Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

Ming-hui Wang; Nan-jing Huang; Zhun Gou

arxiv: 1907.02667 · v1 · pith:HM77TZW2new · submitted 2019-07-05 · 🧮 math.PR

Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

Zhun Gou , Ming-hui Wang , Nan-jing Huang This is my paper

Pith reviewed 2026-05-25 02:23 UTC · model grok-4.3

classification 🧮 math.PR

keywords stochastic differential equationsjump processesstrong solutionspathwise uniquenessnon-Lipschitz conditionsnon-confluent property

0 comments

The pith

Jump-type SDEs have pathwise-unique strong solutions under non-Lipschitz conditions on the coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves existence and pathwise uniqueness for strong solutions of jump-type stochastic differential equations when the coefficients do not satisfy the classical Lipschitz condition. It also supplies a sufficient condition that guarantees the non-confluent property of these solutions. Concrete examples are constructed to show the conditions can be met. A sympathetic reader would care because many applied models involve jumps and rough coefficients where standard theorems give no information.

Core claim

The existence and pathwise uniqueness of strong solutions for jump-type stochastic differential equations are investigated under non-Lipschitz conditions. A sufficient condition is obtained for ensuring the non-confluent property of strong solutions of jump-type stochastic differential equations.

What carries the argument

A set of growth, continuity, and integrability conditions on the drift, diffusion, and jump measure that replace the Lipschitz requirement and close the existence-uniqueness argument.

If this is right

Strong solutions exist for a larger class of jump SDEs than those covered by classical Lipschitz theorems.
Pathwise uniqueness continues to hold without requiring Lipschitz continuity of the coefficients.
The non-confluent property can be verified directly from the same set of conditions used for existence.
Examples confirm that the conditions are checkable on concrete coefficient functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same conditions might be tested numerically by attempting to simulate paths and checking whether they remain distinct.
Similar growth conditions could be examined for mean-field or regime-switching jump equations.
The non-confluent condition may relate to whether the jump measure prevents finite-time merging in applications such as insurance or queueing models.

Load-bearing premise

The coefficients and jump measure obey specific growth, continuity, or integrability conditions that allow the existence and uniqueness proofs to close.

What would settle it

An explicit jump-type SDE whose coefficients satisfy the paper's stated conditions but for which either no strong solution exists or pathwise uniqueness fails.

read the original abstract

In this paper, the existence and pathwise uniqueness of strong solutions for jump-type stochastic differential equations are investigated under non-Lipschitz conditions. A sufficient condition is obtained for ensuring the non-confluent property of strong solutions of jump-type stochastic differential equations. Moreover, some examples are given to illustrate our results.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends non-Lipschitz SDE results to jumps plus a non-confluence condition, but the abstract alone leaves the key conditions and proofs unexamined.

read the letter

The one thing to know is that this paper investigates existence and pathwise uniqueness of strong solutions to jump-type SDEs under non-Lipschitz conditions and gives a sufficient condition for the non-confluent property, with some examples. What looks new is taking the non-Lipschitz approach into the jump setting and adding the non-confluence condition on top. The abstract positions it as a targeted result not directly from prior work. It does fine by including examples to show the results in action. The soft spots start with the fact that we only have the abstract. The specific conditions on the coefficients and the jump measure are not detailed, so it's impossible to judge if they are mild enough or if they actually close the existence argument. The soundness is hard to assess without seeing how they handle the jumps in the proofs. The non-confluence condition is stated as sufficient, but again no details on what it requires. This is for people in stochastic processes who care about SDEs with jumps and weaker regularity. A reader working on similar extensions might get something from it if the full proofs are solid. I would not bring this to reading group on the current information. I would not cite it soon. It should go to peer review because the area is active and a referee can check whether the extension is correct and the condition is useful.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates the existence and pathwise uniqueness of strong solutions to jump-type stochastic differential equations (SDEs) under non-Lipschitz conditions on the coefficients. It derives a sufficient condition ensuring the non-confluent property of these strong solutions and supplies illustrative examples.

Significance. If the central claims hold under verifiable conditions weaker than Lipschitz continuity, the work would extend the existence/uniqueness theory for Lévy-driven SDEs, which is relevant for applications with jumps. The explicit non-confluent condition and the provision of examples are positive features that aid applicability and verification.

major comments (1)

The abstract states that 'a sufficient condition is obtained' for non-confluence but does not identify the precise growth, continuity, or integrability hypotheses on the drift, diffusion, and jump measure. The main existence theorem (presumably in the section presenting the primary result) must be checked to confirm these hypotheses are explicitly stated, non-circular, and sufficient to close the argument without reducing to the Lipschitz case by construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for raising this point about the presentation of hypotheses. We respond to the major comment below.

read point-by-point responses

Referee: The abstract states that 'a sufficient condition is obtained' for non-confluence but does not identify the precise growth, continuity, or integrability hypotheses on the drift, diffusion, and jump measure. The main existence theorem (presumably in the section presenting the primary result) must be checked to confirm these hypotheses are explicitly stated, non-circular, and sufficient to close the argument without reducing to the Lipschitz case by construction.

Authors: The abstract is intentionally concise, as is conventional. The precise hypotheses are stated explicitly in the main result (the existence and pathwise uniqueness theorem). These consist of a one-sided local Lipschitz condition on the drift with an integrable modulus of continuity, a Lipschitz condition on the diffusion coefficient with respect to a suitable distance, and an integrability requirement on the jump coefficient relative to the Lévy measure. The conditions are formulated directly on the coefficients and are therefore non-circular. The proof proceeds via successive approximation and a Yamada-Watanabe-type argument adapted to the jump setting; it does not invoke global Lipschitz continuity at any step and therefore does not reduce to the Lipschitz case by construction. revision: no

Circularity Check

0 steps flagged

No circularity: standard proof paper with independent mathematical arguments

full rationale

The paper is a proof of existence, pathwise uniqueness, and a non-confluence condition for jump SDEs under non-Lipschitz growth/continuity/integrability assumptions on coefficients and the jump measure. These are derived via standard stochastic analysis techniques (e.g., Picard iteration, stopping times, or Gronwall-type estimates adapted to jumps) that do not reduce to the target statements by definition or by fitting. No equations or steps are shown to be self-definitional, and the abstract plus reader assessment indicate no load-bearing self-citations or imported uniqueness theorems. The derivation is self-contained against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background results in stochastic analysis; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

standard math Standard axioms of probability theory and the existence of a filtered probability space supporting the driving Lévy process.
Invoked implicitly by any SDE existence theorem.

pith-pipeline@v0.9.0 · 5567 in / 1181 out tokens · 25453 ms · 2026-05-25T02:23:48.489093+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Assumption 2.1 ... ∫_{0+} ds/ρ(s)=∞ ... Lemma 2.3 (Bihari-LaSalle inequality) ... E[|Δt∧Sδ0|^α]≤p(α)∫ρ(E[|Δs∧Sδ0|^α])ds
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery / initial Peano algebra unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.2 (pathwise uniqueness under non-Lipschitz Assumption 2.3) ... non-confluent property (Theorem 4.1, Assumption 2.5)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · 1 internal anchor

[1]

J. C. Bergquist, R. G. Hulet, W. M. Itano and D. J. Wineland, Obse rvation of quantum jumps in a single atom. Physical Review Letters , 57(14), 1699-1702, 1986

work page 1986
[2]

Biagini, Biagini F

F. Biagini, Biagini F . Stochastic calculus for fractional Brownian mot ion and applications , 2nd Edition, Biagini, Francesca . Stochastic calculus for fractional Bro wnian motion and applications. Springer London, 2008

work page 2008
[3]

Bihari, A generalization of a lemma of Bellman and its application to u niqueness problems of diﬀerential equations, Acta Mathematica Hungarica , 7(1):81–94, 1956

I. Bihari, A generalization of a lemma of Bellman and its application to u niqueness problems of diﬀerential equations, Acta Mathematica Hungarica , 7(1):81–94, 1956

work page 1956
[4]

Binotto, I

G. Binotto, I. Nourdin and D. Nualart. Weak symmetric integrals w ith respect to the fractional Brownian motion, The Annals of Probability , 46(4): 2243-2267, 2018

work page 2018
[5]

Chudley and R.J

C.T. Chudley and R.J. Elliott, Neutron scattering from a liquid on a ju mp diﬀusion model, Proceedings of the Physical Society , 77(2):353–361, 1961

work page 1961
[6]

Czichowsky, R

C. Czichowsky, R. Peyre, W. Schachermayer, et al., hadow price s, fractional Brownian motion, and portfolio optimisation under transaction costs[J], Finance and Stochastics , 22(1): 161-180, 2018. 14

work page 2018
[7]

´Emery, Non conﬂuence des solutions d’une equation stochastique lip schitzienne, In S´ eminaire de Probabilit´ es XV 1979/80, pages 587–589, Springer, Berlin, 1981

M. ´Emery, Non conﬂuence des solutions d’une equation stochastique lip schitzienne, In S´ eminaire de Probabilit´ es XV 1979/80, pages 587–589, Springer, Berlin, 1981

work page 1979
[8]

Fang and T

S. Fang and T. Zhang, A class of stochastic diﬀerential equation s with non-lipschitzian coeﬃcients: pathwise uniqueness and no explosion, Comptes rendus - Math´ omatique, 337(11):737–740, 2003

work page 2003
[9]

Stochastic differential equations with non-lipschitz coefficients: I. Pathwise uniqueness and large deviation

S. Fang and T. Zhang, Stochastic diﬀerential equations with non -lipschitz coeﬃcients: I. pathwise uniqueness and large deviation, arXiv preprint math/0311032 , 2003

work page internal anchor Pith review Pith/arXiv arXiv 2003
[10]

Fang and T

S. Fang and T. Zhang, A study of a class of stochastic diﬀerent ial equations with non-lipschitzian coeﬃcients, Probability Theory and Related Fields , 132(3):356–390, 2005

work page 2005
[11]

Fu and Z

Z. Fu and Z. Li, Stochastic equations of non-negative process es with jumps, Stochastic Processes and their Applications , 120(3):306–330, 2010

work page 2010
[12]

Carr, D.B

Geman, H´ elyette, P. Carr, D.B. Madan, et al. Stochastic Volat ility for Levy Processes, Social Science Electronic Publishing , 13(3):345–382, 2007

work page 2007
[13]

Gleyzes, S

S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deleglise, U.B. Hoﬀ, M. Brune, J. M. Raimond and S. Haroche, Quantum jumps of light recording the birth and death o f a photon in a cavity, Nature, 446(7133):297-300, 2007

work page 2007
[14]

S.W. He, J.G. Wang and J.A Yan, Semimartingale Theory and Stochastic Calculus , 2nd Edition, Routledge, 2018

work page 2018
[15]

Ikeda and S

N. Ikeda and S. Watanabe, Stochastic Diﬀerential Equations and Diﬀusion Processes , 2nd Edition, North-Holland, Armsterdam, 1989

work page 1989
[16]

Kang, B.Z

J.H. Kang, B.Z. Yang and N.J. Huang, Pricing of FX options in the MP T/CIR jump- diﬀusion model with approximative fractional stochastic volatility, Physica A (2019) 121871, https://doi.org/10.1016/j.physa.2019.121871

work page doi:10.1016/j.physa.2019.121871 2019
[17]

Khasminskii and F.C

R.Z. Khasminskii and F.C. Klebaner, Long term behavior of solutio ns of the lotka-volterra system under small random perturbations, Annals of Applied Probability , 11(3):952–963, 2001

work page 2001
[18]

Lan and J.C

G. Lan and J.C. Wu, New suﬃcient conditions of existence, momen t estimations and non con- ﬂuence for sdes with non-lipschitzian coeﬃcients, Stochastic Processes and Their Applications , 124(12):4030–4049, 2014

work page 2014
[19]

Michael, J

Landis, J. Michael, J. G. Schraiber, and M. Liang, Data from: SP hylogenetic analysis using L´ evy processes: ﬁnding jumps in the evolution of continuous traits, Systematic Biology, 62(2):193-204, 2013

work page 2013
[20]

Li and L

Z. Li and L. Mytnik, Strong solutions for stochastic diﬀerentia l equations with jumps, Annales De L Institut Henri Poincar´ e Probabilit´ es Et Statistiques, 47(4):1055–1067, 2010

work page 2010
[21]

X. Mao, G. Marion and E. Renshaw, Environmental brownian nois e suppresses explosions in population dynamics, Stochastic Processes and Their Applications , 97(1):95–110, 2002

work page 2002
[22]

Øksendal and A

B.K. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diﬀusions , 2nd Edition, Springer, Berlin, 2007. 15

work page 2007
[23]

C. Pellegrini, Existence, uniqueness and approximation of the ju mp-type stochastic Sch¨ ordinger equation for two-level systems, Stochastic Processes and their Applications , 120(9): 1722-1747, 2010

work page 2010
[24]

Y. Ren, W. Yin and R. Sakthivel, Stabilization of stochastic diﬀere ntial equations driven by G- Brownian motion with feedback control based on discrete-time sta te observation, Automatica, 95(2018):146-151

work page 2018
[25]

Revathi, R

P. Revathi, R. Sakthivel and Y. Ren, Stochastic functional diﬀ erential equations of Sobolev-type with inﬁnite delay, Statistics and Probability Letters , 109(2016):68-77

work page 2016
[26]

Sakthivel, P

R. Sakthivel, P. Revathi, Y. Ren and G. Shen, Retarded stocha stic diﬀerential equations with inﬁnite delay driven by Rosenblatt process, Stochastic Analysis and Applications , 36(2), 304-323, 2018

work page 2018
[27]

Shao, F.Y

J. Shao, F.Y. Wang and C. Yuan, Harnack inequalities for stocha stic (functional) diﬀerential equations with non-lipschitzian coeﬃcients, Electronic Journal of Probability , 17(27):1–18, 2012

work page 2012
[28]

Shreve, Stochastic Calculus for Finance ll: Continuous-Time Model s, Springer-Verlag New York, 2004

S.E. Shreve, Stochastic Calculus for Finance ll: Continuous-Time Model s, Springer-Verlag New York, 2004

work page 2004
[29]

Tankov, Financial Modelling with Jump Processes , Chapman and Hall/CRC, 2003

P. Tankov, Financial Modelling with Jump Processes , Chapman and Hall/CRC, 2003

work page 2003
[30]

Uppman, Sur le ﬂot d’une equation diﬀerentielle stochastique, In S´ eminaire de Probabilit´ es XVI 1980/81 , pages 268–284, Springer, Berlin, 1982

A. Uppman, Sur le ﬂot d’une equation diﬀerentielle stochastique, In S´ eminaire de Probabilit´ es XVI 1980/81 , pages 268–284, Springer, Berlin, 1982

work page 1980
[31]

Xi and C

F. Xi and C. Zhu, Jump type stochastic diﬀerential equations w ith non-lipschitz coeﬃcients and feller and strong feller properties, Journal of Diﬀerential Equations , 266(8):4668–4711, 2019

work page 2019
[32]

Yamada and S

T. Yamada and S. Watanabe, On the uniqueness of solutions of s tochastic diﬀerential equations, Journal of Mathematics of Kyoto University , 11(1):155–167, 1971

work page 1971
[33]

B.Z. Yang, J. Yue and N.J. Huang, Equilibriam price of variance swa ps under stochastic volatility with L´ evy jumps and stochastic interest rate, International Journal of Theoretical and Applied Finance, 22(4): 1950016 (33 pages), 2019

work page 2019
[34]

Yue and N.J

J. Yue and N.J. Huang, Fractional Wishart processes and ε-fractional Wishart processes with applications, Computers and Mathematics with Applications , 75(2018): 2955-2977. 16

work page 2018

[1] [1]

J. C. Bergquist, R. G. Hulet, W. M. Itano and D. J. Wineland, Obse rvation of quantum jumps in a single atom. Physical Review Letters , 57(14), 1699-1702, 1986

work page 1986

[2] [2]

Biagini, Biagini F

F. Biagini, Biagini F . Stochastic calculus for fractional Brownian mot ion and applications , 2nd Edition, Biagini, Francesca . Stochastic calculus for fractional Bro wnian motion and applications. Springer London, 2008

work page 2008

[3] [3]

Bihari, A generalization of a lemma of Bellman and its application to u niqueness problems of diﬀerential equations, Acta Mathematica Hungarica , 7(1):81–94, 1956

I. Bihari, A generalization of a lemma of Bellman and its application to u niqueness problems of diﬀerential equations, Acta Mathematica Hungarica , 7(1):81–94, 1956

work page 1956

[4] [4]

Binotto, I

G. Binotto, I. Nourdin and D. Nualart. Weak symmetric integrals w ith respect to the fractional Brownian motion, The Annals of Probability , 46(4): 2243-2267, 2018

work page 2018

[5] [5]

Chudley and R.J

C.T. Chudley and R.J. Elliott, Neutron scattering from a liquid on a ju mp diﬀusion model, Proceedings of the Physical Society , 77(2):353–361, 1961

work page 1961

[6] [6]

Czichowsky, R

C. Czichowsky, R. Peyre, W. Schachermayer, et al., hadow price s, fractional Brownian motion, and portfolio optimisation under transaction costs[J], Finance and Stochastics , 22(1): 161-180, 2018. 14

work page 2018

[7] [7]

´Emery, Non conﬂuence des solutions d’une equation stochastique lip schitzienne, In S´ eminaire de Probabilit´ es XV 1979/80, pages 587–589, Springer, Berlin, 1981

M. ´Emery, Non conﬂuence des solutions d’une equation stochastique lip schitzienne, In S´ eminaire de Probabilit´ es XV 1979/80, pages 587–589, Springer, Berlin, 1981

work page 1979

[8] [8]

Fang and T

S. Fang and T. Zhang, A class of stochastic diﬀerential equation s with non-lipschitzian coeﬃcients: pathwise uniqueness and no explosion, Comptes rendus - Math´ omatique, 337(11):737–740, 2003

work page 2003

[9] [9]

Stochastic differential equations with non-lipschitz coefficients: I. Pathwise uniqueness and large deviation

S. Fang and T. Zhang, Stochastic diﬀerential equations with non -lipschitz coeﬃcients: I. pathwise uniqueness and large deviation, arXiv preprint math/0311032 , 2003

work page internal anchor Pith review Pith/arXiv arXiv 2003

[10] [10]

Fang and T

S. Fang and T. Zhang, A study of a class of stochastic diﬀerent ial equations with non-lipschitzian coeﬃcients, Probability Theory and Related Fields , 132(3):356–390, 2005

work page 2005

[11] [11]

Fu and Z

Z. Fu and Z. Li, Stochastic equations of non-negative process es with jumps, Stochastic Processes and their Applications , 120(3):306–330, 2010

work page 2010

[12] [12]

Carr, D.B

Geman, H´ elyette, P. Carr, D.B. Madan, et al. Stochastic Volat ility for Levy Processes, Social Science Electronic Publishing , 13(3):345–382, 2007

work page 2007

[13] [13]

Gleyzes, S

S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deleglise, U.B. Hoﬀ, M. Brune, J. M. Raimond and S. Haroche, Quantum jumps of light recording the birth and death o f a photon in a cavity, Nature, 446(7133):297-300, 2007

work page 2007

[14] [14]

S.W. He, J.G. Wang and J.A Yan, Semimartingale Theory and Stochastic Calculus , 2nd Edition, Routledge, 2018

work page 2018

[15] [15]

Ikeda and S

N. Ikeda and S. Watanabe, Stochastic Diﬀerential Equations and Diﬀusion Processes , 2nd Edition, North-Holland, Armsterdam, 1989

work page 1989

[16] [16]

Kang, B.Z

J.H. Kang, B.Z. Yang and N.J. Huang, Pricing of FX options in the MP T/CIR jump- diﬀusion model with approximative fractional stochastic volatility, Physica A (2019) 121871, https://doi.org/10.1016/j.physa.2019.121871

work page doi:10.1016/j.physa.2019.121871 2019

[17] [17]

Khasminskii and F.C

R.Z. Khasminskii and F.C. Klebaner, Long term behavior of solutio ns of the lotka-volterra system under small random perturbations, Annals of Applied Probability , 11(3):952–963, 2001

work page 2001

[18] [18]

Lan and J.C

G. Lan and J.C. Wu, New suﬃcient conditions of existence, momen t estimations and non con- ﬂuence for sdes with non-lipschitzian coeﬃcients, Stochastic Processes and Their Applications , 124(12):4030–4049, 2014

work page 2014

[19] [19]

Michael, J

Landis, J. Michael, J. G. Schraiber, and M. Liang, Data from: SP hylogenetic analysis using L´ evy processes: ﬁnding jumps in the evolution of continuous traits, Systematic Biology, 62(2):193-204, 2013

work page 2013

[20] [20]

Li and L

Z. Li and L. Mytnik, Strong solutions for stochastic diﬀerentia l equations with jumps, Annales De L Institut Henri Poincar´ e Probabilit´ es Et Statistiques, 47(4):1055–1067, 2010

work page 2010

[21] [21]

X. Mao, G. Marion and E. Renshaw, Environmental brownian nois e suppresses explosions in population dynamics, Stochastic Processes and Their Applications , 97(1):95–110, 2002

work page 2002

[22] [22]

Øksendal and A

B.K. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diﬀusions , 2nd Edition, Springer, Berlin, 2007. 15

work page 2007

[23] [23]

C. Pellegrini, Existence, uniqueness and approximation of the ju mp-type stochastic Sch¨ ordinger equation for two-level systems, Stochastic Processes and their Applications , 120(9): 1722-1747, 2010

work page 2010

[24] [24]

Y. Ren, W. Yin and R. Sakthivel, Stabilization of stochastic diﬀere ntial equations driven by G- Brownian motion with feedback control based on discrete-time sta te observation, Automatica, 95(2018):146-151

work page 2018

[25] [25]

Revathi, R

P. Revathi, R. Sakthivel and Y. Ren, Stochastic functional diﬀ erential equations of Sobolev-type with inﬁnite delay, Statistics and Probability Letters , 109(2016):68-77

work page 2016

[26] [26]

Sakthivel, P

R. Sakthivel, P. Revathi, Y. Ren and G. Shen, Retarded stocha stic diﬀerential equations with inﬁnite delay driven by Rosenblatt process, Stochastic Analysis and Applications , 36(2), 304-323, 2018

work page 2018

[27] [27]

Shao, F.Y

J. Shao, F.Y. Wang and C. Yuan, Harnack inequalities for stocha stic (functional) diﬀerential equations with non-lipschitzian coeﬃcients, Electronic Journal of Probability , 17(27):1–18, 2012

work page 2012

[28] [28]

Shreve, Stochastic Calculus for Finance ll: Continuous-Time Model s, Springer-Verlag New York, 2004

S.E. Shreve, Stochastic Calculus for Finance ll: Continuous-Time Model s, Springer-Verlag New York, 2004

work page 2004

[29] [29]

Tankov, Financial Modelling with Jump Processes , Chapman and Hall/CRC, 2003

P. Tankov, Financial Modelling with Jump Processes , Chapman and Hall/CRC, 2003

work page 2003

[30] [30]

Uppman, Sur le ﬂot d’une equation diﬀerentielle stochastique, In S´ eminaire de Probabilit´ es XVI 1980/81 , pages 268–284, Springer, Berlin, 1982

A. Uppman, Sur le ﬂot d’une equation diﬀerentielle stochastique, In S´ eminaire de Probabilit´ es XVI 1980/81 , pages 268–284, Springer, Berlin, 1982

work page 1980

[31] [31]

Xi and C

F. Xi and C. Zhu, Jump type stochastic diﬀerential equations w ith non-lipschitz coeﬃcients and feller and strong feller properties, Journal of Diﬀerential Equations , 266(8):4668–4711, 2019

work page 2019

[32] [32]

Yamada and S

T. Yamada and S. Watanabe, On the uniqueness of solutions of s tochastic diﬀerential equations, Journal of Mathematics of Kyoto University , 11(1):155–167, 1971

work page 1971

[33] [33]

B.Z. Yang, J. Yue and N.J. Huang, Equilibriam price of variance swa ps under stochastic volatility with L´ evy jumps and stochastic interest rate, International Journal of Theoretical and Applied Finance, 22(4): 1950016 (33 pages), 2019

work page 2019

[34] [34]

Yue and N.J

J. Yue and N.J. Huang, Fractional Wishart processes and ε-fractional Wishart processes with applications, Computers and Mathematics with Applications , 75(2018): 2955-2977. 16

work page 2018