Existence and Uniqueness of Solutions to SDEs with Jumps and Irregular Drifts

Abstract

We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages a space transformation and Itô-Krylov’s formula to effectively eliminate the singular component of the drift, allowing us to obtain a transformed SDEJ that satisfies classical solvability conditions. By applying the inverse transformation proven to be a one-to-one mapping, we retrieve the solution to the original equation. This methodology offers several key advantages. First, it extends the well-known result of Le Gall (1984) from Brownian-driven SDEs to the jump process setting, broadening the range of applicable stochastic models. Second, it provides a robust framework for handling singular drifts, enabling the resolution of equations that would otherwise be intractable. Third, the approach accommodates drifts with quadratic growth, making it particularly relevant for financial modeling, insurance risk assessment, and other applications where such growth behavior is common. Finally, the inclusion of multiple examples illustrates the practical effectiveness of our method, demonstrating its flexibility and applicability to real-world problems.

1. Introduction

Stochastic differential equations without or with jumps (SDEs or SDEJs) appeared to be appropriate models in several disciplines in the sciences, such as finance, insurance, biology, and economics, as well as in interactions with partial differential equations. Many efforts have been made to study the qualitative and quantitative properties of SDEs, in particular, by relaxing the assumptions of the coefficients. This class is usually called SDEs with generalized or singular drift, and this topic has been studied by many researchers (see, among others, Harrison and Shepp (1981) [1]) who investigated a stochastic process defined by the equation $X\left(t\right)=W\left(t\right)+\beta L_0^X\left(t\right)$, where W represents a standard Wiener process and $L_0^X\left(t\right)$ denotes the local time of X at zero.

The authors established that a unique solution $X\left(t\right)$ exists and is adapted to the filtration of $W\left(t\right)$ if and only if $\left|\beta \right|\le 1$; no solutions are found for $\left|\beta \right|>1$. When $\left|\beta \right|\le 1$, setting $\alpha =(\beta +1)/2$, the process $X\left(t\right)$ corresponds to a skew Brownian motion with parameter $\alpha$. This process can be interpreted as a standard Wiener process, where each excursion away from zero is positive with probability $\alpha$ and negative with probability $1-\alpha$, effectively “skewing” the motion at zero. Furthermore, the paper demonstrates that skewed Brownian motion arises as the weak limit of scaled random walks exhibiting exceptional behavior at the origin. Specifically, as $n\to \infty$, the sequence $n^{-1/2}{S}_{\lfloor nt\rfloor }$, where $S_n$ represents such a random walk, converges in distribution to skewed Brownian motion. This result provides a discrete approximation to the continuous skewed Brownian motion, bridging the behavior of discrete random processes with their continuous counterparts.

Portenko (1990) [2] addressed the construction of diffusion processes characterized by a given diffusion matrix and drift vector, particularly when the drift vector exhibits singularities or irregular behavior. The work introduced the concept of generalized diffusion processes and continuous Markov processes, the local characteristics of which are defined in a generalized sense-serving mode as models for diffusion in media with irregular dynamics. By integrating probabilistic and analytic approaches, the author provided a comprehensive framework for understanding diffusion processes in complex, non-regular environments. This work is particularly relevant for specialists in stochastic processes and their applications, offering insights into modeling diffusion in media with irregular characteristics.

In 1981, Stroock and Yor [3] investigated specific martingales associated with Brownian motion. They focused on the process defined by $M_n\left(t\right)={\int }_0^t{B}_s^{n}d{B}_s$, where B denotes standard Brownian motion. The authors explored the conditions under which these martingales are “pure”, meaning they cannot be decomposed into a simpler martingale structure. This work contributed to the understanding of the structural properties of certain stochastic processes and their potential applications in the broader field of probability theory.

Barlow and Perkins (1983) [4] investigated a class of stochastic differential equations (SDEs) that incorporate the local time of the unknown process. Local time, in this context, quantifies the cumulative duration a stochastic process spends at a specific position. The authors focused on establishing conditions under which strong solutions to these SDEs exist and are unique. They demonstrated that, depending on the characteristics of the local time term, the SDEs can exhibit either uniqueness or non-uniqueness in their solutions. This work provided valuable insights into the behavior of stochastic processes influenced by their local times and extends the understanding of SDEs with singular or discontinuous coefficients.

In his relatively old papers (1983 and 1984), Le Gall [5,6] investigated stochastic differential equations (SDEs) in which the local time of the solution process plays a crucial role. Local time, in this context, quantifies the cumulative duration a stochastic process spends at a specific state or position.

Le Gall’s work focused on establishing the existence and uniqueness of solutions to these SDEs, particularly when the drift term is singular or exhibits discontinuities. By leveraging the properties of local times, he provides a framework for analyzing such equations, which are challenging to handle using traditional methods due to their irregular coefficients. This research had significant implications in the study of stochastic processes, especially in understanding phenomena like skewed Brownian motion, where the behavior of the process is influenced by its local time at certain points. Le Gall’s findings offered a mathematical foundation for modeling and analyzing systems where the accumulation of time spent at specific states affects the system’s dynamics.

These results were followed by the work of Engelbert and Schmidt (1985) [7], who investigated stochastic differential equations (SDEs) of the form $d{X}_t=\sigma \left(X_t\right)d{W}_t+b\left(X_t\right)dt$, where the drift term $b\left(X_t\right)$ may exhibit singular behavior or be interpreted in a generalized sense. They focused on establishing conditions for the existence and uniqueness of solutions to these equations, particularly when the drift term is not a conventional function but rather a generalized function or distribution. By employing analytical techniques, the authors derived criteria under which solutions to such SDEs exist and are unique. Their findings extend the classical theory of SDEs to accommodate cases where the drift term is irregular or singular, thereby broadening the applicability of stochastic calculus to more complex systems. This work has significant implications for modeling in fields where systems are influenced by irregular forces or exhibit discontinuous behaviors.

In their series of papers titled “Strong Markov Continuous Local Martingales and Solutions of One-Dimensional Stochastic Differential Equations”, Engelbert and Schmidt (1989, 1991) [8,9] explored the intricate relationship between strong Markov continuous local martingales and the solutions to one-dimensional stochastic differential equations (SDEs) without drift terms. Their research focused on establishing conditions under which these SDEs possess unique solutions and how these solutions exhibit the strong Markov property. By delving into the interplay between the probabilistic properties of local martingales and the analytical structure of SDEs, Engelbert and Schmidt provided a comprehensive framework that enhances the understanding of stochastic processes in one-dimensional settings. Their work offers valuable insights into the behavior of stochastic systems, particularly in scenarios where traditional methods may not readily apply.

Last but not least, Bass and Chen (2005) [10] investigated stochastic differential equations (SDEs) of the form $d{X}_t=a\left(X_t\right)d{W}_t+b\left(X_t\right)dt$, where the diffusion coefficient $a\left(X_t\right)$ may be degenerate (i.e., zero in some regions), and the drift coefficient $b\left(X_t\right)$ can exhibit singular behavior. They focused on establishing conditions under which strong solutions exist and are unique, even when traditional assumptions such as non-degeneracy and smoothness of coefficients are relaxed. Their analysis involved formulating the notion of a solution and proving strong existence and pathwise uniqueness results when a belongs to ${\mathcal{C}}^{1/2}$ and b is a generalized function, such as the distributional derivative of a Hölder function or a function of bounded variation. Additionally, they explore scenarios where the generator of the SDE is in divergence form, providing results on the non-existence of strong solutions and non-pathwise uniqueness, as well as characterizing when a solution is a semi-martingale. This work extended the understanding of one-dimensional SDEs by accommodating more general and potentially singular coefficient functions.

The reviewed papers explored various aspects of stochastic differential equations (SDEs) in the continuous setting, particularly those involving Brownian motion and local time. Barlow and Perkins (1983) [4] examined equations incorporating local time, demonstrating cases of strong existence, uniqueness, and non-uniqueness. Le Gall (1984) [6] investigated one-dimensional SDEs, where the unknown process’s local time played a crucial role, providing conditions for the existence and uniqueness of solutions. Engelbert and Schmidt (1989, 1991) [8,9] extended the classical theory by considering generalized drift terms, analyzing strong Markov properties, and studying local martingales associated with one-dimensional SDEs. Bass and Chen (2005) [10] further developed the theory by addressing SDEs with singular and degenerate coefficients, proving strong existence and pathwise uniqueness under minimal regularity conditions. Collectively, these works enhanced the understanding of SDEs with singular or discontinuous drift terms, highlighting the role of local time in shaping solution behavior in continuous processes driven by Brownian motion.

In this paper, we investigate one-dimensional stochastic differential equations with jumps (SDEJs), extending the classical results of [5,6,7,8,9] to the jump setting. Our approach builds on the transformation technique introduced by Zvonkin (1974) [11], which is commonly used to convert an SDE with a singular drift into an equivalent SDE without drift or, at least, without its singular component. While this technique has been successfully applied in previous works (see, e.g., [5,6,7,8,9]), the introduction of jumps brings additional complexity, as new terms may arise due to the jump noise component. However, we show that these additional terms exhibit regularity properties that enable the resolution of a class of SDEJs with merely measurable generators.

Recent advances in the existence, uniqueness, and approximation of solutions to SDEs with discontinuous drifts have been explored in [12,13], but our methodology differs significantly. Specifically, our contribution is twofold: first, we establish key intermediate results based on Krylov’s estimates, leveraging the techniques and properties of the local time of semi-martingales. Second, we prove an Itô-Krylov-type formula, which provides a foundation for studying the well-posedness of jump-diffusion SDEs driven by a class of Lévy processes with merely measurable drift coefficients. Notably, our approach accommodates drift terms that may exhibit quadratic growth with respect to the Lebesgue measure or even exponential growth relative to the Lévy measure controlling the jumps. Siddiqui et al. [14] studied numerical methods for SDEJs with measurable drifts of possible quadratic growth. By using the space transformation adopted here to remove the measurable drift and apply the Euler–Maruyama scheme, proving a convergence rate of $1/2$, they compared direct and indirect numerical approaches, supporting their findings with illustrative examples.

Compared to the results established in [12,13], our approach provides a more global framework by completely eliminating the singular drift term. In contrast, the methodology adopted in the aforementioned references relies on a local regularization technique, which primarily smooths out discontinuities at specific points rather than addressing the singular drift in its entirety.

Our transformation-based approach ensures that the resulting equation no longer contains irregular drift components, allowing us to study the well-posedness of the stochastic differential equation in a more comprehensive manner. This distinction is particularly important in the jump setting, where discontinuities induced by the noise component and the drift coefficients require a refined analysis. By removing the singularity entirely rather than applying localized regularization, our method enables a broader range of applications and facilitates numerical approximations under more general conditions. As a practical application of our results, we explore indirect numerical approximation schemes for SDEJs with merely measurable and integrable drifts. The advantage of our transformation method is that it eliminates irregular terms, yielding an equivalent equation where all new coefficients are Lipschitz continuous. This feature allows for the application of standard numerical schemes, extending the numerical results obtained in [15] to the jump setting.

In the following, we formally introduce the problem under investigation, outlining the key mathematical framework and assumptions necessary for our analysis. We begin by defining the class of stochastic differential equations with jumps (SDEJs) under consideration, specifying the nature of the driving noise, the properties of the drift and diffusion coefficients, and the role of the jump component.

Our objective is to study the well-posedness of these equations in the presence of irregular, merely measurable drift terms, which may exhibit quadratic or even exponential growth under certain measures. To achieve this, we employ transformation techniques that regularize the drift, allowing us to analyze the resulting equations using classical tools.

Furthermore, we highlight the main challenges associated with this problem, particularly the impact of discontinuities introduced by the jump component and the need for refined estimates to handle the singularities in the drift term. These challenges motivate our approach, which builds upon Krylov’s estimates and Itô-Krylov-type formulas, as well as recent advancements in stochastic calculus.

With this formulation in place, we proceed to establish the necessary theoretical foundations, laying the groundwork for the existence and uniqueness results to be developed in the subsequent sections.

Let $[0,T]$ be a bounded time interval, $E=\mathbb{R}\setminus \left\{0\right\}$, endowed with the Borel $\sigma$-algebra $\mathcal{B}\left(E\right)$, on which a positive measure $\nu$ is defined such that $\nu \left(E\right)$ is finite and $(\Omega ,\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{t\in [0,T]},\mathbb{P})$ is a filtered probability space supporting the following two independent stochastic processes:

• $W={\left\{W_t\right\}}_{t\in [0,T]}$ is one-dimensional standard Brownian motion.
• $N(\mathrm{d}s,\mathrm{d}e)$ is a time-homogeneous Poisson random measure with compensator $\nu \left(\mathrm{d}e\right)\mathrm{d}s$ on $([0,T]\times E,\mathcal{B}\left([0,T]\right)\otimes \mathcal{B}\left(E\right)).$

We denote by $\tilde{N}(\mathrm{d}s,\mathrm{d}e)=N(\mathrm{d}s,\mathrm{d}e)-\nu \left(\mathrm{d}e\right)\mathrm{d}s$ the compensated jump measure, and we take ${\left\{{\mathcal{F}}_t\right\}}_{t\in [0,T]}$ as being generated by W and $\tilde{N}$, completed with $\mathbb{P}$-null sets and made right continuous.

Our goal in this paper is to investigate the existence and uniqueness of solutions of a class of SDEs with jumps and that involve the local time of the unknown process.

Concretely, we are concerned with $\mathbb{R}$-valued stochastic integral equations with jumps of a quadratic type of the form

$${\mathfrak{X}}_t={\mathfrak{X}}_0+{\int }_0^t{b}_f(s,{\mathfrak{X}}_s,\phi ({\mathfrak{X}}_{s-},·))ds+{\int }_0^t\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e),$$

(1)

for the special generators $b_f$ of the following form:

$$b_f(s,x,\phi ):=b(s,x)+f\left(x\right){\sigma }^2\left(x\right)\mathrm{independent}\mathrm{of}\phi ,$$

or

$$b_f(s,x,\phi ):=b(s,x)+f\left(x\right){\sigma }^2\left(x\right)-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left(x\right),$$

where, for any u in ${\mathcal{L}}_{\nu }^1$ (to be defined in the next section),

$${\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right):={\int }_E\left({\displaystyle \frac{\mathrm{F}(x+u(e\left)\right)-\mathrm{F}\left(x\right)}{\mathrm{F}\prime \left(x\right)}}-u\left(e\right)\right)\nu \left(\mathrm{d}e\right),$$

and the function $\mathrm{F}$ is defined for every $x\in \mathbb{R}$ by

$$\mathrm{F}\left(x\right)={\int }_{-\infty }^{x}exp\left(-2{\int }_{-\infty }^{y}f\left(t\right)\mathrm{d}t\right)\mathrm{d}y,$$

(2)

for a given measurable and integrable function $f.$

Specific assumptions on b, f, $\sigma$, and $\phi$ will be given when studying each case.

Throughout the paper, we will consistently refer to Equation (1) as $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ for clarity and technical convenience. This notation is particularly useful, as we will be working with various coefficients, making it essential to distinguish between different forms of the equation in our analysis. This equation has an irregular or singular drift coefficient since f will be assumed to be merely measurable and integrable, and, sometimes, we will need to add the boundedness, as well as the integral operator ${\mathcal{I}}_{\mathrm{F}}$.

The structure of this paper is as follows: Section 2 introduces the necessary notations, defines key concepts, and establishes fundamental preliminary results that will be used throughout the study. Section 3 presents Krylov’s estimates, Itô-Krylov’s formula, and a priori estimates for potential solutions of $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$. These results are then leveraged to rigorously prove the existence and uniqueness of solutions to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$. Section 4 focuses on the application of these theoretical results to the solvability of stochastic differential equations with jumps (SDEJs). Several illustrative examples are discussed, featuring different generators of a quadratic form in x. These examples highlight cases that were not addressed in the previous literature, demonstrating the broader applicability and novelty of our approach.

2. Notations and Preliminary Results

This section establishes the fundamental mathematical framework for the study. We introduce the necessary notations, define key concepts, and present essential lemmas that will be used throughout the paper. In particular, we provide definitions of function spaces, stochastic processes, and transformation techniques relevant to our problem. Additionally, we prove some preliminary results that lay the groundwork for the subsequent analysis. For any real number $p>0$, we introduce the following spaces:

• ${\mathcal{S}}^p$: The set of càdlàg-adapted processes $\mathfrak{X}$ such that $\mathbb{E}\left[{sup}_{0\le t\le T}\right|{\mathfrak{X}}_t{|}^p]$ is finite.
• ${\mathcal{L}}_{\nu }^p$: The set of real-valued measurable functions u defined on E such that

  $${\parallel u\parallel }_{\nu ,p}:=({\int }_E{\left|u\left(e\right)\right|}^p{\nu \left(\mathrm{d}e\right))}^{{\displaystyle \frac{1}{p}}},$$

  is finite.
• ${\mathcal{M}}_W^p$: The space of ${\mathcal{F}}_t$-predictable processes $\phi$ satisfying

  $${\int }_0^T\mathbb{E}\left[|{\phi }_s{|}^p\right]\mathrm{d}s<\infty .$$
• ${\mathcal{L}}_W^p$: The space of ${\mathcal{F}}_t$-predictable processes ${\phi }_{·}$ satisfying

  $${\int }_0^T{\left|{\phi }_s\right|}^p\mathrm{d}s<\infty \mathrm{P}\text{-}\mathrm{a}.\mathrm{s}.$$
• ${\mathcal{M}}_N^p$: The space of ${\mathcal{F}}_t$-predictable processes U satisfying

  $${\int }_0^T\mathbb{E}\left[\parallel U_s{\parallel }_{\nu ,p}^p\right]\mathrm{d}s<\infty .$$
• ${\mathcal{L}}_N^p$: The space of ${\mathcal{F}}_t$-predictable processes U satisfying

  $${\int }_0^T{∥U_s∥}_{\nu ,p}^p\mathrm{d}s<\infty \mathrm{P}\text{-}\mathrm{a}.\mathrm{s}.$$
• ${\mathcal{W}}_1^2\left(\mathbb{R}\right)$: The space of continuous functions g from $\mathbb{R}$ to $\mathbb{R}$ such that $g\prime$ is continuous and $g″$ is integrable on $\mathbb{R}$.
• ${\mathcal{W}}_{1,loc}^2\left(\mathbb{R}\right)$:= The Sobolev space of functions g defined on $\mathbb{R}$ such that both g and its generalized derivatives $g\prime$ and $g″$ are locally integrable on $\mathbb{R}$.

Definition 1. 

Given an initial condition ${\mathfrak{X}}_0$ and measurable function $f:\mathbb{R}⟶\mathbb{R}$, a stochastic process ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$, starting from x, is a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ if ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ satisfies the integral Equation (1), $\mathbb{P}$-a.s. for all $t\in [0,T]$ such that the stochastic processes ${\left({\mathfrak{X}}_s\right)}_{0\le s\le T}$, ${\left(\sigma \left({\mathfrak{X}}_s\right)\right)}_{0\le s\le T}$, ${\left(\phi ({\mathfrak{X}}_{s-},·)\right)}_{0\le s\le T}$, ${\left(b(s,{\mathfrak{X}}_s)\right)}_{0\le s\le T}$ and $(f\left({\mathfrak{X}}_{·}\right){\displaystyle \frac{\mathrm{d}{\left[\mathfrak{X}\right]}_{·}^c}{\mathrm{d}s}})$ belong, respectively, to the spaces ${\mathcal{S}}^2$, ${\mathcal{M}}_W^2$, ${\mathcal{M}}_N^2$, ${\mathcal{M}}_W^2$, and ${\mathcal{M}}_W^1.$

The next lemma plays a crucial role in the proof of Theorem 2 below. In fact, the transformation $\mathrm{F}$ allows us to eliminate the singular part of the drift $b_f$ in the $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$.

Lemma 1. 

The function $\mathrm{F}$ defined in Equation (2) satisfies

$$\mathrm{F}″\left(x\right)+2f\left(x\right)\mathrm{F}\prime \left(x\right)=0,\mathit{for}a.e.x$$

(3)

and has the following properties:

(i) 

$\mathrm{F}$ and ${\mathrm{F}}^{-1}$ have quasi-isometry; that is, for any $x,y\in \mathbb{R}$ and ${\left|\mathrm{f}\right|}_1={\int }_{\mathbb{R}}\left|f\left(x\right)\right|\mathrm{d}x$,

$$\begin{array}{ccccc}{e}^{-2{\left|\mathrm{f}\right|}_1}\left|x-y\right|& \le & \left|\mathrm{F}\left(x\right)-\mathrm{F}\left(y\right)\right|& \le & e^{2{\left|\mathrm{f}\right|}_1}\left|x-y\right|,\\ e^{-2{\left|\mathrm{f}\right|}_1}\left|x-y\right|& \le & \left|{\mathrm{F}}^{-1}\left(x\right)-{\mathrm{F}}^{-1}\left(y\right)\right|& \le & e^{2{\left|\mathrm{f}\right|}_1}\left|x-y\right|.\end{array}$$

(4)

(ii) 

$\mathrm{F}$ is a one-to-one function. Both $\mathrm{F}$ and its inverse function ${\mathrm{F}}^{-1}$ belong to ${\mathcal{W}}_1^2\left(\mathbb{R}\right)$.

Proof of Lemma 1. 

(i) By definition, the function $\mathrm{F}$ and its inverse ${\mathrm{F}}^{-1}$ are continuous, one-to-one, strictly increasing functions; moreover, $\mathrm{F}″\left(x\right)+2f\left(x\right)\mathrm{F}\prime \left(x\right)=0$ for a.e. $x\in \mathbb{R}$. In addition $\mathrm{F}\prime \left(x\right)=exp(-2{\int }_{-\infty }^{x}f\left(t\right)\mathrm{d}t)$; hence, for every $x\in \mathbb{R}$,

$$e^{-2{\left|\mathrm{f}\right|}_1}\le \mathrm{F}\prime \left(x\right)\le e^{2{\left|\mathrm{f}\right|}_1}\mathrm{and}{e}^{-2{\left|\mathrm{f}\right|}_1}\le \left({\mathrm{F}}^{-1}\right)\prime \left(x\right)\le e^{2{\left|\mathrm{f}\right|}_1}.$$

(5)

(ii) Using inequality (5), one can show that both $\mathrm{F}$ and ${\mathrm{F}}^{-1}$ are ${\mathcal{C}}^1\left(\mathbb{R}\right)$. Since the second generalized derivative $\mathrm{F}″$ satisfies Equation (3) for almost all x, we obtain that $\mathrm{F}″$ belongs to $L^1\left(\mathbb{R}\right)$. Therefore, $\mathrm{F}$ belongs to the space ${\mathcal{W}}_1^2\left(\mathbb{R}\right)$. By using, again, assertion (i), one can easily check that ${\mathrm{F}}^{-1}$ also belongs to ${\mathcal{W}}_1^2\left(\mathbb{R}\right)$. □

Lemma 2. 

For a given real number x and a measurable function u in ${\mathcal{L}}_{\nu }^1$,

(i) 

the operator ${\mathcal{I}}_{\mathrm{F}}$ given by

$${\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right):={\int }_E\left({\displaystyle \frac{\mathrm{F}(x+u(e\left)\right)-\mathrm{F}\left(x\right)}{\mathrm{F}\prime \left(x\right)}}-u\left(e\right)\right)\nu \left(\mathrm{d}e\right)$$

is well-defined. Moreover,

$$\left|{\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right)\right|\le \left(1+e^{4{\left|\mathrm{f}\right|}_1}\right){∥u∥}_{\nu ,1}.$$

(6)

(ii) 

If, moreover, $f\le 0$-a.e., then ${\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right)\ge 0$ for all $x\in \mathbb{R}$.

Proof of Lemma 2. 

(i) From the quasi-isometry properties of the function $\mathrm{F}$ defined in Equation (2), for all $x\in \mathbb{R}$, we have

$$\begin{array}{ccc}\hfill \left|{\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right)\right|& \le & {\int }_E\left|{\displaystyle \frac{\mathrm{F}(x+u(e\left)\right)-\mathrm{F}\left(x\right)}{\mathrm{F}\prime \left(x\right)}}-u\left(e\right)\right|\nu \left(\mathrm{d}e\right)\hfill \\ & \le & {\int }_E\left|{\displaystyle \frac{\mathrm{F}(x+u(e\left)\right)-\mathrm{F}\left(x\right)}{\mathrm{F}\prime \left(x\right)}}\right|\nu \left(\mathrm{d}e\right)+{∥\phi ∥}_{\nu ,1}\hfill \\ & \le & \left({\displaystyle \frac{e^{2{\left|\mathrm{f}\right|}_1}}{{min}_x\mathrm{F}\prime \left(x\right)}}+1\right){∥\phi ∥}_{\nu ,1}.\hfill \end{array}$$

Thus,

$$\left|{\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right)\right|\le \left(1+e^{4{\left|\mathrm{f}\right|}_1}\right){∥\phi ∥}_{\nu ,1}\mathrm{for}\mathrm{all}x\in \mathbb{R}.$$

Hence, the operator ${\mathcal{I}}_{\mathrm{F}}$ is well-defined.

(ii) Observe, also, that for each $x\in \mathbb{R}$, we can write

$$\begin{array}{ccc}\hfill \mathrm{F}\prime \left(x\right){\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right)& =& {\int }_E\left(\mathrm{F}(x+u\left(e\right))-\mathrm{F}\left(x\right)-\mathrm{F}\prime \left(x\right)u\left(e\right)\right)\nu \left(\mathrm{d}e\right)\hfill \\ & =& {\int }_E{\int }_x^{x+u\left(e\right)}\left(\mathrm{F}\prime \left(y\right)-\mathrm{F}\prime \left(x\right)\right)\mathrm{d}y\nu \left(\mathrm{d}e\right)\hfill \\ & =& {\int }_E{\int }_x^{x+u\left(e\right)}\left(\mathrm{F}\prime \left(y\right)-\mathrm{F}\prime \left(x\right)\right)11_{\left\{u\right(e)>0\}}\mathrm{d}y\nu \left(\mathrm{d}e\right)\hfill \\ & & +{\int }_E{\int }_{x+u\left(e\right)}^x\left(\mathrm{F}\prime \left(x\right)-\mathrm{F}\prime \left(y\right)\right)11_{\left\{u\right(e)<0\}}\mathrm{d}y\nu \left(\mathrm{d}e\right).\hfill \end{array}$$

The last two terms in the above inequality are non-negative since $\mathrm{F}\prime$ is positive and increasing whenever f is non-positive. □

Corollary 1. 

Assuming that inequality (6) holds, for a given real number x and a predictable process $\left({\phi }_s\left(e\right)\right)$ on $[0,T]\times E$, such that

$${\int }_0^T{∥{\phi }_s∥}_{\nu ,1}\mathrm{d}s<+\infty \mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.$$

we have

$${\int }_0^T\left|{\mathcal{I}}_{\mathrm{F}}\left({\phi }_s\right)\left(x\right)\right|\mathrm{d}s\le \left(1+e^{4{\left|\mathrm{f}\right|}_1}\right){\int }_0^T{\int }_E\left|{\phi }_s\left(e\right)\right|\nu \left(\mathrm{d}e\right)\mathrm{d}s\mathbb{P}\text{-}\mathrm{a}.\mathrm{s}.\forall x\in \mathbb{R}.$$

Moreover, if φ in ${\mathcal{M}}_N^2$, then there exists a constant $C_{f,\nu }$ (depending only on f and ν) such that

$${\int }_0^T\mathbb{E}{\left|{\mathcal{I}}_{\mathrm{F}}\left({\phi }_s\right)\left(x\right)\right|}^2\mathrm{d}s\le C_{f,\nu }{\int }_0^T\mathbb{E}{∥{\phi }_s∥}_{\nu ,2}^2\mathrm{d}s.$$

(7)

Proof of Inequality (7).

By squaring both sides of inequality (6) and using Hölder’s inequality, for all $0\le s\le T$, we obtain

$$|{\mathcal{I}}_{\mathrm{F}}\left({\phi }_s\right)\left(x\right){|}^2\le \left(1+e^{4{\left|\mathrm{f}\right|}_1}\right)\nu \left(E\right){\int }_E|{\phi }_s\left(e\right){|}^2\nu \left(\mathrm{d}e\right)=C_{f,\nu }{\parallel {\phi }_s\parallel }_{\nu ,2}^2\mathrm{since}\nu \mathrm{is}\mathrm{finite}.$$

By taking the expectation and integrating over $[0,T]$ on both sides of the above inequality, we obtain the desired result. □

3. SDEs with Jumps and Globally Integrable Drift

In this section, we develop the mathematical foundation necessary to establish the existence and uniqueness of solutions to the stochastic differential equation with jumps (SDEJs) given by $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$. To achieve this, we present and utilize key theoretical tools, including Krylov’s estimates, Itô-Krylov’s formula, and a priori estimates for potential solutions. These results play a crucial role in handling irregularities in the drift term and ensuring well-posedness under minimal regularity assumptions.

-

Krylov’s estimates: These estimates provide bounds on the moments of stochastic processes, ensuring control over the growth of solutions and aiding in the derivation of stability properties.

-

Itô-Krylov’s formula: A generalized stochastic calculus tool that allows us to handle non-smooth functions in the context of SDEJs, enabling rigorous derivations of solution properties.

-

A priori estimates: These estimates are essential for proving the boundedness and continuity of solutions, ensuring their existence under controlled conditions.

By systematically applying these techniques, we rigorously prove the existence and uniqueness of solutions to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$. The uniqueness result guarantees that under our framework, the solution obtained is the only possible one, reinforcing the reliability of our approach. Together, these findings provide a strong analytical foundation for the study, allowing us to confidently apply our methodology in subsequent sections to solve specific examples of SDEJs. More precisely, the goal of this section is to study the existence and uniqueness of solutions to one kind of quadratic SDE with jumps described by $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$, where the drift is of the form

$$b_f(s,x,\phi ):=b(s,x)+f\left(x\right){\sigma }^2\left(x\right)-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left(x\right).$$

3.1. Krylov’s Estimates and Itô-Krylov’s Formula for Singular SDEJs

In this subsection, we focus on two fundamental analytical tools: Krylov’s estimates and the Itô-Krylov formula, which play a crucial role in handling stochastic differential equations with jumps (SDEJs) that exhibit singularities in their drift terms. These results provide essential bounds and structural properties that enable the rigorous study of SDEJs under minimal regularity assumptions.

Let us recall Tanaka’s formula. $({\mathfrak{X}}_t,t\ge 0)$ is a real-valued semi-martingale such that ${\sum }_{0\le s\le t}|\Delta {\mathfrak{X}}_s|$ is finite $\mathbb{P}$-a.s. for each $t>0$, and $(L_t^x\left({\mathfrak{X}}_{·}\right),t\ge 0)$ is the local time of X at the level x. If g is the difference of two convex functions, then, for each $t\ge 0$, with a probability of 1, we have

$$\begin{array}{ccc}\hfill g\left({\mathfrak{X}}_t\right)& =& g\left({\mathfrak{X}}_0\right)+{\int }_0^t{g}_{\ell }\prime \left({\mathfrak{X}}_{s-}\right)\mathrm{d}{\mathfrak{X}}_s+{\displaystyle \frac{1}{2}}{\int }_{\mathbb{R}}{L}_t^x\left({\mathfrak{X}}_{·}\right)g_{\ell }″\left(\mathrm{d}x\right)\hfill \\ & & +\sum _{0\le s\le t}g\left({\mathfrak{X}}_s\right)-g\left({\mathfrak{X}}_{s-}\right)-g_{\ell }\prime \left({\mathfrak{X}}_{s-}\right)\Delta {\mathfrak{X}}_s),\hfill \end{array}$$

(meaning that the probability of the set of all $\omega \in \Omega$ for which the above formula does not hold equals 0). Here, $g_{\ell }$ stands for the left, first derivatives of g, and $g_{\ell }$ is a signed measure that is the second derivative of g in the generalized function sense.

3.1.1. Krylov’s Estimates

Krylov’s estimates are powerful probabilistic tools that provide upper bounds on the expected values of functionals of stochastic processes. Specifically, for SDEJs with irregular drifts, these estimates ensure that solutions remain well-controlled despite the presence of singularities. By deriving appropriate moment estimates, we establish conditions under which the solutions remain bounded in expectation, which is a key step in proving their existence and uniqueness.

Proposition 1. 

Let ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ be a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ in the sense of Definition 1. Put

$$\eta =2\underset{0\le t\le T}{sup}\left|{\mathfrak{X}}_t\right|+2{\int }_0^T\left|b(s,{\mathfrak{X}}_s)\right|\mathrm{d}s+2\left(1+e^{4{\left|\mathrm{f}\right|}_1}\right){\int }_0^T{∥\phi ({\mathfrak{X}}_{s-},·)∥}_{\nu ,1}\mathrm{d}s;$$

then, for any measurable and integrable function ϕ on $\mathbb{R}$, we have

$$\mathbb{E}{\int }_0^T\left|\varphi \right|\left({\mathfrak{X}}_s\right)\mathrm{d}{\left[{\mathfrak{X}}_{·}\right]}_s^c\le 2{\left|\varphi \right|}_1{e}^{2{\left|\mathrm{f}\right|}_1}\mathbb{E}\left[\eta \right].$$

(8)

Proof. 

Observe, first, that $\mathbb{E}[{\sum }_{0<s\le T}|\Delta {\mathfrak{X}}_s\left|\right]={\int }_0^T{\int }_E\mathbb{E}\left[\right|\phi ({\mathfrak{X}}_{s-},e)\left|\right]\nu \left(\mathrm{d}e\right)\mathrm{d}s$, which is finite since ${\left(\phi ({\mathfrak{X}}_{s-},·)\right)}_{0\le s\le T}$ belongs to ${\mathcal{M}}_N^2$; hence, ${\sum }_{0\le s\le T}|\Delta {\mathfrak{X}}_s|$ is finite $\mathbb{P}$-a.s. Let a be a real number, set, for notational simplicity, ${\psi }_a\left(x\right)={(x-a)}^{-}$. According to Tanaka’s formula, we have

$$\begin{array}{ccc}\hfill {\psi }_a\left({\mathfrak{X}}_t\right)& =& {\psi }_a\left({\mathfrak{X}}_0\right)+{\int }_0^{t}11_{\{{\mathfrak{X}}_{s-}<a\}}\mathrm{d}{\mathfrak{X}}_s+{\displaystyle \frac{1}{2}}{L}_t^a\left({\mathfrak{X}}_{·}\right)\hfill \\ & & +{\int }_0^t{\int }_E\left({\psi }_a({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-{\psi }_a\left({\mathfrak{X}}_{s-}\right)-11_{\{{\mathfrak{X}}_{s-}<a\}}\phi ({\mathfrak{X}}_{s-},e)\right)\nu \left(\mathrm{d}e\right)\mathrm{d}s\hfill \\ & =& {\psi }_a\left({\mathfrak{X}}_0\right)+M_t+{\displaystyle \frac{1}{2}}{L}_t^a\left({\mathfrak{X}}_{·}\right)\hfill \\ & & +{\int }_0^{t}11_{\{{\mathfrak{X}}_{s-}<a\}}\left(b(s,{\mathfrak{X}}_s)+f\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_{s-}\right)\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t{\int }_E\left({\psi }_a({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-{\psi }_a\left({\mathfrak{X}}_{s-}\right)-11_{\{{\mathfrak{X}}_{s-}<a\}}\phi ({\mathfrak{X}}_{s-},e)\right)\nu \left(\mathrm{d}e\right)\mathrm{d}s\hfill \end{array}$$

where

$$M_t={\int }_0^{t}11_{\{{\mathfrak{X}}_{s-}<a\}}\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_{E}11_{\{{\mathfrak{X}}_{s-}\le a\}}\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)$$

is a martingale. Since the map $x⟼{\psi }_a\left(x\right)$ is one-Lipschitz, it follows that

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{2}}{L}_t^a\left({\mathfrak{X}}_{·}\right)& \le & |{\mathfrak{X}}_t-{\mathfrak{X}}_0|-M_t\hfill \\ & & +{\int }_0^t|b(s,{\mathfrak{X}}_s)+f\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)|11_{\{{\mathfrak{X}}_{s-}\le a\}}\mathrm{d}s\hfill \\ & & +{\int }_0^T\left|{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_s\right)\right|\mathrm{d}s+2{\int }_0^T{∥\phi ({\mathfrak{X}}_{s-},·)∥}_{\nu ,1}\mathrm{d}s,\hfill \end{array}$$

hence,

$$\begin{array}{ccc}\hfill 0\le L_t^a\left({\mathfrak{X}}_{·}\right)& \le & 2\eta -2M_t+2{\int }_{-\infty }^a\left|f\left(x\right)\right|L_t^x\left({\mathfrak{X}}_{·}\right)\mathrm{d}x\hfill \\ & =& 2(\eta -M_t)+2{\int }_{-\overline{\mathfrak{X}}}^a\left|f\left(x\right)\right|L_t^x\left({\mathfrak{X}}_{·}\right)\mathrm{d}x\hfill \end{array}$$

where $\overline{\mathfrak{X}}={max}_{0\le s\le T}{\mathfrak{X}}_s$; we have also used the fact that local time has its support in $[-\overline{\mathfrak{X}},+\overline{\mathfrak{X}}]$, as per estimation (6), the occupation time density formula, and inequality (8). Now, the application of Gronwall’s Lemma gives

$$\begin{array}{ccc}\hfill 0\le L_t^a\left({\mathfrak{X}}_{·}\right)& \le & 2(\eta -M_t)e^{2{\int }_{-\overline{\mathfrak{X}}}^a\left|f\left(x\right)\right|dx}.\hfill \end{array}$$

(9)

Now, by taking the expectation on both sides of inequality (9), and then considering the supremum over the space variable a, we obtain

$$0\le \underset{a}{sup}\mathbb{E}\left[L_t^a\left({\mathfrak{X}}_{·}\right)\right]\le 2\mathbb{E}\left[\eta \right]e^{2{\left|\mathrm{f}\right|}_1}.$$

To complete the proof, let $\varphi$ be a measurable and integrable function on $\mathbb{R}$. The time occupation formula shows that

$$\begin{array}{ccc}\hfill \mathbb{E}{\int }_0^T\left|\varphi \right|\left({\mathfrak{X}}_s\right)\mathrm{d}{\left[{\mathfrak{X}}_{·}\right]}_s^c& =& \mathbb{E}{\int }_{-\infty }^{\infty }\left|\varphi \right|\left(a\right)L_T^a\left({\mathfrak{X}}_{·}\right)\mathrm{d}a\hfill \\ & \le & \underset{a}{sup}\mathbb{E}\left[L_T^a\left({\mathfrak{X}}_{·}\right)\right]{\int }_{-\infty }^{\infty }\left|\varphi \right|\left(a\right)\mathrm{d}a\hfill \\ & \le & 2{\left|\varphi \right|}_1{e}^{2{\left|\mathrm{f}\right|}_1}\mathbb{E}\left[\eta \right].\hfill \end{array}$$

Proposition 1 is proved since $\mathbb{E}\left[\eta \right]$ is finite, thanks to Definition 1. □

3.1.2. Itô-Krylov’s Formula

The Itô-Krylov formula is a generalization of the classical Itô formula, adapted for cases where the drift term lacks smoothness. This extension is particularly useful when dealing with discontinuous or non-differentiable coefficients, as it allows us to handle weak solutions and derive crucial estimates for their behavior. In the context of singular SDEJs, Itô-Krylov’s formula enables us to perform stochastic calculus operations that would otherwise be intractable under standard Itô calculus.

By applying these tools, we develop key intermediary results that will later be used to establish the existence and uniqueness of solutions to the singular SDEJs under consideration. These estimates and formulae provide the necessary analytical structure to control the behavior of the solutions and ensure their well-posedness in challenging stochastic environments.

Theorem 1. 

Let ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ be a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ in the sense of Definition 1. Then, for any function g belonging to the space ${\mathcal{W}}_{1,loc}^2\left(\mathbb{R}\right)$, considering a probability of 1, we have

$$\begin{array}{ccc}\hfill g\left({\mathfrak{X}}_t\right)& =& g\left({\mathfrak{X}}_0\right)+{\int }_0^{t}g\prime \left({\mathfrak{X}}_{s-}\right)\mathrm{d}{\mathfrak{X}}_s+{\displaystyle \frac{1}{2}}{\int }_0^{t}g″\left({\mathfrak{X}}_s\right){\left|\sigma \left({\mathfrak{X}}_s\right)\right|}^2\mathrm{d}s\hfill \\ & & +\sum _{0<s\le t}\left(g\left({\mathfrak{X}}_s\right)-g\left({\mathfrak{X}}_{s-}\right)-g\prime \left({\mathfrak{X}}_{s-}\right)\Delta {\mathfrak{X}}_s\right),\hfill \end{array}$$

(10)

which can be written as

$$\begin{array}{ccc}\hfill g\left({\mathfrak{X}}_t\right)& =& g\left({\mathfrak{X}}_0\right)+{\int }_0^{t}g\prime \left({\mathfrak{X}}_{s-}\right)\mathrm{d}{\mathfrak{X}}_s+{\displaystyle \frac{1}{2}}{\int }_0^{t}g″\left({\mathfrak{X}}_s\right){\left|\sigma \left({\mathfrak{X}}_s\right)\right|}^2\mathrm{d}s\hfill \\ & & +{\int }_0^t{\int }_E\left(g\left({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e)\right)-g\left({\mathfrak{X}}_{s-}\right)-g\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\right)N(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

Proof. 

Let us first note that the functions of ${\mathcal{W}}_{1,loc}^2\left(\mathbb{R}\right)$ have a representative that belongs to ${\mathcal{C}}^1\left(\mathbb{R}\right)$, and this representative will be considered from now on. For $R>|{\mathfrak{X}}_0|$, let us define the stopping time ${\tau }_R$ by

$${\tau }_R:=inf\left\{t>0:max\left(|{\mathfrak{X}}_{t-}|,\underset{e\in E}{sup}\left|{\mathfrak{X}}_{t-}+\phi ({\mathfrak{X}}_{t-},e)\right|\right)\ge R\right\}.$$

Since ${\tau }_R$ tends to infinity as R tends to infinity, it is then enough to establish Equation (10) by replacing t with $t\wedge {\tau }_R$. The stochastic integral ${\int }_0^{t\wedge {\tau }_R}g\prime \left({\mathfrak{X}}_{s-}\right)\mathrm{d}{\mathfrak{X}}_s$ is well-defined since $g\prime$ is continuous and ${\left({\mathfrak{X}}_s\right)}_{0\le s\le T}$ is a càdlàg semi-martingale. Moreover, the jump term

$${\int }_0^{t\wedge {\tau }_R}{\int }_E\left(g\left({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e)\right)-g\left({\mathfrak{X}}_{s-}\right)-g\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\right)N(\mathrm{d}e,\mathrm{d}s)$$

is also well-defined since

$$\begin{array}{ccc}& & {\int }_0^{t\wedge {\tau }_R}{\int }_E\left|\left(g\left({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e)\right)-g\left({\mathfrak{X}}_{s-}\right)-g\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\right)\right|\nu \left(\mathrm{d}e\right)\mathrm{d}s\hfill \\ & \le & \left(M_R+\underset{0\le t\le T}{max}\left|g\prime \left({\mathfrak{X}}_s\right)\right|\right){\int }_0^{t\wedge {\tau }_R}{∥\phi ({\mathfrak{X}}_{s-},·)∥}_{\nu ,1}\mathrm{d}s,\hfill \end{array}$$

and the fact that g and $g\prime$ are locally Lipschitz-continuous functions.

Using Proposition 1, the term ${\int }_0^{t\wedge {\tau }_R}g″\left({\mathfrak{X}}_s\right){\left|\sigma \left({\mathfrak{X}}_s\right)\right|}^2\mathrm{d}s$ is well-defined since

$$\begin{array}{ccc}\hfill \mathbb{E}\left|{\int }_0^{t\wedge {\tau }_R}g″\left({\mathfrak{X}}_s\right){\left|\sigma \left({\mathfrak{X}}_s\right)\right|}^2\mathrm{d}s\right|& =& \mathbb{E}\left|{\int }_{\mathbb{R}}{L}_{t\wedge {\tau }_R}^x\left({\mathfrak{X}}_{·}\right)g″\left(x\right)\mathrm{d}x\right|\hfill \\ & \le & \underset{a}{sup}\mathbb{E}\left[L_T^a\left({\mathfrak{X}}_{·}\right)\right]{\int }_{-R}^R\left|g″\left(x\right)\right|dx.\hfill \end{array}$$

Now, let $g_n$ be a sequence of ${\mathcal{C}}^2$-class functions obtained via a classical regularization using convolution to satisfy the following:

(i) $g_n$ converges uniformly to g in the interval $[-R,R]$.

(ii) $g_n$ converges uniformly to $g\prime$ in the interval $[-R,R]$

(iii) $g_n$ converges in $L^1\left([-R,R]\right)$ to $g″$.

The classical Itô’s formula applied to $g_n\left({\mathfrak{X}}_{t\wedge {\tau }_R}\right)$ gives

$$\begin{array}{ccc}\hfill g_n\left({\mathfrak{X}}_{t\wedge {\tau }_R}\right)& =& g_n\left({\mathfrak{X}}_0\right)+{\int }_0^{t\wedge {\tau }_R}{g}_n\prime \left({\mathfrak{X}}_{s-}\right)\mathrm{d}{\mathfrak{X}}_s+{\displaystyle \frac{1}{2}}{\int }_0^{t\wedge {\tau }_R}{g}_n″\left({\mathfrak{X}}_s\right){\left|\sigma \left({\mathfrak{X}}_s\right)\right|}^2\mathrm{d}s\hfill \\ & & +{\int }_0^{t\wedge {\tau }_R}{\int }_E\left(g_n\left({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e)\right)-g_n\left({\mathfrak{X}}_{s-}\right)-g_n\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\right)N(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

(11)

Now, we pass the limit on n in Equation (11), and then use the above properties $\left(i\right)$, $\left(ii\right)$, and $\left(iii\right)$ and Proposition 1 to obtain

$$\begin{array}{ccc}\hfill g\left({\mathfrak{X}}_{t\wedge {\tau }_R}\right)& =& g\left({\mathfrak{X}}_0\right)+{\int }_0^{t\wedge {\tau }_R}g\prime \left({\mathfrak{X}}_{s-}\right)\mathrm{d}{\mathfrak{X}}_s+{\displaystyle \frac{1}{2}}{\int }_0^{t\wedge {\tau }_R}g″\left({\mathfrak{X}}_s\right){\left|\sigma \left({\mathfrak{X}}_s\right)\right|}^2\mathrm{d}s\hfill \\ & & +{\int }_0^{t\wedge {\tau }_R}{\int }_E\left(g\left({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e)\right)-g\left({\mathfrak{X}}_{s-}\right)-g\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\right)N(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

This achieves the proof. □

Application of Theorem 1

In the following, we present an application of Theorem 1, which demonstrates how a stochastic differential equation with jumps (SDEJs) involving singular or irregular terms can be transformed into an equivalent SDEJ with more regular and well-behaved coefficients. This transformation is particularly useful for both theoretical analysis and practical computations. From this point onward, various forms of the Itô-Krylov formula will be employed frequently throughout the remainder of the paper to facilitate such transformations and related derivations.

For given coefficients a, $\sigma$, and $\phi$, let ${\mathfrak{X}}_{·}$ be a solution to the SDEJs

$${\mathfrak{X}}_t=x+{\int }_0^{t}a(s,{\mathfrak{X}}_s,\phi ({\mathfrak{X}}_{s-},·))\mathrm{d}s+{\int }_0^t\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e).$$

(12)

Let $\mathrm{F}$ be the function defined in Equation (2); from Lemma 1, $\mathrm{F}$ belongs to ${\mathcal{W}}_{1,loc}^2\left(\mathbb{R}\right)$; hence, Itô-Krylov’s formula, given by Equation (10), with Theorem 1 applied to $\mathrm{F}\left({\mathfrak{X}}_t\right)$, leads to

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\mathfrak{X}}_t\right)& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\left(\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)\left(a(s,{\mathfrak{X}}_s,\phi ({\mathfrak{X}}_{s-},·))\right)+{\displaystyle \frac{1}{2}}\mathrm{F}″\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)\hfill \\ & & +\sum _{0<s\le t}\left(\mathrm{F}\left({\mathfrak{X}}_s\right)-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)-\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)\Delta {\mathfrak{X}}_s\right).\hfill \end{array}$$

Furthermore,

$$\begin{array}{ccc}& & \sum _{0<s\le t}\left(\mathrm{F}\left({\mathfrak{X}}_s\right)-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)-\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)\Delta {\mathfrak{X}}_s\right)\hfill \\ & =& {\int }_0^t{\int }_E\left(\mathrm{F}\left({\mathfrak{X}}_s\right)-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)-\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)\phi ({\mathfrak{X}}_{s-},e)\right)N(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

Remember, also, that

$$\mathrm{F}\prime \left(x\right){\mathcal{I}}_{\mathrm{F}}\left(u\right)\left(x\right)={\int }_E\left(\mathrm{F}\left(x+u\left(e\right)\right)-\mathrm{F}\left(x\right)-\mathrm{F}\prime \left(x\right)u\left(e\right)\right)\nu \left(\mathrm{d}e\right).$$

This implies

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\mathfrak{X}}_t\right)& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\left(a(s,{\mathfrak{X}}_s,\phi ({\mathfrak{X}}_s,·))+{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_s\right)\right)\mathrm{d}s\hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_0^t\mathrm{F}″\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)\mathrm{d}s+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +{\int }_0^t{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

(13)

In particular, we have

case 1: 

if

$$a(s,x)=b_f(s,x)=b(s,x)+f\left(x\right){\sigma }^2\left(x\right),$$

we obtain

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\mathfrak{X}}_t\right)& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\left(b\left(s,{\mathfrak{X}}_s\right)+{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_s\right)\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +{\int }_0^t{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

Now, let us set

$$\begin{array}{ccc}\hfill \overline{b}(s,x,\overline{\phi }(·,·)):& =& \mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)(b(s,{\mathrm{F}}^{-1}\left(x\right))+{\mathcal{I}}_{\mathrm{F}}\left(\overline{\phi }\right)\left({\mathrm{F}}^{-1}\left(x\right)\right)),\hfill \\ \hfill \overline{\sigma }\left(x\right):& =& \mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)\sigma \left({\mathrm{F}}^{-1}\left(x\right)\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \overline{\phi }(x,e):& =& \mathrm{F}({\mathrm{F}}^{-1}\left(x\right)+\phi ({\mathrm{F}}^{-1}\left(x\right),e))-x.\hfill \end{array}$$

case 2: 

if $a(s,x,\phi (x,·)):=b_f(s,x,\phi (x,·))=b(s,x)+f\left(x\right){\sigma }^2\left(x\right)-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left(x\right)$, we obtain

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\mathfrak{X}}_t\right)& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)b\left(s,{\mathfrak{X}}_s\right)\mathrm{d}s+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +{\int }_0^t{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e),\hfill \end{array}$$

which can be written as

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\mathfrak{X}}_t\right)& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)b\left(s,{\mathfrak{X}}_s\right)\mathrm{d}s+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +{\int }_0^t{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

(14)

For

$$\overline{b}(s,x):=\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)b(s,{\mathrm{F}}^{-1}\left(x\right)),\overline{\sigma }\left(x\right):=\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)\sigma \left({\mathrm{F}}^{-1}\left(x\right)\right)$$

and

$$\overline{\phi }(x,e):=\mathrm{F}({\mathrm{F}}^{-1}\left(x\right)+\phi ({\mathrm{F}}^{-1}\left(x\right),e))-x.$$

That is, in both cases, ${\mathfrak{Z}}_{·}$ is a solution to $\mathrm{Eq}({\mathfrak{Z}}_0,\overline{b},\overline{\sigma },\overline{\phi })$.

These notations will be used repeatedly for the remainder of this paper.

3.2. SDEJs with Integrable Generator

The main result of this section is given by Theorem 2 below. For ${\mathfrak{Z}}_0=\mathrm{F}\left({\mathfrak{X}}_0\right)$, we shall refer to the following SDE with jumps as $\mathrm{Eq}({\mathfrak{Z}}_0,\overline{b},\overline{\sigma },\overline{\phi })$, but without the singular term:

$${\mathfrak{Z}}_t={\mathfrak{Z}}_0+{\int }_0^t\overline{b}\left(s,{\mathfrak{Z}}_s\right)\mathrm{d}s+{\int }_0^t\overline{\sigma }\left({\mathfrak{Z}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\overline{\phi }({\mathfrak{Z}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)$$

Theorem 2. 

Assume that f is an integrable function on the whole space $\mathbb{R}$; then, ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ if and only if ${\left({\mathfrak{Z}}_t=\mathrm{F}\left({\mathfrak{X}}_t\right)\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}({\mathfrak{Z}}_0,\overline{b},\overline{\sigma },\overline{\phi }).$

Proof. 

If ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$, then Equation (14) shows that ${\left({\mathfrak{Z}}_t\right)}_{0\le t\le T}$ satisfies $\mathrm{Eq}({\mathfrak{Z}}_0,\overline{b},\overline{\sigma },\overline{\phi })$. Indeed, for $f\in L^1\left(\mathbb{R}\right)$, let ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ be a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi ))$ in the sense of Definition 1; then, from inequality (4), we have

$$\begin{array}{ccc}\hfill {\int }_0^T|\overline{b}(s,{\mathfrak{Z}}_s){|}^2\mathrm{d}s& =& {\int }_0^T|\mathrm{F}\prime \left({\mathfrak{X}}_s\right)b(s,{\mathfrak{X}}_s){|}^2\mathrm{d}s\hfill \\ & \le & e^{{4\left|\mathrm{f}\right|}_1}{\int }_0^T{\left|b(s,{\mathfrak{X}}_s)\right|}^2\mathrm{d}s.\hfill \end{array}$$

Similarly, we have

$$\begin{array}{ccc}\hfill {\int }_0^T|\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\sigma \left({\mathfrak{X}}_s\right){|}^2\mathrm{d}s& \le & e^{{4\left|\mathrm{f}\right|}_1}{\int }_0^T|\sigma \left({\mathfrak{X}}_s\right){|}^2\mathrm{d}s.\hfill \end{array}$$

The jump term

$${\int }_0^T{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}s)$$

is well-defined as soon as the term

$${\int }_0^T{\int }_E\left|\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\right|\nu \left(\mathrm{d}e\right)\mathrm{d}s$$

is finite. However, thanks, again, to inequality (4), we have

$$\begin{array}{ccc}\hfill {\int }_0^T{\int }_E{\left|\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\right|}^2\nu \left(\mathrm{d}e\right)\mathrm{d}s& \le & e^{4{\left|\mathrm{f}\right|}_1}{\int }_0^T{\int }_E{\left|\phi ({\mathfrak{X}}_{s-},e)\right|}^2\nu \left(\mathrm{d}e\right)\mathrm{d}s.\hfill \end{array}$$

It follows that from the above estimates, ${\mathfrak{Z}}_{·}=\mathrm{F}\left({\mathfrak{X}}_{·}\right)\in {\mathcal{S}}^2$, $\overline{b}(·,{\mathfrak{Z}}_{·})=\mathrm{F}\prime \left({\mathfrak{X}}_{·}\right)b\left(s,{\mathfrak{X}}_{·}\right)\in {\mathcal{M}}_W^2$, $\overline{\sigma }\left({\mathfrak{Z}}_{·}\right)=\mathrm{F}\prime \left({\mathfrak{X}}_{·}\right)\sigma \left({\mathfrak{X}}_{·}\right)\in {\mathcal{M}}_W^2$ and $\overline{\phi }({\mathfrak{Z}}_{·-},e):=\mathrm{F}({\mathfrak{X}}_{·-}+\phi ({\mathfrak{X}}_{·-},e))-\mathrm{F}\left({\mathfrak{X}}_{·-}\right)\in {\mathcal{M}}_N^2$; that is, ${\left({\mathfrak{Z}}_t\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}(\mathrm{F}\left(x\right),\overline{b},\overline{\sigma },\overline{\phi })$ in the sense of Definition 1.

• Conversely, let ${\left({\mathfrak{Z}}_t\right)}_{0\le t\le T}$ be a solution to $\mathrm{Eq}({\mathfrak{Z}}_0,\overline{b},\overline{\sigma },\overline{\phi })$; then, Itô-Krylov’s formula given in Equation (10) applied to ${\mathfrak{X}}_t={\mathrm{F}}^{-1}\left({\mathfrak{Z}}_t\right)$ (since ${\mathrm{F}}^{-1}$ belongs to ${\mathcal{W}}_{1,loc}^2\left(\mathbb{R}\right)$) shows that

$$\begin{array}{ccc}\hfill {\mathrm{F}}^{-1}\left({\mathfrak{Z}}_t\right)& =& {\mathrm{F}}^{-1}\left({\mathfrak{Z}}_0\right)+{\int }_0^t\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{b}\left(s,{\mathfrak{Z}}_s\right)\mathrm{d}s\hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_0^t\left({\mathrm{F}}^{-1}\right)″\left({\mathfrak{Z}}_s\right){\overline{\sigma }}^2\left({\mathfrak{Z}}_s\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t{\int }_E\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\phi }({\mathfrak{Z}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)\hfill \\ & & +{\int }_0^t\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\sigma }\left({\mathfrak{Z}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +\sum _{0<s\le t}\left({\mathrm{F}}^{-1}\left({\mathfrak{Z}}_s\right)-{\mathrm{F}}^{-1}\left({\mathfrak{Z}}_{s-}\right)-\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\Delta {\mathfrak{Z}}_s\right).\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_t& =& {\mathfrak{X}}_0+{\int }_0^t\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{b}\left(s,{\mathfrak{Z}}_s\right)\mathrm{d}s+{\displaystyle \frac{1}{2}}{\int }_0^t\left({\mathrm{F}}^{-1}\right)″\left({\mathfrak{Z}}_s\right){\overline{\sigma }}^2\left({\mathfrak{Z}}_s\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\sigma }\left({\mathfrak{Z}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\phi }({\mathfrak{Z}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)\hfill \\ & & +{\int }_0^t{\int }_E\left({\mathrm{F}}^{-1}({\mathfrak{Z}}_{s-}+\overline{\phi }({\mathfrak{Z}}_{s-},e))-{\mathrm{F}}^{-1}\left({\mathfrak{Z}}_{s-}\right)-\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\phi }({\mathfrak{Z}}_{s-},e)\right)N(\mathrm{d}s,\mathrm{d}e)\hfill \\ & =& {\mathfrak{X}}_0+{\int }_0^t\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{b}\left(s,{\mathfrak{Z}}_s\right)\mathrm{d}s+{\displaystyle \frac{1}{2}}{\int }_0^t\left({\mathrm{F}}^{-1}\right)″\left({\mathfrak{Z}}_s\right){\overline{\sigma }}^2\left({\mathfrak{Z}}_s\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\sigma }\left({\mathfrak{Z}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +{\int }_0^t{\int }_E\left({\mathrm{F}}^{-1}({\mathfrak{Z}}_{s-}+\overline{\phi }({\mathfrak{Z}}_{s-},e))-{\mathrm{F}}^{-1}\left({\mathfrak{Z}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e)\hfill \\ & & +{\int }_0^t{\int }_E\left({\mathrm{F}}^{-1}({\mathfrak{Z}}_{s-}+\overline{\phi }({\mathfrak{Z}}_{s-},e))-{\mathrm{F}}^{-1}\left({\mathfrak{Z}}_{s-}\right)-\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\phi }({\mathfrak{Z}}_{s-},e)\right)\nu \left(\mathrm{d}e\right)\mathrm{d}s.\hfill \end{array}$$

(15)

Notice that

$$\left({\mathrm{F}}^{-1}\right)\prime \left(x\right)={\displaystyle \frac{1}{\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)}}\mathrm{and}\left({\mathrm{F}}^{-1}\right)″\left(x\right)=-{\displaystyle \frac{\mathrm{F}″\left({\mathrm{F}}^{-1}\left(x\right)\right)}{{(\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right))}^3}}.$$

(16)

Set

$$b\left(s,{\mathfrak{X}}_s\right)=\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{b}\left(s,{\mathfrak{Z}}_s\right),\sigma \left({\mathfrak{X}}_s\right)=\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\sigma }\left({\mathfrak{Z}}_s\right),$$

and

$$\phi ({\mathfrak{X}}_{s-},e)={\mathrm{F}}^{-1}({\mathfrak{Z}}_{s-}+\overline{\phi }({\mathfrak{Z}}_{s-},e))-{\mathrm{F}}^{-1}\left({\mathfrak{Z}}_{s-}\right).$$

This implies, thanks to identity (16), that

$$\begin{array}{ccc}\hfill \left({\mathrm{F}}^{-1}\right)″\left({\mathfrak{Z}}_{s-}\right){\overline{\sigma }}^2\left({\mathfrak{Z}}_s\right)& =& -{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{F}″\left({\mathfrak{X}}_s\right)}{{(\mathrm{F}\prime \left({\mathfrak{X}}_s\right))}^3}}{\displaystyle \frac{{\sigma }^2\left({\mathfrak{X}}_s\right)}{{(\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right))}^2}}\hfill \\ & \stackrel{\mathrm{d}s\mathrm{a}.\mathrm{e}.}{=}& -{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{F}″\left({\mathfrak{X}}_s\right)}{\mathrm{F}\prime \left({\mathfrak{X}}_s\right)}}{\sigma }^2\left({\mathfrak{X}}_s\right)=f\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right).\hfill \end{array}$$

(17)

where, we have used the fact that $\mathrm{F}$ satisfies Equation (3). In addition, we have

$$\begin{array}{ccc}& & {\int }_E\left({\mathrm{F}}^{-1}({\mathfrak{Z}}_{s-}+\overline{\phi }({\mathfrak{Z}}_{s-},e))-{\mathrm{F}}^{-1}\left({\mathfrak{Z}}_{s-}\right)-\left({\mathrm{F}}^{-1}\right)\prime \left({\mathfrak{Z}}_{s-}\right)\overline{\phi }({\mathfrak{Z}}_{s-},e)\right)\nu \left(\mathrm{d}e\right)\hfill \\ & =& {\int }_E\left(\phi ({\mathfrak{X}}_{s-},e)-{\displaystyle \frac{1}{\mathrm{F}\prime \left({\mathfrak{X}}_s\right)}}\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\right)\nu \left(\mathrm{d}e\right)\hfill \\ & =& -{\int }_E\left({\displaystyle \frac{\mathrm{F}\left({\mathfrak{X}}_s\right)-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)-\mathrm{F}\prime \left({\mathfrak{X}}_s\right)\phi ({\mathfrak{X}}_{s-},e)}{\mathrm{F}\prime \left({\mathfrak{X}}_s\right)}}\right)\nu \left(\mathrm{d}e\right)\hfill \\ & =& -{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_{s-}\right).\hfill \end{array}$$

(18)

By substituting expressions (17) and (18) in Equation (15), we end up with

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_t& =& {\mathfrak{X}}_0+{\int }_0^t\left(b(s,{\mathfrak{X}}_s)+f\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_{s-}\right)\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

Observe, again, that thanks to inequalities (4) and (5), we show easily that

$$\left|{\mathfrak{X}}_s\right|\le e^{{2\left|\mathrm{f}\right|}_1}\left|{\mathfrak{Z}}_s\right|,\left|\sigma \left({\mathfrak{X}}_s\right)\right|\le e^{{2\left|\mathrm{f}\right|}_1}\left|{\mathfrak{Z}}_s\right|\mathrm{and}\left|\phi ({\mathfrak{X}}_{s-},e)\right|\le e^{{2\left|\mathrm{f}\right|}_1}\left|\overline{\phi }({\mathfrak{Z}}_{s-},e)\right|.$$

Consequently,

$${\left({\mathfrak{X}}_s\right)}_{0\le s\le T}:={\left({\mathrm{F}}^{-1}\left({\mathfrak{Z}}_s\right)\right)}_{0\le s\le T}$$

is a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ in the sense of Definition 1. □

Let

$$b:[0,T]\times \mathbb{R}⟶\mathbb{R},\sigma :\mathbb{R}⟶\mathbb{R}\mathrm{and}\phi :[0,T]\times \mathbb{R}\times E⟶\mathbb{R}$$

satisfy the following assumptions:

(A_3.1)

There exists a $L>0$ such that, for all $s\in [0,T]$, $x,y\in \mathbb{R}$,

$$\left|b(s,x)-b\left(s,y\right)\right|+\left|\sigma \left(x\right)-\sigma \left(x\right)\right|\le L\left|x-y\right|,$$

(A_3.2)

There exists a function $\rho :E⟼{\mathbb{R}}^{+}$ with ${\int }_E{\rho }^2\left(e\right)\nu \left(\mathrm{d}e\right)<+\infty$ such that, for any $x,y\in \mathbb{R}$ and $e\in E$,

$$\left|\phi (x,e)-\phi \left(y,e\right)\right|\le \rho \left(e\right)\left|x-y\right|\mathrm{and}\left|\phi (0,e)\right|\le \rho \left(e\right).$$

(A_3.3)

The functions b and $\sigma$ are bounded functions.

• It is clear that the above conditions imply that b, $\sigma$, and $\phi$ satisfy the global linear growth conditions; that is, there exists $C>0$ such that, for all $0\le s\le T,x\in \mathbb{R}$,

$$\left|b\right(s,x\left)\right|+\left|\sigma \right(x\left)\right|\le C(1+|x\left|\right)\mathrm{and}\left|\phi \right(x,e\left)\right|\le \rho \left(e\right)(1+|x\left|\right).$$

Theorem 3. 

Let f be a bounded and integrable function on $\mathbb{R}$ and b, σ, and φ satisfy $\left({\mathbf{A}}_{3.1}\right)-\left({\mathbf{A}}_{3.3}\right)$; then, Equation (1) has a unique solution.

Proof. 

From Theorem 2, ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b_f,\sigma ,\phi )$ if and only if ${\left({\mathfrak{Z}}_t=\mathrm{F}\left({\mathfrak{X}}_t\right)\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),\overline{b},\overline{\sigma },\overline{\phi })$ where

$$\begin{array}{ccc}\hfill \overline{b}(s,x):& =& \mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)b(s,{\mathrm{F}}^{-1}\left(x\right)),\hfill \\ \hfill \overline{\sigma }\left(x\right):& =& \mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)\sigma \left({\mathrm{F}}^{-1}\left(x\right)\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \overline{\phi }(x,e):& =& \mathrm{F}({\mathrm{F}}^{-1}\left(x\right)+\phi ({\mathrm{F}}^{-1}\left(x\right),e))-x.\hfill \end{array}$$

It remains to be proven that $\overline{b}$, $\overline{\sigma }$, and $\overline{\phi }$ satisfy $\left({\mathbf{A}}_{3.1}\right)-\left({\mathbf{A}}_{3.3}\right)$.

Lipschitz property of $\overline{b}$:

$$\begin{array}{ccc}\hfill |\overline{b}(s,x)-\overline{b}(s,y)|& =& |\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)b(s,{\mathrm{F}}^{-1}\left(x\right))-\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(y\right)\right)b(s,{\mathrm{F}}^{-1}\left(y\right))|\hfill \\ & \le & |\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)||b(s,{\mathrm{F}}^{-1}\left(x\right))-b(s,{\mathrm{F}}^{-1}\left(y\right))|\hfill \\ & & +|\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)-\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(y\right)\right)|\left|b(s,{\mathrm{F}}^{-1}\left(y\right))\right|.\hfill \end{array}$$

Remember that $\mathrm{F}\prime$ is bounded thanks to inequality (5), and the function b is bounded by assumption ${\mathbf{A}}_{3.3}$; therefore,

$$|\overline{b}(s,x)-\overline{b}(s,y)|\le C|{\mathrm{F}}^{-1}\left(x\right)-{\mathrm{F}}^{-1}\left(y\right)|\le C{e}^{{2\left|f\right|}_1}|x-y|,$$

here, we have used the Lipschitz properties of b, $\mathrm{F}\prime$, and inequality (5).

Lipschitz property of $\overline{\sigma }$:

$$\begin{array}{ccc}\hfill |\overline{\sigma }\left(x\right)-\overline{\sigma }\left(y\right)|& =& |\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)\sigma \left({\mathrm{F}}^{-1}\left(x\right)\right)-\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(y\right)\right)\sigma \left({\mathrm{F}}^{-1}\left(y\right)\right)|\hfill \\ & \le & |\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)||\sigma \left({\mathrm{F}}^{-1}\left(x\right)\right)-\sigma \left({\mathrm{F}}^{-1}\left(y\right)\right)|\hfill \\ & & +|\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)-\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(y\right)\right)|\left|\sigma \left({\mathrm{F}}^{-1}\left(y\right)\right)\right|.\hfill \end{array}$$

Again, according to the Lipschitz properties of $\sigma$, $\mathrm{F}\prime$, and inequality (5), we deduce that

$$|\overline{\sigma }\left(x\right)-\overline{\sigma }\left(y\right)|\le C\left|{\mathrm{F}}^{-1}\left(x\right))-{\mathrm{F}}^{-1}\left(y\right)\right|\le C{e}^{{2\left|f\right|}_1}|x-y|.$$

Lipschitz property of $\overline{\phi }$:

$$|\overline{\phi }(x,e)-\overline{\phi }(y,e)|=|\mathrm{F}({\mathrm{F}}^{-1}\left(x\right)+\phi ({\mathrm{F}}^{-1}\left(x\right),e))-\mathrm{F}({\mathrm{F}}^{-1}\left(y\right)+\phi ({\mathrm{F}}^{-1}\left(y\right),e))|+|x-y|.$$

The Lipschitz properties of $\mathrm{F}$ and $\phi$ imply

$$\begin{array}{ccc}\hfill |\overline{\phi }(x,e)-\overline{\phi }(y,e)|& \le & e^{{2\left|f\right|}_1}|{\mathrm{F}}^{-1}\left(x\right)-{\mathrm{F}}^{-1}\left(y\right)|+|x-y|\hfill \\ & & +e^{{2\left|f\right|}_1}\left|\phi ({\mathrm{F}}^{-1}\left(x\right),e))-\phi ({\mathrm{F}}^{-1}\left(y\right),e)\right|\hfill \\ & \le & e^{{2\left|f\right|}_1}(1+C)|{\mathrm{F}}^{-1}\left(x\right)-{\mathrm{F}}^{-1}\left(y\right)|+|x-y|\hfill \\ & \le & \left(e^{{4\left|f\right|}_1}(1+C)+1\right)|x-y|,\hfill \end{array}$$

again, we made use of the Lipschitz property of ${\mathrm{F}}^{-1}$ and inequality (5). □

4. Solvability of Stochastic Differential Equations with Jumps

In this section, we apply the theoretical results developed in Section 3 to analyze the solvability of stochastic differential equations with jumps (SDEJs). Our focus is on demonstrating the practical applicability of the existence and uniqueness results by considering multiple examples featuring different generators of a quadratic form in x due to the presence of $f\left(x\right){\sigma }^2\left(x\right)$. These examples illustrate how our approach extends beyond the existing methods in the literature and provides new insights into handling SDEJs with irregular dynamics.

We leverage the transformation techniques and estimates established in the previous section to construct explicit and approximate solutions under various conditions. The influence of discontinuous drifts and jump processes on the solvability of the equations is carefully examined. Furthermore, we compare our approach to traditional methods, highlighting cases where prior techniques fail to provide solutions, thus emphasizing the robustness and generality of our framework.

To systematically study these equations, we set ${\mathfrak{X}}_0$ as the initial condition and proceed to solve several SDEJs in the following examples. Each example presents a specific scenario where our method proves effective in overcoming the analytical challenges associated with singular drifts and jump processes. Through these case studies, we provide a deeper understanding of how the theoretical results can be applied to real-world stochastic systems.

1.

Example 1: $\mathrm{Eq}({\mathfrak{X}}_0,f{\sigma }^2-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left(x\right),\sigma ,\phi )$.

This example corresponds to the case where $b=0$, consequently, via Theorem 2 $\mathrm{Eq}({\mathfrak{X}}_0,f{\sigma }^2-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left(x\right),\sigma ,\phi )$, admits a (unique) solution if and only if $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),0,\overline{\sigma },\overline{\phi })$ admits a (unique) solution. Clearly, $\overline{\sigma }$ and $\overline{\phi }$ satisfy $\left({\mathbf{A}}_{3.1}\right)-\left({\mathbf{A}}_{3.3}\right)$ since $\mathrm{F}\prime$ is bounded; thus, $\mathrm{Eq}(\mathrm{F}\left(x\right),0,\overline{\sigma },\overline{\phi })$ admits a (unique) solution, and so is $\mathrm{Eq}({\mathfrak{X}}_0,f{\sigma }^2-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right),\sigma ,\phi ).$

2.

Example 2: $\mathrm{Eq}({\mathfrak{X}}_0,b+f{\sigma }^2,\sigma ,\phi )$:

Equation (13) corresponding to the coefficients $b+f{\sigma }^2$ and $\sigma$ shows that ${\left({\mathfrak{X}}_t\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}({\mathfrak{X}}_0,b+f{\sigma }^2,\sigma ,\phi )$ if and only if the stochastic process ${\left({\mathfrak{Z}}_t=\mathrm{F}\left({\mathfrak{X}}_t\right)\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),\overline{b},\overline{\sigma },\overline{\phi })$ where

$$\begin{array}{ccc}\hfill \overline{b}(s,x):& =& \mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)b(s,{\mathrm{F}}^{-1}\left(x\right))+{\mathcal{I}}_{\mathrm{F}}\left(\overline{\phi }\right)\left(x\right),\hfill \\ \hfill \overline{\sigma }\left(x\right):& =& \mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)\sigma \left({\mathrm{F}}^{-1}\left(x\right)\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \overline{\phi }(x,e):& =& \mathrm{F}({\mathrm{F}}^{-1}\left(x\right)+\phi ({\mathrm{F}}^{-1}\left(x\right),e))-x.\hfill \end{array}$$

Now, thanks to assumptions $\left({\mathbf{A}}_{3.1}\right)$–$\left({\mathbf{A}}_{3.3}\right)$, $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),\overline{b},\overline{\sigma },\overline{\phi })$ has a unique solution whenever the coefficients $\overline{b}$, $\overline{\sigma }$, and $\overline{\phi }$ satisfy $\left({\mathbf{A}}_{3.1}\right)$–$\left({\mathbf{A}}_{3.3}\right)$. This can be verified easily under these assumptions.

3.

Example 3: $\mathrm{Eq}({\mathfrak{X}}_0,b+f{\sigma }^2+{\int }_E\phi (·,e)\nu \left(de\right),\sigma ,\phi )$

Consider the following SDEJ:

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_t& =& x+{\int }_0^t\left(b\left({\mathfrak{X}}_s\right)+f\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)+{\int }_E\phi ({\mathfrak{X}}_s,e)\nu \left(\mathrm{d}e\right)\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e).\hfill \end{array}$$

By taking $a(x,\phi ):=b\left(x\right)+f\left(x\right){\sigma }^2\left(x\right)+{\int }_E\phi (x,e)\nu \left(\mathrm{d}e\right)$ in (13), we obtain

$$\begin{array}{ccc}\hfill \mathrm{F}\left({\mathfrak{X}}_t\right)& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)b\left({\mathfrak{X}}_s\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\nu \left(\mathrm{d}e\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\mathrm{F}\prime \left({\mathfrak{X}}_{s-}\right)\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s\hfill \\ & & +{\int }_0^t{\int }_E\left(\mathrm{F}({\mathfrak{X}}_{s-}+\phi ({\mathfrak{X}}_{s-},e))-\mathrm{F}\left({\mathfrak{X}}_{s-}\right)\right)\tilde{N}(\mathrm{d}s,\mathrm{d}e),\hfill \end{array}$$

or, equivalently,

$$\begin{array}{ccc}\hfill {\mathfrak{Z}}_t& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t\left(\overline{b}\left({\mathfrak{Z}}_s\right)+{\int }_E\overline{\phi }({\mathfrak{Z}}_{s-},e)\nu \left(\mathrm{d}e\right)\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\overline{\sigma }\left({\mathfrak{Z}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\overline{\phi }({\mathfrak{Z}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)\hfill \end{array}$$

since the coefficients $\overline{b}$, $\overline{\sigma }$, and $\overline{\phi }$ satisfy $\left({\mathbf{A}}_{3.1}\right)$–$\left({\mathbf{A}}_{3.3}\right)$, then the singular SDEJ

$$\mathrm{Eq}(\mathrm{F}\left(x\right),\overline{b}+{\int }_E\overline{\phi }(·,e)\nu \left(de\right),\overline{\sigma },\overline{\phi })$$

has a unique solution.

4.

Example 4: $\mathrm{Eq}({\mathfrak{X}}_0,f{\sigma }^2,\sigma ,\phi )$

Consider the following SDEJ:

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_t& =& {\mathfrak{X}}_0+{\int }_0^{t}f\left({\mathfrak{X}}_s\right){\sigma }^2\left({\mathfrak{X}}_s\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\sigma \left({\mathfrak{X}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e),\hfill \end{array}$$

By using, again, the space transformation $\mathrm{F}$, we obtain another SDEJ:

$$\begin{array}{ccc}\hfill {\mathfrak{Z}}_t& =& \mathrm{F}\left({\mathfrak{X}}_0\right)+{\int }_0^t{\int }_0^t\overline{b}\left({\mathfrak{Z}}_s\right)\mathrm{d}s\hfill \\ & & +{\int }_0^t\overline{\sigma }\left({\mathfrak{Z}}_s\right)\mathrm{d}{W}_s+{\int }_0^t{\int }_E\overline{\phi }({\mathfrak{Z}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e),\hfill \end{array}$$

where

$$\overline{b}\left(x\right)={\int }_E\left(\mathrm{F}\prime \left({\mathrm{F}}^{-1}\left(x\right)\right)\left({\mathrm{F}}^{-1}(x+\overline{\phi }(x,e))-{\mathrm{F}}^{-1}\left(x\right)\right)-\overline{\phi }(x,e)\right)\nu \left(\mathrm{d}e\right).$$

That is, ${\mathfrak{Z}}_{·}$ is a solution to $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),\overline{b},\overline{\sigma },\overline{\phi })$. Now, simple computations show that $\overline{b}$, $\overline{\sigma }$, and $\overline{\phi }$ satisfy $\left({\mathbf{A}}_{3.1}\right)-\left({\mathbf{A}}_{3.3}\right)$; hence, $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),\overline{b},\overline{\sigma },\overline{\phi })$ has a unique solution. Finally, $\mathrm{Eq}({\mathfrak{X}}_0,f{\sigma }^2,\sigma ,\phi )$ has a unique solution.

5.

Example 5: $\mathrm{Eq}({\mathfrak{X}}_0,-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right),0,\phi )$.

Consider the following equation:

$$\begin{array}{ccc}\hfill {\mathfrak{X}}_t& =& {\mathfrak{X}}_0-{\int }_0^t{\mathcal{I}}_{\mathrm{F}}\left(\phi \right)\left({\mathfrak{X}}_s\right)\mathrm{d}s+{\int }_0^t{\int }_E\phi ({\mathfrak{X}}_{s-},e)\tilde{N}(\mathrm{d}s,\mathrm{d}e)\hfill \end{array}$$

This equation corresponds to the case where $\sigma \equiv 0$; clearly, this model is covered by Example 1 but deserves particular attention. Indeed, applying Itô’s formula to $\mathrm{F}\left({\mathfrak{X}}_t\right)$ shows that ${\left({\mathfrak{Z}}_t=\mathrm{F}\left({\mathfrak{X}}_t\right)\right)}_{0\le t\le T}$ is a solution to $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),0,0,\overline{\phi })$, where

$$\overline{\phi }(x,e):=\mathrm{F}\left({\mathrm{F}}^{-1}\left(x\right)+\phi ({\mathrm{F}}^{-1}\left(x\right),e))-x\right).$$

Therefore, thanks to Theorem 2, $\mathrm{Eq}({\mathfrak{X}}_0,-{\mathcal{I}}_{\mathrm{F}}\left(\phi \right),0,\phi )$ is equivalent to $\mathrm{Eq}(\mathrm{F}\left({\mathfrak{X}}_0\right),0,0,\overline{\phi })$, which, in turn, admits a unique solution as soon as $\overline{\phi }$ satisfies $\left({\mathbf{A}}_{3.2}\right)$.

5. Conclusions

This paper examines the well-posedness of jump-diffusion stochastic differential equations (SDEs) driven by stochastic processes with jumps and irregular coefficients. The drift coefficient may exhibit quadratic behavior with respect to the Lebesgue measure and/or exponential growth relative to the Lévy measure governing the jumps. An intermediate result, leveraging Krylov’s estimate, is also established. These findings facilitate the study of indirect approximation schemes for SDEs with jumps and drifts that are merely measurable and integrable. The transformed equations lack singular terms, and all resulting coefficients are Lipschitz-continuous, enabling an extension of the results presented in [15] to the context of jumps.

In future research, a continuation of the results obtained in [14] will be studied; we aim to further investigate the efficiency of various numerical schemes for solving irregular stochastic differential equations with jumps (ISDEJs) that have integrable and measurable drifts. Building on the approach of applying the space transformation $\mathrm{F}$, we will explore its impact on different classes of ISDEJs with discontinuous drifts. This transformation eliminates the singularity in the drift, converting the problem into a standard SDEJ that can be analyzed and, in some cases, solved explicitly.

We will systematically apply and evaluate numerical schemes, including the Euler-Maruyama, Milstein, and Runge-Kutta methods, in relation to the transformed SDEJ to approximate solutions with high accuracy. The inverse transformation ${\mathrm{F}}^{-1}$ will then be used to retrieve approximations for the original equation. A comparative analysis of these schemes will be conducted to determine their efficiency, convergence properties, and practical applications. Additionally, further refinement of the transformation technique will be explored in financial modeling, particularly in stock price forecasting, with results visualized through simulations, graphs, and figures.