarrow\mathds{T}$ by
$\rho(t)=\sup\\{s\in\mathds{T}:s<t\\}$. Of course both $\sigma(t)$ and
$\rho(t)$ are in $\mathds{T}$ when $t\in\mathds{T}$. This is because we assume
that $\mathds{T}$ is a closed subset of $\mathbf{R}$. We say that $t$ is
_right-scattered_ (_left-scattered_ , _right-dense_ , _left-dense_), if
$\sigma(t)>t$ ($\rho(t)<t$, $\sigma(t)=t$, $\rho(t)=t$) hold.
$\mu(t)=\sigma(t)-t$ is called _graininess_ , $\nu(t)=t-\rho(t)$ is called
_backward graininess_. The set of right-scattered points of a time scale is at
most countably infinite. For $a,b\in\mathds{T}$ with $a\leq b$ we define the
closed interval in $\mathds{T}$ by $[a,b]_{\mathds{T}}=[t\in\mathds{T}:a\leq
t\leq b]$. Other types of intervals are defined similarly. We introduce the
set $\mathds{T}_{k}$ if $\mathds{T}$ has a right-scattered minimum $t_{2}$,
then $\mathds{T}_{k}=\mathds{T}-\\{t_{2}\\}$, otherwise
$\mathds{T}_{k}=\mathds{T}$. If $t\in\mathds{T}_{k}$, then the
$\nabla$-_derivative_ of $f$ at the point $t$ is defined to be the number
$f^{\nabla}(t)$(provided it exists) with the property that for each
$\epsilon>0$, there is a neighborhood $U$(in $\mathds{T}$)of $t$ such that
$|f(\rho(t))-f(s)-f^{\nabla}(t)[\rho(t)-s]|\leq\epsilon|\rho(t)-s|,\quad\forall
s\in U$. Now suppose that $f:\mathds{T}\rightarrow\mathbf{R}$. Continuity of
$f$ is defined in the usual manner. A function $f$ is called right-dense
continuous (rd-continuous) on $\mathds{T}$ iff it is continuous at every
right-dense point and the left-sided limit exists at every left-dense point.
It is similar to the notation of left-dense continuous (ld-continuous). Note
that, on the right-scattered points, $f(t)\neq f(t+)$ for continuous functions
on time scales. Denote $\lim_{\sigma(s)\uparrow t}f(s)$ by $f(t-)$ if this
limit exists, $\lim_{\rho(s)\downarrow t}f(s)$ by $f(t+)$. If $t$ is left-
scattered then $f(t-)=f(\rho(t))$, right-scattered then $f(t+)=f(\sigma(t))$.

Let $A$ be an increasing right-continuous function of finite variation defined
on $\mathds{T}$. We denote by $\mu^{A}_{\nabla}$ the Lebesgue $\nabla$-measure
associated with $A$. For any $\mu^{A}_{\nabla}$-measure function
$f:\mathds{T}\rightarrow\mathbf{R}$, we write $\int_{a}^{t}f_{s}\nabla A_{s}$
for the integral of $f$ with respect to the measure $\mu_{\nabla}^{A}$ on
$\left(0,t\right]$. It is seen that the function
$t\mapsto\int_{a}^{t}f_{s}\nabla A_{s}$ is c$\grave{a}$dl$\grave{a}$g. For
details, we can refer to [26].

###  2.1 Stochastic calculus on time scales

As time scales unify continuous and discrete analysis, we assume that the
readers are familiar with the concepts and basic properties of the stochastic
process in both discrete-time and continuous-time. We denote by
$\mathbf{R}^{k}$ the k-dimensional Euclidean space, equipped with the standard
inner product $(\cdot,\cdot)$, and the Euclidean norm $|\cdot|$. We also
denote by $\mathbf{R}^{k\times d}$ the collection of all $k\times d$ real
matrices, and for matrix $z=(z_{ij})_{k\times d}$. We denote
$z_{i}:=(z_{i1},\cdots,z_{id})^{T}$ and $|z|:=\sqrt{tr(zz^{T})}$, where
$z^{T}$ represents the transpose of $z$. In our study, we will be concerned
with bounded time scales in which $a=\inf\mathds{T}$ and $b=\sup\mathds{T}$
and $a,b\in\mathds{T}$ are both finite. For simplicity, we consider
$a=0,b=T>0$. By convention, we write $\rho(0)=0$. The basic definitions of
process on time scales are analogous to that of classical discrete-time
processes or continuous-time processes, such as
$\\{\mathcal{F}_{t}\\}_{t\in\mathds{T}}$-filtration,
$\mathcal{M}_{\mathds{T}}$-the space of uniformly integrable martingales,
$\mathcal{M}_{\mathds{T}}^{2}$-the space of square integrable martingales with
$E\\{\sup|M|_{t}^{2}\\}<\infty$. We add the subscript $\mathds{T}$ to spaces
on time scales. Most results of martingales also hold in the time scale cases.
We are interested in the distinctions between them. We assume that we are
working on a probability space
$(\Omega,\mathcal{F},\mathbf{F}=(\mathcal{F}_{t})_{t\in\mathds{T}},\mathbf{P})$
with the filtration $\\{\mathcal{F}_{t}\\}_{t\in\mathds{T}}$ satisfying the
usual conditions ($\mathcal{F}_{t}\neq\mathcal{F}_{t+}$ on right-scattered
points even it is continuous on time scales, since right-continuous has no
meaning at right-scattered points). For simplicity,
$\mathcal{F}_{T}=\bigvee_{t\in\mathds{T}}\mathcal{F}_{t}$.

We denote by $\mathcal{M}^{2}_{\mathds{T}}$ the space of square integrable
$\\{\mathcal{F}_{t}\\}_{t\in\mathds{T}}$-martingales on time scales and
consider $M\in\mathcal{M}^{2}_{\mathds{T}}$. Since $M^{2}$ is a submartingale,
following the Doob-Meryer decomposition theorem on time scales [26], there
exists uniquely a natural increasing process $(\langle
M\rangle_{t})_{t\in\mathds{T}}$, such that $M^{2}-\langle M\rangle$ is an
$\mathbf{F}$-martingale. The natural increasing process $\langle M\rangle$ is
called characteristic of the martingale $M$.

We define the quadratic co-variation $[M,N]$ of two processes $M,N$ similar to
[26], Definition 3.13. If $N=M$ we write $[M]_{t}$ for $[M,M]_{t}$ and call it
the quadratic variation of $M$. For partitions $\\{t_{i}\\}$ of
$[s,t]_{\mathds{T}}$ with $\max_{i}(\rho(t_{i})-t_{i-1})\leq 2^{-n}$, we have

$E[(M_{t}-M_{s})^{2}|\mathcal{F}_{s}]=E[\sum_{i}(M_{t_{i}}-M_{t_{i-1}})^{2}|\mathcal{F}_{s}]\rightarrow
E[[M]_{t}-[M]_{s}|\mathcal{F}_{s}].$

It means, $M^{2}-[M]$ is an $\mathbf{F}$-martingale, which implies that
$[M]-\langle M\rangle$ is also a $\mathbf{F}$-martingale.

As the one-dimensional case was given in [17], the multi-dimension Brownian
motion can be constructed as the classical product space method [24]. Now we
give a result about multi-dimension Brownian motion on time scales.

######  Lemma 2.1.

Let
$\\{W_{t}=(W_{t}^{1}.W_{t}^{2},\cdots,W_{t}^{d}),\mathcal{F}_{t};t\in\mathds{T}\\}$
be a d-dimensional Brownian motion. The processes

$M_{t}^{i}=W_{t}^{i}-W_{0}^{i},~{}\mathcal{F}_{t};~{}t\in\mathds{T},~{}1\leq
i\leq d,$

are continuous, square-integrable martingales, with $\langle
M^{i},M^{j}\rangle=t\delta_{ij};1\leq i,j\leq d$.
$[M^{i},M^{j}]=\delta_{ij}\\{\lambda([t_{0},t]\cap\mathds{T})+\sum_{t_{0}\leq
a_{n}<b_{n}\leq t}(W_{b_{n}}-W_{a_{n}})^{2}\\},\forall t\in\mathds{T}$, where
$\lambda$ denotes the classical Lebesgue measure and
$\cup_{n=1}^{\infty}(a_{n},b_{n})=\left[t_{0},\infty\right)\setminus\mathds{T}$
is the expression for the open subset
$\left[t_{0},\infty\right)\setminus\mathds{T}$ of $\mathbf{R}$ as the
countable of disjoint open intervals[18]. Furthermore, the vector of
martingales $M=(M^{1},\cdots,M^{d})$ is independent of $\mathcal{F}_{0}$.

######  Remark 2.2.

The Levy martingale characterization of Brownian motion failed on time scales,
that’s a continuous martingale with $\langle X^{i},X^{j}\rangle=t\delta_{ij}$,
cannot be a Brownian motion.

Our stochastic integral on time scales is based on Nguyen Huu Du & Nguyen
Thanh Dieu [26], in which they established the stochastic $\nabla$-integral
w.r.t. square integral martingales, and extended to special semimartingales
[27]. Consider the integral with respect to Brownian motion $W$ on time scales
(square integral on time scales): let
$\mathcal{L}_{\mathds{T}}^{2}(\left(0,T\right];W)$ be the space of all real-
valued, predictable processes $\phi=\\{\phi_{t}\\}_{t\in\mathds{T}}$
satisfying $\|\phi\|^{2}_{t,W}=E\int_{0}^{t}|\phi_{s}|^{2}\nabla\langle
W\rangle_{s}=E\int_{0}^{t}|\phi_{s}|^{2}\nabla s<\infty$. By using [[26],
Definition 3.6], we can define the integral: $I_{t}(X)=\int_{0}^{t}X_{s}\nabla
W_{s},\quad X\in\mathcal{L}_{\mathds{T}}^{2}(\left(0,T\right];W)$. The space
$\mathcal{L}_{\mathds{T}}^{2}(\left(0,T\right];W)$ is actually the $L^{2}$
space under the measure given by $\nabla\langle W\rangle\times dP$. Now we
define the Multi-dimension Stochastic Integral:

######  Definition 2.3.

Let $\\{W_{t},t\in\mathds{T}\\}$ be a $d$-dimension Brownian motion on time
scales, $\\{X_{t},t\in\mathds{T}\\}$ is $k\times d$ matrix process, $\forall
1\leq i\leq n,1\leq j\leq d$,
$X^{i,j}\in\mathcal{L}^{2}_{\mathds{T}}(\left(0,T\right];W^{j})$. Define:

$I_{t,n}(X)=\int_{0}^{t}X_{s}\nabla
W_{s}=\int_{0}^{t}\left(\begin{array}[]{lll}X_{11}&\cdots&X_{1d}\\\
\vdots&&\vdots\\\ X_{n1}&\cdots&X_{nd}\\\
\end{array}\right)\left(\begin{array}[]{l}\nabla W_{s}^{1}\\\ \vdots\\\ \nabla
W_{s}^{d}\\\ \end{array}\right).$ (8)

We denote the multi-dimensional stochastic integral $I_{t,k}(X)$ for $X$,
where $X$ is $(\mathcal{F}_{t})_{t\in\mathds{T}}$-predictable,
$\mathbf{R}^{k\times d}$-valued processes such that
$E\int_{0}^{T}|X_{s}|^{2}\nabla t<\infty$, denote the set by
$\mathcal{L}^{2}_{\mathds{T}}((\left(0,T\right];\mathbf{R}^{k\times d})$
(short for $\mathcal{L}^{2}_{\mathds{T},k\times d}$). We denote the integral
also by $I_{t}(X)$ for short. The i-th component of $I_{t}(X)$ is:

$I_{t}^{i}(X)=\sum_{j=1}^{d}\int_{0}^{t}X_{s}^{ij}\nabla W_{s}^{j},\quad 1\leq
i\leq k,\quad t\in\mathds{T}.$

Clearly that $I_{t}(X)$ belongs to $\mathcal{M}^{2}_{\mathds{T},k}$, the
$\mathbf{R}^{k}$-valued square-integral martingale on time scales. Therefore
we also introduce

  * •

$\mathcal{M}_{\mathds{T},k}^{2,c}$-the continuous $\mathbf{R}^{k}$-valued
square-integral martingale processes on time scale with
$M_{0}=0\quad\mathbf{P}-a.s.$,

  * •

$\mathcal{M}_{\mathds{T},k}^{2,*}$-the subset of
$\mathcal{M}_{\mathds{T},k}^{2,c}$, such that for each
$M\in\mathcal{M}_{\mathds{T},k}^{2,*}$, there exists
$X\in\mathcal{L}_{\mathds{T},k\times d}^{2}$, and $M_{t}=I_{t}(X)$

for any integer $k$. We will study properties of these spaces next section.
For more properties about the stochastic integral, readers could see [26].

######  Remark 2.4.

Suman [30] also defined a stochastic integral on time scale, the
$\Delta$-integral. For the stochastic integral, the integrands are actually
predictable in order to establish Itô type integral. On this occasion, the
stochastic integral has martingale property. (For the discrete case, the
martingale transform is the discrete-time version of the stochastic integral
with respect to a martingale.)

##  3 Martingale decomposition theorem on time scales

The fundamental tool on time scales analysis is the countable dense subset.
The countable dense subset will play a role analogous to the dyadic rational
numbers that played in the classical analysis from discrete-time to
continuous-time. For any $\delta>0$, consider a partition of
$[0,T]_{\mathds{T}}$ inductively by letting $t_{0}=0$ and for $i=1,2,\cdots$,
set:

$t_{i}=\left\\{\begin{array}[]{lll}\sup B_{i}&if~{}B_{i}\neq\emptyset\\\
\sigma(t_{i-1})&if~{}B_{i}=\emptyset\\\ \end{array}\right.\quad where\quad
B_{i}=(t_{i-1},t_{i-1}+\delta]_{\mathds{T}}\bigcap[0,T]_{\mathds{T}}.$ (9)

The partition is given in [19], [26]. On time scales, the interval
$|t_{i+1}-t_{i}|$ will not converge to zero. A more specific result was given
by David Grow [17]. Now we provide the optional sampling theorem on time
scales to show the basic analysis method on general time scales.

######  Lemma 3.1 (Optional Sampling Theorem on Time Scales).

If $X$ is a right-continuous martingale (submartingale) on bounded time scales
$\mathds{T}$ with a last element $X_{T}$ and $S_{1},S_{2}$ are two bounded
stopping times with $S_{1}\leq S_{2}$ on $\mathds{T}$, then

$E[X_{S_{2}}|\mathcal{F}_{S_{1}}]=X_{S_{1}}(\geq
X_{S_{1}})\quad\mathbf{P}-a.s..$

###### Proof.

Let $[0,T]_{\mathds{T}}$ be a time scale, and let
$\Pi_{n}=\\{t_{0},t_{1},\cdots,t_{n}\\}\subseteq[0,T]_{\mathds{T}}$ be a
partition (as the same in [19]) of $\mathds{T}$, where
$0=t_{0}<t_{1}<t_{2}<\cdots<t_{n}=T$. Consider the sequence of random times

$S_{1}^{n}(\omega;P_{n})=\rho(t_{i+1})\quad if\quad t_{i}\leq
S_{1}(\omega)<t_{i+1},$ (10)

and the similarly defined sequences $\\{S_{2}^{n}\\}$. These are stopping
times. For every fixed integer $n\geq 1$, both $S_{1}^{n}$ and $S_{2}^{n}$
take on a countable number of values and we also have $S_{1}^{n}\leq
S_{2}^{n}$. Therefore, by the discrete optional sampling theorem we have
$\int_{A}X_{S_{1}^{n}}dP\leq\int_{A}X_{S_{2}^{n}}dP$ for every
$A\in\mathcal{F}_{S_{1}^{n}}$. $S_{1}\leq S_{1}^{n}$ implies
$\mathcal{F}_{S_{1}}\subset\mathcal{F}_{S_{1}^{n}}$, the preceding inequality
also holds for every $A\in\mathcal{F}_{S_{1}}$.

The discrete martingale results show that the sequence of random variables
$\\{X_{S_{1}^{n}}\\}$ is uniformly integrable, and the same is of course true
for $\\{X_{S_{2}^{n}}\\}$.
$X_{S_{2}}=\lim_{n\rightarrow\infty}X_{S_{2}^{n}}(\omega)$ and
$X_{S_{1}}=\lim_{n\rightarrow\infty}X_{S_{1}^{n}}(\omega)$ hold for a.e.
$\omega\in\Omega$. It follows from uniform integrability that $X_{S_{1}}$,
$X_{S_{2}}$ are integrable, and that
$\int_{A}X_{S_{1}}dP\leq\int_{A}X_{S_{2}}dP$ holds for every
$A\in\mathcal{F}_{S_{1}}$. ∎

Via the identity $M_{t}=E[M_{T}|\mathcal{F}_{t}]$, each
$M\in\mathcal{M}_{\mathds{T}}^{2}$ can be identified with its terminal value
$M_{T}\in L^{2}(\Omega,\mathcal{F}_{T},\mathbf{P})$ (in general the terminal
variables can be extended $M_{\infty}$ if exists).
$\mathcal{M}_{\mathds{T}}^{2}$ becomes a Hilbert space isomorphic to
$L^{2}(\Omega,\mathcal{F}_{T},\mathbf{P})$ , if endowed with the inner product

$(M,N)_{\mathcal{M}^{2}_{\mathds{T}}}:=E[M_{T}N_{T}],\quad\|M\|_{\mathcal{M}^{2}_{\mathds{T}}}=\|M_{T}\|_{L^{2}}.$

Indeed, if $(M^{n})$ is a Cauchy sequence for
$\|\cdot\|_{\mathcal{M}^{2}_{\mathds{T}}}$, then the sequence $(M^{n}_{T})$ is
Cauchy in $L^{2}(\Omega,\mathcal{F}_{T},\mathbf{P})$ and so goes to a limit
$M_{T}$ in this space; then if $M$ is the martingale with terminal variable
$M_{T}$, it belongs to $\mathcal{M}^{2}_{\mathds{T}}$ and
$\|M^{n}-M\|\rightarrow 0$. The set of all continuous elements of on time
scales $\mathcal{M}^{2}_{\mathds{T}}$ denoted by
$\mathcal{M}^{2,c}_{\mathds{T}}$, is a closed subspace of the Hilbert space
$\mathcal{M}^{2}_{\mathds{T}}$.

In this part, we study some properties of the spaces
$\mathcal{M}^{2}_{\mathds{T},k}$ on time scales. We may define a measure
$\mu_{p}$ on
$(\mathds{T}\times\Omega,\mathcal{B}(\mathds{T})\bigotimes\mathcal{F}_{\rho(t)})$
by

$\mu_{p}(A)=E(\int_{\mathds{T}}1_{A}(s,\omega)\nabla\langle W\rangle_{s}).$

We recall $\mathcal{L}_{\mathds{T},k\times d}^{2}$ the space of all
$\mathbf{R}^{k\times d}$-valued, predictable processes
$\phi=\\{\phi_{t}\\}_{t\in\mathds{T}}$ satisfying

$\|\phi\|^{2}_{t,W}=E\int_{0}^{t}|\phi_{s}|^{2}\nabla s<\infty.$

Similar to Lemma 2.2 chapter 3 in Karatzas’s book [24], we have

######  Lemma 3.2.

The $\mathcal{L}^{2}_{\mathds{T},k\times d}$ space is a closed space with the
norm $\|\cdot\|_{T,W}$.

###### Proof.

We define a Hilbert space

$\mathcal{H}_{T}=L^{2}(\mathds{T}\times\Omega,\mathcal{B}(\mathds{T})\otimes\mathcal{F}_{\rho(T)},\mu_{p})$

Obviously, $\mathcal{L}^{2}_{\mathds{T},k\times d}$ is a subspace of
$\mathcal{H}_{T}$. Now we prove that it is closed. Suppose that
$\\{X^{m}\\}_{m=1}^{\infty}$ is a convergent sequence in
$\mathcal{L}^{2}_{\mathds{T},k\times d}$ with limit $X\in\mathcal{H}_{T}$.
Thus the sequence has a convergent subsequence which converges almost surely
under $\mu_{p}$, also denoted by $\\{X^{m}\\}_{m=1}^{\infty}$. Therefore

$\mu_{p}\\{(t,\omega):\lim_{n\rightarrow\infty}X_{t}^{m}(\omega)\neq
X_{t}(\omega);\quad t\in\mathds{T}\\}=0,$

$X\in\mathcal{H}_{T}$, thus $X$ is
$\mathcal{B}(\mathds{T})\otimes\mathcal{F}_{\rho(T)}$-measurable.

Restricted on $[0,t]_{\mathds{T}}$ for $0<t\leq T$, repeating the above
procedure, by the uniqueness of convergence, we can get that $X$ is
$\mathcal{B}([0,t]_{\mathds{T}})\otimes\mathcal{F}_{\rho(t)}$-measurable.
Therefore $X$ is predictable and belongs to
$\mathcal{L}^{2}_{\mathds{T},k\times d}$. The proof is complete. ∎

We can define an inner product on $\mathcal{L}_{\mathds{T},k\times d}^{2}$ by
$(X,Y)_{1}=E\int_{0}^{t}X_{s}Y_{s}\nabla s$. Recall the inner product on
$L^{2}(\Omega,\mathcal{F}_{T},\mathbf{P};\mathbf{R}^{k})$ (simply denoted by
$L^{2}_{k}(\mathcal{F}_{T})$), $(I_{1},I_{2})_{2}=E[I_{1}I_{2}]$. Now we
consider the mapping $X\mapsto I_{T}(X)$ from $\mathcal{L}_{\mathds{T},k\times
d}^{2}$ to $L^{2}_{k}(\mathcal{F}_{T})$. This mapping preserves inner
products:

$(X,Y)_{1}=E\int_{0}^{T}X_{t}Y_{t}\nabla
t=E[I_{T}(X)I_{T}(Y)]=(I_{T}(X),I_{T}(Y))_{2}.$

Denote
$\mathcal{R}_{k}(\mathcal{F}_{T})\triangleq\\{I_{T}(X);X\in\mathcal{L}_{\mathds{T},k\times
d}^{2}\\}$. Since $\mathcal{L}_{\mathds{T},k\times d}^{2}$ is closed, any
convergent sequence in $\mathcal{R}_{k}(\mathcal{F}_{T})$ is also Cauchy, its
preimage sequence in $\mathcal{L}_{\mathds{T},k\times d}^{2}$ must have a
limit in $\mathcal{L}_{\mathds{T},k\times d}^{2}$. It follows that
$\mathcal{R}_{k}(\mathcal{F}_{T})$ is closed in $L^{2}_{k}(\mathcal{F}_{T})$.
Let us denote by $\mathcal{M}^{2,*}_{\mathds{T}}$ the subset of
$\mathcal{M}^{2,c}_{\mathds{T},k}$ which consists of stochastic integrals

$I_{t}(X)=\int_{0}^{t}X_{s}\nabla W_{s};\quad 0\leq t\leq T,\quad
t\in\mathds{T},$

of processes $X\in\mathcal{L}_{\mathds{T},k\times d}^{2}$:

$\mathcal{M}^{2,*}_{\mathds{T},k}\triangleq\\{I(X);X\in\mathcal{L}_{\mathds{T},k\times
d}^{2}\\}\subseteq\mathcal{M}^{2,c}_{\mathds{T},k}\subseteq\mathcal{M}^{2}_{\mathds{T},k}.$
(11)

Now we state the ”fundamental decomposition theorem” for the martingales w.r.t
Brownian motion on time scales.

######  Theorem 3.3.

For every $M\in\mathcal{M}_{\mathds{T},k}^{2}$, with
$M_{0}=0,~{}\mathbf{P}-a.s.$, we have the decomposition

$M_{t}=I_{t}(X)+N_{t},\quad\forall t\in\mathds{T},$

where $X\in\mathcal{L}^{2}_{\mathds{T},k\times d}$,
$I_{t}(X)\in\mathcal{M}_{\mathds{T},k}^{2,*},~{}N\in\mathcal{M}_{\mathds{T},k}^{2}$
with $N_{0}=0$ and $N$ is orthogonal to every element of
$\mathcal{M}_{\mathds{T},k}^{2,*}$.

###### Proof.

Uniqueness: Suppose that there exists a process
$Y\in\mathcal{L}_{\mathds{T},k\times d}^{2}$ such that $M_{t}=I_{t}(Y)+N_{t}$,
where $N\in\mathcal{M}_{\mathds{T},k}^{2}$ has the property

$\langle I(X),N\rangle_{t}=0,\quad\forall X\in\mathcal{L}_{\mathds{T},k\times
d}^{2}.$

Such a decomposition is unique (up to indistinguishability); indeed, if we
have $M=I(Y^{\prime})+N^{\prime}=I(Y^{\prime\prime})+N^{\prime\prime}$ with
$Y^{\prime},Y^{\prime\prime}\in\mathcal{L}_{\mathds{T},k\times d}^{2}$ and
both $N^{\prime}$ and $N^{\prime\prime}$ satisfy the property, then

$Z\triangleq N^{\prime\prime}-N^{\prime}=I(Y^{\prime}-Y^{\prime\prime})$

is in $\mathcal{M}_{\mathds{T},k}^{2}$ with $Z_{0}=0$ and $\langle
Z\rangle=\langle Z,I(Y^{\prime}-Y^{\prime\prime})\rangle=0$. Then
$P[Z_{t}=0,\forall t\in\mathds{T}]=1$ from the Optional Sampling Theorem.
(Similar results as Problem1.5.12 in Karatzas[24], applying optional sampling
theorem to the martingale $Z^{2}-\langle Z\rangle$.) The decomposition is
unique up to indistinguishability.

Existence:

Since $\mathcal{R}_{k}(\mathcal{F}_{T})$ is a closed subspace of
$L^{2}_{k}(\mathcal{F}_{T})$, we can denote its orthogonal complement by
$\mathcal{R}_{k}^{\perp}(\mathcal{F}_{T})$. The random variable $M_{T}$ is in
$L^{2}_{k}(\mathcal{F}_{T})$, so it admits the decomposition

$M_{T}=I_{T}(Y)+N_{T},$ (12)

where $Y\in\mathcal{L}_{\mathds{T},k\times d}^{2}$ and $N_{T}\in
L^{2}_{k}(\mathcal{F}_{T})$ satisfies

$E\\{N_{T}I_{T}(X)\\}=0;\quad\forall X\in\mathcal{L}_{\mathds{T},k\times
d}^{2}.$

We construct a martingale through $N_{T}$ by $N_{t}=E(N_{T}|\mathcal{F}_{t})$.
Obviously $N\in\mathcal{M}_{\mathds{T}}^{2}$. Taking conditional expectation
under $\mathcal{F}_{t}$ on $M_{T}$, we obtain

$M_{t}=I_{t}(Y)+N_{t}.$

It remains to show that $N$ is orthogonal to every square-integrable
martingale of the form $\\{I(X);X\in\mathcal{L}_{\mathds{T},k\times
d}^{2}\\}$, or equivalently, that
$\\{N_{t}I_{t}(X),\mathcal{F}_{t}\\}_{t\in\mathds{T}}$ is a martingale. Note
that each martingale has a right continuous modification. So now we suppose
that $N$ is right continuous.

According to lemma 3.1, we only need to prove that
$E[N_{S}I_{S}(X)]=E[N_{0}I_{0}(X)]=0$ holds for every stoping time $S$ of the
filtration $\\{\mathcal{F}_{t}\\}_{t\in\mathds{T}}$, with $S<T$ (since
$I_{0}(X)=0$). The integral has $I_{S}(X)=I_{T}(\tilde{X})$, where
$\tilde{X}_{t}(\omega)=X_{t}(\omega)1_{\\{t\leq S(\omega)\\}}$ is a process in
$\mathcal{L}_{\mathds{T},k\times d}^{2}$. Therefore, by the Optional Sampling
Theorem, we have

$E[N_{S}I_{S}(X)]=E[E(N_{T}|\mathcal{F}_{S})I_{S}(X)]=E[N_{T}I_{T}(\tilde{X})]=0.$

The proof is complete. ∎

######  Remark 3.4.

  1. 1.

Note $f(t)\neq f(t-)$ at left-scattered points for continuous function on time
scales, $N\in\mathcal{M}_{\mathds{T},k}^{2,c}$ is continuous, but $\nabla
N_{t}\neq 0$ at left-scattered points.

  2. 2.

Let $\mathds{T}=\mathbf{R},k=1$ (Karatzas [24], Proposition 4.14 one-dimension
decomposition) $\mathcal{M}_{\mathbf{R}}^{2,c}$ and
$\mathbf{M}_{\mathbf{R}}^{2,*}$ are actually coincide, the component $N_{t}$
in the decomposition is actually $\nabla N_{t}=0$. The predictable process
space is isomorphic to the adapted process space [14].

  3. 3.

For Brownian motion on general time scales, even on the augmentation
filtration of the filtration generated by $W$,
$\mathcal{M}_{\mathds{T},k}^{2,c}$ and $\mathcal{M}_{\mathds{T},k}^{2,*}$ do
not coincide, see the following example.

  4. 4.

Let $\mathds{T}=\mathds{N}$ (Follmer, Hans, [16]Theorem 10.18), it is the
discrete time version of the Kunita-Watanabe decomposition w.r.t a sequence of
normal distribution random variables.

######  Example 3.5.

For a martingale $B_{t}^{2}-t$ on time scales
$\mathds{T}=[0,1]\cup\\{3,4,5\\}$, we have

$B_{t}^{2}-t=(B_{t}-B_{s})^{2}-(t-s)+2B_{s}(B_{t}-B_{s})+B_{s}^{2}-s,$

so for $(s,t)_{\mathds{T}}\bigcap\mathbf{R}=\emptyset$,

  * •

$N_{t}=(B_{t}-B_{s})^{2}-(t-s)$ is the orthogonal martingale part w.r.t.
Brownian interal.

  * •

$I_{t}=2B_{s}(B_{t}-B_{s})$ is the Itô integral part w.r.t Brownian motion at
left-scattered point. In this case $N_{t}$ is orthogonal to $I_{t}$.

##  4 BS$\nabla$E driven by Brownian Motion on time scales

Denote $(\Omega,\mathcal{F},\mathbf{P})$ be a probability space equipped with
a complete filtration $\mathcal{F}=(\mathcal{F}_{t})_{t\in\mathds{T}}$
generated by a d-dimensional Brownian Motion $W_{t}$ on time scales, and
augmented by all the $\mathbf{P}$-null sets in $\mathcal{F}$.
$E[X_{t}|\mathcal{F}_{s}]$ or $E^{\mathcal{F}_{s}}$ denotes the conditional
expectation with respect to the filtration $\mathcal{F}_{s}$.

For simplicity, we consider the general BSDE on time scales

$\left\\{\begin{array}[]{ll}Y_{t}=Y_{T}+\int_{t}^{T}g(u,Y_{u-},Z_{u})\nabla
u-\int_{t}^{T}Z_{u}\nabla W_{u}-(N_{T}-N_{t}),\\\ Y_{T}=\xi.\\\
\end{array}\right.$ (13)

We call $g$ the driver of the BSDE on time scales and the pair $(\xi,g)$ the
data of the BSDE on time scales.

  * •

$L^{2}_{k}(\mathcal{F}_{t})$, short for
$L^{2}(\Omega,\mathcal{F}_{t},\mathbf{P};R^{k})$, the space of
$\mathbf{R}^{k}$-valued random vectors $\xi$ that are
$\mathcal{F}_{t}$-measurable and satisfy $E[\|\xi\|^{2}]<\infty$,

  * •

$\mathcal{S}^{2}_{\mathds{T}}$, short for
$\mathcal{S}^{2}_{\mathds{T}}(0,T;\mathbf{R}^{k})$, the set of
$\mathbf{R}^{k}$-valued, adapted and continuous processes
$(\phi_{t})_{t\in[0,T]_{\mathds{T}}}$ such that
$E[sup_{t\in[0,T]_{\mathds{T}}}|\phi_{t}|^{2}]<\infty,$

  * •

$M^{2}_{\mathds{T}}$, short for $M^{2}_{\mathds{T}}(0,T;\mathbf{R}^{k\times
d})$, the set of $\mathbf{R}^{k\times d}$-valued,
$\mathcal{F}_{t}$-progressively measurable, predictable processes
$(\phi_{t})_{t\in[0,T]_{\mathds{T}}}$ such that

$E\\{\int_{0}^{T}|\phi_{t}|^{2}\nabla t\\}<\infty,$

  * •

$\mathcal{M}_{\mathds{T},k}^{2,\perp}$-subset of
$\mathcal{M}_{\mathds{T},k}^{2,c}$, the set of all the $k$-dimensional
martingales, such that each martingale is orthogonal to that in
$\mathcal{M}_{\mathds{T},k}^{2,*}$,

for any integer $k$.

Obviously, $\mathcal{S}^{2}_{\mathds{T}}$ is a Banach space and
$M^{2}_{\mathds{T}}$ is a Hilbert space. A solution to a BSDE on time scales
is a triple of process $(Y,Z,N)$ satisfying the above equation, such that $Y$
is $\mathbf{R}^{k}$-valued, continuous and adapted, $Z$ is
$\mathbf{R}^{k\times d}$-valued and predictable, $N$ is a martingale process
strongly orthogonal to $W$. For terminal condition $\xi$, and generator $g$,
we make the following assumptions:

Assumption(H):

  * •

(H1) $g$ is defined as
$\Omega\times\mathds{T}\times\mathbf{R}^{k}\times\mathbf{R}^{k\times
d}\rightarrow\mathbf{R}^{k}$, such that
$\forall(y,z)\in\mathbf{R}^{k}\times\mathbf{R}^{k\times d}$, $g(\cdot,y,z)$ is
$\mathcal{F}_{t}$ progressively measurable. The deterministic integral part
could also be $g(\omega,t,Y_{t},Z_{t})$ or $g(\omega,t-,Y_{t-},Z_{t-})$. For
simplicity, we only consider the $g(\cdot,Y_{t-},Z_{t})$ case, which
corresponding to the implicit discrete backward difference equation (4).

  * •

(H2)

$\int_{0}^{T}|g(\cdot,0,0)|^{2}\nabla t\in L^{2}_{k}(\mathcal{F}_{T}).$

  * •

(H3) we also assume $g$ satisfies Lipschitz condition w.r.t. $(y,z)$: there
exists a constant $L>0$, $\forall y,y^{\prime}\in\mathbf{R}^{k}$,
$z,z^{\prime}\in\mathbf{R}^{k\times d}$

$|g(t,y,z)-g(t,y^{\prime},z^{\prime})|\leq L(|y-y^{\prime}|+|z-z^{\prime}|).$

  * •

(H4) $\xi\in L^{2}_{k}(\mathcal{F}_{T})$

for any inter $k$.

######  Theorem 4.1.

(Main Result) Assuming (H) is satisfied, BS$\nabla$E(13) admits a unique
triple of solution $(Y,Z,N)\in\mathcal{S}^{2}_{\mathds{T}}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$.

######  Remark 4.2.

$\mathds{T}=\mathbf{R}$, Theorem 4.1 degenerates to the classical BSDE.

For simplicity, we first consider $g$ real-valued and not contain $(y,z)$.

######  Lemma 4.3.

For a fixed $\xi\in L^{2}_{k}(\mathcal{F}_{T})$ and
$g_{0}({\cdot}):\Omega\times\mathds{T}\rightarrow\mathbf{R}^{k}$ is a
progressively measurable and $\mathcal{F}_{t}$-predictable process which
satisfies

$E\int_{0}^{T}|g_{0}(s)|^{2}\nabla s<\infty.$

Then for BSDE(13) with parameters $\xi$ and $g_{0}$, there exists a triple
solution $(Y,Z,N)\in\mathcal{S}^{2}_{\mathds{T}}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$. Moreover,

$\begin{split}&|Y_{t}|^{2}+E^{\mathcal{F}_{t}}\int_{t}^{T}\\{\frac{\beta}{2}|Y_{s-}|^{2}+|Z_{s}|^{2}\\}e_{\beta}(s,t)\nabla
s+\int_{t}^{T}e_{\beta}(s,t)\nabla[N_{s}]\\\
\leq&E^{\mathcal{F}_{t}}|\xi|^{2}e_{\beta}(T,t)+\frac{2}{\beta}E^{\mathcal{F}_{t}}\int_{t}^{T}|g_{0}(s)|^{2}e_{\beta}(s,t)\nabla
s.\\\ \end{split}$ (14)

$\beta$ is a positive constant.

###### Proof.

Existence:

We denote

$M_{t}=E^{\mathcal{F}_{t}}[\xi+\int_{0}^{T}g_{0}(s)\nabla s].$

Apparently, $M_{t}$ is a square integrable martingale. Following the
Martingale Decomposition Theorem on time scales, there exists a unique
predictable process $Z_{t}\in\mathcal{L}^{2}_{\mathds{T},k\times d}$ and
$N\in\mathcal{M}_{\mathds{T},k}^{2,\perp}$, s.t.
$M_{t}=M_{0}+\int_{0}^{t}Z_{s}\nabla W_{s}+N_{t}$,
$M_{T}=M_{0}+\int_{\left(0,T\right]}Z_{s}\nabla W_{s}+N_{T}$. Then
$M_{t}=M_{T}-\int_{t}^{T}Z_{s}\nabla W_{s}-(N_{T}-N_{t})$. As also
$M_{T}=\xi+\int_{0}^{T}g_{0}(s)\nabla s$, denote
$Y_{t}=M_{t}-\int_{0}^{t}g_{0}(s)\nabla s=M_{T}+\int_{t}^{T}g_{0}(s)\nabla
s-\int_{t}^{T}Z_{s}\nabla W_{s}-(N_{T}-N_{t})$. It can be seen that
$Y\in\mathcal{S}^{2}_{\mathds{T}}$, we get the existence.

Uniqueness:

For any fixed $\beta$, with $1+\beta\mu(t)>0$, the $\Delta$-exponential
function $e_{\beta}(t,t_{0})$, defined by [5], is the solution $y(\cdot)$ of
the initial value problem

$\nabla y(t)=\beta y(t-),\forall t>t_{0},t\in\mathds{T},y(t_{0})=1.$ (15)

Applying Ito’s formula to $|Y|_{s}^{2}e_{\beta}(s,t)$ on $[t,T]_{\mathds{T}}$,
where

$Y_{t}=\xi+\int_{t}^{T}g_{0}(s)\nabla s-\int_{t}^{T}Z_{s}\nabla
W_{s}-N_{T}+N_{t},$

or the differential form:

$\nabla Y_{t}=-g_{0}(t)\nabla t+Z_{t}\nabla W_{t}+\nabla N_{t}.$

We have

$\begin{split}|Y_{T}|^{2}e_{\beta}(T,t)&=|Y_{t}|^{2}+\int_{t}^{T}\beta
e_{\beta}(s,t)Y_{s-}^{2}\nabla s+2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}\nabla
Y_{s}\\\ &+\int_{t}^{T}e_{\beta}(s,t)\nabla[Y]_{s}+S\\\
&=|Y_{t}|^{2}+\int_{t}^{T}\beta e_{\beta}(s,t)Y_{s-}^{2}\nabla
s-2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}g_{0}(s)\nabla s\\\
&+2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}Z_{s}\nabla
W_{s}+2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}\nabla N_{s}\\\
&+\int_{t}^{T}e_{\beta}(s,t)Z_{s}^{2}\nabla[W]_{s}+\int_{t}^{T}e_{\beta}(s,t)\nabla[N]_{s}+S,\\\
\end{split}$

where

$\begin{split}S=&\sum_{s\in\left(t,T\right]}\\{|Y_{s}|^{2}e_{\beta}(s,t)-|Y_{s-}|^{2}e_{\beta}(s,t)-2|Y_{s-}|e_{\beta}(s,t)(|Y_{s}|-|Y_{s-}|)\\\
&-e_{\beta}(s,t)(|Y_{s}|-|Y_{s-}|)^{2}\\}\\\ =&0.\\\ \end{split}$

Thus we have

$\begin{split}&|Y_{t}|^{2}+\int_{t}^{T}\beta e_{\beta}(s,t)Y_{s-}^{2}\nabla
s+\int_{t}^{T}e_{\beta}(s,t)Z_{s}^{2}\nabla[W]_{s}+\int_{t}^{T}e_{\beta}(s,t)\nabla[N_{s}]\\\
=&|\xi|^{2}e_{\beta}(T,t)+2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}g_{0}(s)\nabla
s-2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}Z_{s}\nabla W_{s}\\\
&-2\int_{t}^{T}e_{\beta}(s,t)Y_{s-}\nabla N_{s}.\\\ \end{split}$

Since $g_{0}(s)$ is $\mathcal{F}_{s}$-predictable, the above integrals with
respect to $\nabla W$ and $\nabla N$ all belong to
$\mathcal{M}_{\mathds{T},k}^{2}$. By taking the expectation with respect to
$\mathcal{F}_{t}$, we get

$\begin{split}&|Y_{t}|^{2}+E^{\mathcal{F}_{t}}\int_{t}^{T}[\beta|Y_{s-}|^{2}+|Z_{s}|^{2}]e_{\beta}(s,t)\nabla
s+E^{\mathcal{F}_{t}}\int_{t}^{T}e_{\beta}(s,t)\nabla[N]_{s}\\\
=&E^{\mathcal{F}_{t}}|\xi|^{2}e^{\beta(T-t)}+E^{\mathcal{F}_{t}}\int_{t}^{T}2Y_{s-}g_{0}(s)e_{\beta}(s,t)\nabla
s\\\
\leq&E^{\mathcal{F}_{t}}|\xi|^{2}e^{\beta(T-t)}+E^{\mathcal{F}_{t}}\int_{t}^{T}[\frac{\beta}{2}|Y_{s-}|^{2}+\frac{2}{\beta}|g_{0}(s)|^{2}]e_{\beta}(s,t)\nabla
s\\\ \end{split}$ (16)

So we get the inequality.

Now we suppose there are two solutions $(Y^{1},Z^{1},N^{1})$ and
$(Y^{2},Z^{2},N^{2})$. Define $\delta Y_{t}=Y_{t}^{1}-Y_{t}^{2}$, $\delta
Z_{t}=Z_{t}^{1}-Z_{t}^{2}$, $\delta N_{t}=N_{t}^{1}-N_{t}^{2}$, then

$\left\\{\begin{array}[]{ll}\delta Y_{t}=-\int_{t}^{T}\delta Z_{s}\nabla
W_{s}-\delta N_{T}+\delta N_{t},\\\ \delta Y_{T}=0.\end{array}\right.$

then $(\delta Y,\delta Z,\delta N)$ satisfy BSDE(13) with $\xi=0$ and $g=0$.
By the inequality

$|\delta Y_{0}|^{2}+E\int_{0}^{T}[\frac{\beta}{2}|\delta Y_{s-}|^{2}+|\delta
Z_{s}|^{2}]e_{\beta}(s,0)\nabla s+\int_{0}^{T}e_{\beta}(s,0)\nabla[\delta
N]_{s}\leq 0$

By the continuity of $\delta Y_{s},\delta Y_{s}=0,\delta N_{s}=C-\delta
N_{0}=0,a.s.-\mathbf{P}$. The uniqueness proved. ∎

Now we introduce a new norm on $\mathcal{S}^{2}_{\mathds{T}}$:

$\|\phi(\cdot)\|_{\beta,T}=\\{E\int_{0}^{T}|\phi_{s}|^{2}e_{\beta}(s,0)\nabla
s\\}^{\frac{1}{2}}$

and a new norm on $\mathcal{M}_{\mathds{T},k}^{2,\perp}$:
$\|N\|_{\beta,T}=\\{E\int_{0}^{T}e_{\beta}(s,0)\nabla[N]_{s}\\}^{\frac{1}{2}}$
for any positive integer $k$. Apparently, for each $\beta>0$,
$\|\cdot\|_{\beta,T}$ is equivalent to $\|\cdot\|_{0,T}$ which is the original
norm on the corresponding space. Now we start to prove Theorem 4.1:

###### Proof.

For any fixed
$(y(\cdot),z(\cdot),n(\cdot))\in\mathcal{S}^{2}_{\mathds{T}}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$, it follows from
the lemma 4.3 that it admits a unique triple solution
$(Y(\cdot),Z(\cdot),N(\cdot))\in\mathcal{S}^{2}_{\mathds{T}}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$ satisfies

$Y_{t}=\xi+\int_{t}^{T}g(s,y_{s},z_{s})\nabla s-\int_{t}^{T}Z_{s}\nabla
W_{s}-N_{T}+N_{t},\quad\forall t\in[0,T]_{\mathds{T}}.$

Hence, we can define an operator

$(Y.,Z.,N.)=I[(y.,z.,n.)]:\mathcal{S}^{2}_{\mathds{T}}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}\rightarrow\mathcal{S}^{2}_{\mathds{T}}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}.$

We can prove that $I$ forms a contraction mapping on the Banach space
$\mathcal{S}_{\mathds{T}}^{2}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$. Take any
$(y^{1}(\cdot),z^{1}(\cdot),n^{1}(\cdot)),(y^{2}(\cdot),z^{2}(\cdot),n^{2}(\cdot))\in\mathcal{S}_{\mathds{T}}^{2}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$, we denote

$(Y^{1},Z^{1},N^{1})=I[(y^{1},z^{1},n^{1})],\quad(Y^{2},Z^{2},N^{2})=I[(y^{2},z^{2},n^{2})].$

and $\delta y=y^{1}-y^{2}$, $\delta z=z^{1}-z^{2}$, $\delta n=n^{1}-n^{2}$. By
equation(14), we get

$\begin{split}&E|\delta Y_{0}|^{2}+E\\{\int_{0}^{T}\\{\frac{\beta}{2}|\delta
Y_{s-}|^{2}+|\delta Z_{s}|^{2}\\}e_{\beta}(s,0)\nabla
s+\int_{0}^{T}e_{\beta}(s-,0)\nabla[\delta N]_{s}\\}\\\
\leq&\frac{2}{\beta}E\int_{0}^{T}e_{\beta}(s,0)|g(s,y^{1}_{s-},z^{1}_{s})-g(s,y^{2}_{s-},z^{2}_{s})|^{2}\nabla
s\\\ \leq&\frac{2}{\beta}E\int_{0}^{T}e_{\beta}(s,0)|L|\delta y_{s-}|+L|\delta
z_{s}||^{2}\nabla s\\\
\leq&\frac{2}{\beta}E\int_{0}^{T}e_{\beta}(s,0)[2L^{2}|\delta
y_{s-}|^{2}+2L^{2}|\delta z_{s}|^{2}]\nabla s.\\\ \end{split}$

We get

$\begin{split}&E\\{\int_{0}^{T}\\{\frac{\beta}{2}|\delta Y_{s-}|^{2}+|\delta
Z_{s}|^{2}\\}e_{\beta}(s,0)\nabla s+\int_{0}^{T}e_{\beta}(s-,0)\nabla[\delta
N]_{s}\\}\\\ \leq&\frac{4L^{2}}{\beta}E\int_{0}^{T}e_{\beta}(s,0)[|\delta
y_{s-}|^{2}+|\delta z_{s}|^{2}]\nabla s.\end{split}$ (17)

Let $\beta=8(1+L^{2})$,

$\|\delta Y\|_{\beta,T}^{2}+\|\delta Z\|_{\beta,T}^{2}+\|\delta
N\|_{\beta,T}^{2}\leq\frac{1}{2}(\|\delta y\|_{\beta,T}^{2}+\|\delta
z\|_{\beta,T}^{2}).$

We can get that the operator $I$ is contractive. Hence, in this case, there
exists a unique fixed point $(Y,Z,N)\in\mathcal{S}_{\mathds{T}}^{2}\times
M^{2}_{\mathds{T}}\times\mathcal{M}_{\mathds{T},k}^{2,\perp}$ for the map $I$.
The proof is complete. ∎

###  4.1 The Linear case

######  Example 4.4.

(Nguyen Huu Du 2011[26])(exponential martingale) Let $M$ be a bounded
martingale satisfying $\nabla^{*}M_{s}=M_{s}-M_{\rho(s)}\neq-1$ for any
$s\in\left(0,T\right]_{\mathds{T}},s\in\mathds{T}.$ Set

$\tilde{M}_{t}=M_{t}-\sum_{s\leq t,s,t\in\mathds{T}}(M_{s}-M_{s-}),$
$\mathcal{E}_{t}=\\{\prod_{s\left(0,t\right]}(1+\nabla^{*}M_{s})\\}\exp\bigg{\\{}\tilde{M}_{t}-\frac{1}{2}[\tilde{M}_{t}]\bigg{\\}}.$

Then, $\mathcal{E}_{t}$ satisfies Doleans exponential equation

$\mathcal{E}_{t}=1+\int_{\left(0,t\right]}\mathcal{E}_{s_{-}}\nabla M_{s}.$

######  Lemma 4.5.

Let $a,b,c$ be predictable bounded processes, let $\mathcal{E}$ be the
exponential martingale of the martingale $M_{t}=\int_{0}^{t}b_{s}\nabla
W_{s}$, and define

$\gamma_{t}=\exp(\int_{0}^{t}a_{s}\nabla
s),~{}\Gamma_{t}=\gamma_{t}\mathcal{E}_{t}$

Suppose that

  * •

$\mathcal{E}_{t}$ is a positive uniformly integrable martingale;

  * •

$E[(sup_{t}\gamma_{t})\mathcal{E}_{T}^{2}]<\infty$.

If the linear backward equation

$-\nabla Y_{t}=(a_{t}Y_{t-}+b_{t}Z_{t}+c_{t})\nabla t-Z_{t}\nabla W_{t}-\nabla
N_{t},~{}Y_{T}=\xi$

has a solution $(Y,Z,N)$, then $Y$ is given by

$Y_{t}=E[\xi\frac{\Gamma_{T}}{\Gamma_{t}}+\int_{t}^{T}\frac{\Gamma_{s}}{\Gamma_{t}}c_{s}\nabla
s|\mathcal{F}_{t}].$

###### Proof.

By Girsanov’s theorem [22],

$\tilde{W}_{t}=W_{t}-\int_{0}^{t}b_{s}\nabla s$

is a Q-martingale, the probability measure Q which has density
$\mathcal{E}_{T}$ with respect to $\mathbf{P}$ on $\mathcal{F}_{T}$. Under the
measure Q, the equation becomes

$-\nabla Y_{t}=(a_{t}Y_{t-}+c_{t})\nabla t-Z_{t}\nabla\tilde{W}_{t}-\nabla
N_{t},~{}Y_{T}=\xi.$

Applying Itô formula to the process $\gamma Y$, we obtain

$\gamma_{t}Y_{t}=\gamma_{0}Y_{0}+\int_{0}^{t}\gamma_{s}Z_{s}\nabla\tilde{W}_{s}+\int_{0}^{t}\gamma_{s}\nabla
N_{s}-\int_{0}^{t}c_{s}\gamma_{s}\nabla s+S$

where
$S=\sum_{s\in\left(0,t\right]}(\gamma_{s}Y_{s}-\gamma_{s}Y_{s-}-(\gamma_{s}(Y_{s}-Y_{s-})))=0$.
Since $\int_{0}^{t}\gamma_{s}Z_{s}\nabla\tilde{W}_{s}$ and
$\int_{0}^{t}\gamma_{s}\nabla N_{s}$ are martingales,

$\gamma_{t}Y_{t}=E^{Q}[\xi\gamma_{T}+\int_{t}^{T}c_{s}\gamma_{s}\nabla
s|\mathcal{F}_{t}],$

and consequently

$\Gamma_{t}Y_{t}=E[\xi\Gamma_{T}+\int_{t}^{T}c_{s}\Gamma_{s}\nabla
s|\mathcal{F}_{t}].$

This concludes the proof. ∎

We have the one-dimension comparison result

######  Theorem 4.6.

Let $(Y^{1},Z^{1},N^{1})$, $(Y^{2},Z^{2},N^{2})$ be the corresponding
solutions to $(g^{1}(t,Y_{t-}^{1},Z_{t}^{1}),\xi^{1})$,
$(g^{2}(t,Y_{t-}^{2},Z_{t}^{2}),\xi^{2})$ and suppose $g^{1},g^{2}$ satisfy
conditions (H2)(H3). Assume that $\xi_{1}\geq\xi_{2}$ and satisfy (H4),
$g^{1}(t,Y_{t-}^{2},Z_{t}^{2})\geq
g^{2}(t,Y_{t-}^{2},Z_{t}^{2})~{}a.s.-\mathbf{P}$. Then, for any $t$, we have
$Y_{t}^{1}\geq Y_{t}^{2}~{}a.s.-\mathbf{P}$.

###### Proof.

We define $\delta Y_{t}=Y_{t}^{1}-Y_{t}^{2}$, $\delta
Z_{t}=Z_{t}^{1}-Z_{t}^{2}$, $\delta N_{t}=N_{t}^{1}-N_{t}^{2}$. Then $\delta
Y$ solves the following linear BSDE on time scales

$-\nabla\delta Y_{t}=(a_{t}\delta Y_{t-}+b_{t}\delta Z_{t}+c_{t})\nabla
t-\delta Z_{t}\nabla W_{t}-\nabla\delta N_{t},~{}\delta
Y_{T}=\xi^{1}-\xi^{2},$

where

$a_{t}=\frac{g^{1}(t,Y_{t}^{1},Z_{t}^{1})-g^{1}(t,Y_{t}^{2},Z_{t}^{1})}{\delta
Y_{t}}1_{\delta Y_{t}\neq 0},$
$b_{t}=\frac{g^{1}(t,Y_{t}^{2},Z_{t}^{1})-g^{1}(t,Y_{t}^{2},Z_{t}^{2})}{\delta
Y_{t}}1_{\delta Z_{t}\neq 0},$
$c_{t}=g^{1}(t,Y^{2}_{t-},Z_{t}^{2})-g^{2}(t,Y_{t-}^{2},Z_{t}^{2}).$

According to the linear BSDE on time scales, if the coefficients
$a_{t},b_{t},c_{t}$ verify the suitable conditions, we have

$\Gamma_{t}\delta Y_{t}=E[\Gamma_{T}(\xi^{1}-\xi^{2})+\int_{t}^{T}\delta
c_{s}\Gamma_{s}\nabla s|\mathcal{F}_{t}].$

Assume now that $\xi^{1}\geq\xi^{2}$, and for any $t$, $c_{t}\geq
0~{}a.s.-\mathbf{P}$. Then, for any $t$, $Y_{t}^{1}\geq
Y_{t}^{2}~{}a.s.-\mathbf{P}$. ∎

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