On the anisotropic stable JCIR process

Martin Friesen; Peng Jin

REVIEW 1 major objections 5 minor 45 references

On the anisotropic stable JCIR process

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The anisotropic stable JCIR process admits a density, is strong Feller, and mixes exponentially in total variation.

desk verdict Strong paper: multidimensional stable JCIR process gets a genuine density, strong Feller property, and exponential TV ergodicity without any diffusion component, under an explicit boundary condition that is honestly stated. read the letter →

arxiv 1908.05473 v1 pith:RDVAFJKK submitted 2019-08-15 math.PR

Martin Friesen , Peng Jin This is my paper · ORCID

classification math.PR MSC 60H1060J2560J3537A25

keywords stableJCIRprocessaffineheatkernelanisotropicBesovspacestrongFellerpropertyexponentialergodicitytotalvariationdistancespectrallypositive

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper studies the anisotropic stable JCIR process, a multidimensional affine process whose coordinates are driven by independent spectrally positive stable noises and a common subordinator. It proves that, under a boundary-nonattainment condition, the transition semigroup has a genuine density, that the density depends continuously on the starting point in $L^1$, and hence that the process is strong Feller. In the subcritical case it further proves exponential convergence to the unique invariant measure in total variation. These are, the authors argue, the first total-variation ergodicity results for multidimensional affine processes that do not use smoothing from a diffusion component. A reader building models in finance or branching theory should care because density existence and exponential mixing are the statements that justify numerical pricing, filtering, and long-run simulation.

What carries the argument

The load-bearing object is the weighted anisotropic Besov space $B^{\lambda,a}_{1,\infty}(R^m)$ with anisotropy $a_i=\alpha/\alpha_i$ and weight $\rho_\delta(y)=\min\{\delta,y_1^{1/\alpha_1},\dots,y_m^{1/\alpha_m}\}$. The proof builds a short-time approximation $X^\varepsilon(t)$ of the process and applies a discrete integration by parts to test functions, obtaining the Besov estimate (4.3). Condition (A) enters through Proposition 4.2, giving $P[\min_i X^x_i(t)\le\varepsilon]\le C\varepsilon$, so the process does not hit the boundary and the weight can be removed; a convolution argument removes the extra moment assumption on the large jumps. This machinery supplies both the density and its continuity in the starting point, and the local Dobrushin condition needed for Harris-type ergodicity.

What would settle it

Take $m=1$, $b=0$, and let the subordinator have Lévy measure $\nu(dz)=z^{-1-\vartheta}\mathbf{1}_{z>1}\,dz$ with $\vartheta\in(\alpha-1,1)$. Then condition (2.1) fails because $\int_0^\infty(1-e^{-\xi z})\nu(dz)$ stays bounded as $\xi\to\infty$ instead of growing like $\xi^\vartheta$. One can test the claimed conclusions directly: check whether $P[X^x(t)=0]>0$ for some $x,t$, or whether $x\mapsto p_t(x,\cdot)$ fails to be continuous in $L^1$ at $x=0$.

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Extended reading notes

Core claim

The central claim is Theorem 2.4: if condition (A) holds, then $P_t(x,dy)=p_t(x,y)dy$ and the map $R^m_+\ni x\mapsto p_t(x,\cdot)\in L^1(R^m_+)$ is continuous for every $t>0$, so the anisotropic stable JCIR process is strong Feller. In the subcritical case, under the log-moment condition (2.5), Theorem 2.5 gives the explicit bound $\|P_t(x,\cdot)-\pi\|_{TV}\le C(1+\log(1+|x|)+\int\log(1+|y|)\pi(dy))e^{-\delta t}$. In dimension one, Theorem 2.1 upgrades the density to a smooth, jointly continuous heat kernel. The authors obtain these as consequences of an a-priori bound on the heat kernel in a weighted anisotropic Besov norm, together with a boundary estimate showing that the process almost surely stays in the interior.

Load-bearing premise

Condition (A), which says that in every coordinate the inward drift plus the small jumps of the subordinator grow at least like $\xi^{\vartheta_k}$, so the process never hits the boundary; if it fails, the density and strong Feller conclusions need not hold.

Editorial extensions

If this is right

The transition kernel has a density, so probabilities and option prices for the model can be represented by integrals against $p_t(x,y)dy$ instead of abstract measures.
The strong Feller property follows, meaning the semigroup maps bounded measurable functions into continuous functions at positive times.
In the subcritical case the law converges to the invariant measure in total variation at rate $e^{-\delta t}$, with the prefactor growing only logarithmically in the starting point.
The one-dimensional stable JCIR heat kernel is smooth in space with all derivatives bounded, so Fourier-based pricing and spectral methods are justified there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The multidimensional proof does not use the affine structure, so the same short-time approximation plus anisotropic Besov strategy should transfer to non-affine Markov processes with anisotropic jumps; testing it on an explicit non-affine example would be a direct extension.
Condition (A)'s exponent $\vartheta_k$ is likely sharp: setting $\vartheta_k$ at the lower endpoint $\alpha_k-1$ should make the boundary-nonattainment rate fail or degrade, producing a threshold phenomenon that could be checked numerically.
The paper leaves aside coordinates with $\sigma_i=0$; its own remark suggests a coordinatewise nondegeneracy condition on the subordinator should recover the results, so filling in that case is a concrete follow-up.
In interest-rate models, exponential convergence in total variation means the pricing kernel itself converges to equilibrium, not just moments, so the stated rate $\delta$ could be used to give a quantitative burn-in time for simulations.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a strong paper. It proves the multidimensional stable JCIR process has a density with L1-continuous dependence on the starting point, hence is strong Feller, and in the subcritical case converges exponentially fast in total variation — the first such result for a multidimensional affine process without any diffusion component. The price is an explicit boundary-non-attainment condition (A), and the main external dependencies are the authors' own preprints, not all of which are proved here.

What is actually new: the anisotropic Besov regularity estimate with the boundary weight ρδ, the one-dimensional smoothness theorem via characteristic-function decay, and the Harris-type argument that turns a local Dobrushin condition into TV ergodicity. The proofs are written out in substantial detail — the Fourier argument in Section 3, the four-step compactness argument for Theorem 2.4, the convolution trick that removes the moment assumption, and the Lyapunov verification in Appendix B. The paper is honest about what it borrows: Section 4.1 credits [19,20] and the Debussche–Fournier framework, and the appendix states the anisotropic version of the density criterion is due to [20]. That is reproducible, parameter-free work, and the self-citations are appropriate.

The soft spots are in proportion. Condition (A) is genuinely load-bearing: it drives Proposition 4.2, which excludes a boundary atom and justifies inverting the volatility in the short-time approximation. The authors give examples where b_k = 0 but enough small jumps in direction k make (A) hold, so the condition is not vacuous. The larger concern is that Proposition 4.2 uses the pathwise comparison X_k(t) ≥ Y_k(t) from [21, Prop. 4.2], which is not proved in this paper. That comparison is plausible and likely true, but a referee should check it, because if it fails the density and strong Feller theorem lose their boundary control. The other preprints [18–20,22] are used for moment bounds and the Wasserstein ergodicity, and those are less delicate. Minor: the 'first result' claim rests on the authors' literature survey; I did not find a missing reference, but it is a claim about absence.

For whom: anyone working on affine processes, CBI processes, or ergodicity without diffusion smoothing. It deserves a serious referee. I would send it to review and ask the referee to verify the comparison principle and the moment assumption in Theorem 4.1. If those hold — and I expect they do — the paper is a solid advance.

Referee Report

1 major / 5 minor

Summary. This paper studies the anisotropic stable JCIR process, an m-dimensional affine process on R^m_+ whose coordinates are driven by independent spectrally positive alpha_k-stable noises, a linear drift with nonnegative off-diagonal coefficients, and a subordinator J with jumps in R^m_+. The main results are: (i) Theorem 2.1, existence of a jointly continuous and smooth heat kernel for the one-dimensional stable JCIR process under condition (2.1); (ii) Theorem 2.4, existence of a density p_t(x,dy) whose dependence on the initial point x is L^1-continuous under condition (A), hence the strong Feller property; and (iii) Theorem 2.5, exponential ergodicity in total variation in the subcritical case under condition (A) and the log-moment condition (2.5). The proof combines the affine transform formula with a short-time approximation, anisotropic Besov regularity, boundary non-attainment estimates, a convolution trick to remove moment conditions on the big jumps, and a Harris-type theorem verified through a local Dobrushin condition and a Lyapunov function.

Significance. If the proofs are correct, this is the first exponential ergodicity result in total variation for multidimensional affine processes that does not rely on smoothing by a diffusion component. The method is not specific to the affine structure; the Besov-regularity and convolution arguments are applicable to other Markov processes with anisotropic jumps. The assumptions are stated directly in terms of the model primitives, with no free parameters, and the technical estimates are largely proved in full in the paper and its appendices. The one-dimensional smoothness theorem (Theorem 2.1) and the explicit exponential rate in Theorem 2.5 are concrete, falsifiable statements.

major comments (1)

[Section 4.2, Proposition 4.2] The proof of Proposition 4.2 uses the pathwise comparison result [21, Proposition 4.2] to conclude X^x_k(t) >= Y^{x_k}_k(t) for all k. This comparison is load-bearing: it is the mechanism by which condition (A) is transferred from the projected one-dimensional processes to the multidimensional process, and it is also used in Proposition C.3 to justify the invertibility of sigma(X(t-epsilon)). However, [21] is an authors' preprint and the comparison is neither stated nor proved in the present manuscript. Please provide the precise statement and either a proof in an appendix or a reference to a published version; without it the proof chain for Theorems 2.4 and 2.5 is not self-contained.

minor comments (5)

[Section 4.2, proof of Proposition 4.2] The index set in 'k in {1,...,d}' should be {1,...,m}.
[Section 2.4, Example 2.7] The condition 'nu_k in (alpha_k-1, 1)' should read 'vartheta_k in (alpha_k-1, 1)'.
[Section 4.1, Theorem 4.1] The Besov space B^{lambda,a}_{1,infty}(R^m_+) is used although the norm in (4.2) was defined for functions on R^m; please state explicitly that functions are extended by zero outside R^m_+.
[Section 3, proof of Theorem 2.1] The constants in Proposition 3.1 depend on t_0; for the claimed joint continuity in t, a sentence explaining how to make the estimate uniform on compact time intervals would be helpful.
[Section 5.1, Corollary 5.2] The step combining Proposition A.2 with weak convergence to conclude that integral |y| pi(dy) is finite is stated in one sentence; please spell out the truncation argument, for example using integral (|y| wedge R) pi(dy) <= liminf_t E|X^z(t)| <= C(1+|z|) and then R -> infinity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the heat-kernel regularity, strong Feller property, and exponential ergodicity are derived from explicit SDE/affine-transform estimates and do not reduce to their inputs.

full rationale

The derivation chain for Theorems 2.4 and 2.5 is self-contained in the relevant sense: condition (A) is a stated hypothesis, not a fitted or renamed version of the conclusions; the density p_t(x,y) is exhibited through characteristic-function estimates in the one-dimensional case and through a Besov-space regularity proof in the multidimensional case, with the singular boundary part eliminated by Proposition 4.2. Proposition 4.2 is not circular: it uses the pathwise comparison result [21, Prop. 4.2] for CBI processes, a parameter-free theorem with assumptions that do not include the present density or strong-Feller conclusions, then applies the already-proved one-dimensional Theorem 2.1 to the coordinatewise dominated processes. Likewise, the convolution decomposition (4.6) is derived from the affine characteristic-function representation rather than assumed, and the exponential ergodicity proof independently verifies the two conditions of the Harris-type Theorem D.1, using [22] only for moment and Wasserstein estimates that are separate from the total-variation conclusion. The self-citations to [18]–[22] are load-bearing in the sense of providing lemmas, but none of those lemmas is equivalent to the target result, and no fitted parameter is relabelled as a prediction. The residual concern that some cited prior works are author preprints is a verification risk, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper's contribution sits on a well-established affine-process base: the model is a special case of the multi-type CBI class of [3], the affine transform formula is from [13], and the main regularity engine is the short-time approximation of [11,19,20]. The genuinely new inputs are the boundary condition (A), the boundary weight ρδ, the L1-continuity argument, and the Harris-type verification. No new entities are postulated and no constants are fitted to data; no theorem output enters as an input assumption.

assumptions (6)

standard math Unique R^m_+-valued strong solution for (1.3) and affine transform formula (1.4) with Riccati equations (1.5).
Taken from Barczy-Li-Pap [3] and Duffie-Filipovic-Schachermayer [13]; invoked throughout Sections 2 to 4, for instance in the Theorem 2.1 proof via (2.2) and (2.3).
domain assumption Condition (A): for each k, b_k ξ + ∫_{R^m_+}(1−e^{−ξz_k})ν(dz) ≥ Cξ^{ϑ_k} for ξ ≥ M, with ϑ_k ∈ (α_k−1,1].
Main hypothesis of Theorems 2.1, 2.4 and 2.5; drives boundary non-attainment in Proposition 4.2 and the decay estimates in Proposition 3.1.
domain assumption Subcriticality of β (all eigenvalues have negative real parts) and the log-moment condition (2.5): ∫_{|z|>1} log(1+|z|)ν(dz) < ∞.
Hypotheses of Theorem 2.5; used for existence of the invariant measure from [28], the Lyapunov estimate in Lemma B.1, and the exponential contraction in Proposition 5.3.
standard math Comparison principle: X_k^x(t) ≥ Y_k^{x_k}(t) pathwise for the diagonal process (4.4), from [21, Proposition 4.2].
Used in Proposition 4.2 to reduce boundary estimates to one-dimensional marginals; cited from the authors' preprint [21], whose assumptions (β_kj ≥ 0, k≠j) are part of the model setup.
standard math Moment bounds and exponential contraction from [22, Proposition 6.1] and Proposition A.2.
Used in Proposition 5.3 and Lemma 4.3 for tail control E|X^x(t)| ≤ C(1+|x|) and for the coupling decay |E[Y^x(h−1)−Y^y(h−1)]| ≤ C|x−y|e^{−ch}.
standard math Harris-type ergodicity criterion (Theorem D.1): a Foster-Lyapunov drift condition plus a local Dobrushin condition imply exponential ergodicity in total variation.
Quoted from Hairer [24] and Kulik [35], stated in Appendix D, and used as the backbone of Section 5.2.

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Pith. "Pith review of On the anisotropic stable JCIR process." pith.science (2026). https://pith.science/paper/RDVAFJKK

@misc{pith2026190805473,
  author       = {Pith},
  title        = {Pith review of: On the anisotropic stable JCIR process},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RDVAFJKK}},
  note         = {Machine review of arXiv:1908.05473}
}

read the original abstract

We investigate the anisotropic stable JCIR process which is a multi-dimensional extension of the stable JCIR process but also a multi-dimensional analogue of the classical JCIR process. We prove that the heat kernel of the anisotropic stable JCIR process exists and it satisfies an a-priori bound in a weighted anisotropic Besov norm. Based on this regularity result we deduce the strong Feller property and prove, for the subcritical case, exponential ergodicity in total variation. Also, we show that in the one-dimensional case the corresponding heat kernel is smooth.

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Works this paper leans on

45 extracted references · 44 canonical work pages

[20]

, Existence of densities for stochastic diﬀerential equation s driven by L´ evy processes with anisotropic jumps, arXiv:1810.07504 [math.PR] (2018)

work page Pith review arXiv 2018
[21]

, Boundary behavior of multi-type continuous-state branchi ng processes with immigration , arXiv:1902.01162 [math.PR] (2019)

work page Pith review arXiv 2019
[1]

6, Springer, Cham; Bocconi University Press, Milan, 2015

Aur´ elien Alfonsi,Aﬃne diﬀusions and related processes: simulation, theory an d applications, Boc- coni & Springer Series, vol. 6, Springer, Cham; Bocconi University Press, Milan, 2015. MR 3363174

work page 2015
[2]

M´ aty´ as Barczy, Leif D¨ oring, Zenghu Li, and Gyula Pap,Stationarity and ergodicity for an aﬃne two-factor model, Adv. in Appl. Probab. 46 (2014), no. 3, 878–898. MR 3254346

work page 2014
[3]

M´ aty´ as Barczy, Zenghu Li, and Gyula Pap,Stochastic diﬀerential equation with jumps for multi- type continuous state and continuous time branching proces ses with immigration , ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 1, 129–169. MR 3340375

work page 2015
[4]

R. F. Bass and M. Cranston, The Malliavin calculus for pure jump processes and applicat ions to local time, Ann. Probab. 14 (1986), no. 2, 490–532. MR 832021

work page 1986
[5]

Krzysztof Bogdan, Pawe/suppress l Sztonyk, and Victoria Knopova,Heat kernel of anisotropic nonlocal operators, arXiv:1704.03705v1 [math.AP] (2017)

work page Pith review arXiv 2017
[6]

The martingale problem for a class of nonlocal operators of diagonal type

Jamil Chaker, The martingale problem for anisotropic nonlocal operators , arXiv:1802.05888v1 [math.PR], to appear in: Math. Nachrichten (2018)

work page Pith review arXiv 2018

Show all 45 references

[7]

Chazal, R

M. Chazal, R. Loeﬀen, and P. Patie, Option pricing in a one-dimensional aﬃne term structure model via spectral representations, SIAM J. Financial Math. 9 (2018), no. 2, 634–664. MR 3805842

2018
[8]

Marie Chazal, Ronnie Loeﬀen, and Pierre Patie, Smoothness of continuous state branching with immigration semigroups, J. Math. Anal. Appl. 459 (2018), no. 2, 619–660. MR 3732548

2018
[9]

Serguei Dachkovski, Anisotropic function spaces and related semi-linear hypoe lliptic equations , Math. Nachr. 248/249 (2003), 40–61. MR 1950714

2003
[10]

Stefano De Marco, Smoothness and asymptotic estimates of densities for SDEs w ith locally smooth coeﬃcients and applications to square root-type diﬀusions , Ann. Appl. Probab. 21 (2011), no. 4, 1282–1321. MR 2857449

2011
[11]

Arnaud Debussche and Nicolas Fournier, Existence of densities for stable-like driven SDE’s with H¨ older continuous coeﬃcients, J. Funct. Anal. 264 (2013), no. 8, 1757–1778. MR 3022725

2013
[12]

Theory Related Fields 158 (2014), no

Arnaud Debussche and Marco Romito, Existence of densities for the 3D Navier-Stokes equa- tions driven by Gaussian noise , Probab. Theory Related Fields 158 (2014), no. 3-4, 575–596. MR 3176359

2014
[13]

Darrell Duﬃe, Damir Filipovi´ c, and Walter Schachermayer, Aﬃne processes and applications in ﬁnance, Ann. Appl. Probab. 13 (2003), no. 3, 984–1053. MR 1994043

2003
[14]

, Financial Analysts Journal 57 (2001), no

Darrell Duﬃe and Nicolae Grleanu, Risk and valuation of collateralized debt obligations. , Financial Analysts Journal 57 (2001), no. 1, 41–59

2001
[15]

Xan Duhalde, Cl´ ement Foucart, and Chunhua Ma, On the hitting times of continuous-state branching processes with immigration , Stochastic Process. Appl. 124 (2014), no. 12, 4182–4201. MR 3264444

2014
[16]

Econometrics 176 (2013), no

Damir Filipovi´ c, Eberhard Mayerhofer, and Paul Schneider, Density approximations for multivari- ate aﬃne jump-diﬀusion processes , J. Econometrics 176 (2013), no. 2, 93–111. MR 3084047

2013
[17]

4, 1819–1844

Cl´ ement Foucart and Ger´ onimo Uribe Bravo,Local extinction in continuous-state branching pro- cesses with immigration , Bernoulli 20 (2014), no. 4, 1819–1844. MR 3263091

2014
[18]

Martin Friesen, Peng Jin, Jonas Kremer, and Barbara R¨ udiger, Exponential ergodicity for stochastic equations of nonnegative processes with jumps , arXiv:1902.02833 [math.PR] (2019)

2019 arXiv
[19]

Martin Friesen, Peng Jin, and Barbara R¨ udiger,Existence of densities for multi-type CBI processes , arXiv:1810.00400 [math.PR] (2018)

2018 arXiv
[22]

, Stochastic equation and exponential ergodicity in wassers tein distances for aﬃne processes , arXiv:1901.05815 [math.PR] (2019)

2019 arXiv
[23]

Zongfei Fu and Zenghu Li, Stochastic equations of non-negative processes with jumps , Stochastic Process. Appl. 120 (2010), no. 3, 306–330. MR 2584896

2010
[24]

Martin Hairer, Convergence of Markov procesess , http://www.hairer.org/notes/Convergence.pdf (2016)

2016
[25]

21 (2017), no

Ying Jiao, Chunhua Ma, and Simone Scotti, Alpha-CIR model with branching processes in sovereign interest rate modeling , Finance Stoch. 21 (2017), no. 3, 789–813. MR 3663644 ON THE ANISOTROPIC STABLE JCIR PROCESS 30

2017
[26]

Ying Jiao, Chunhua Ma, Simone Scotti, and Chao Zhou, The alpha-heston stochastic volatility model, arXiv:1812.01914 [q-ﬁn.MF] (2018)

2018 arXiv
[27]

Peng Jin, Jonas Kremer, and Barbara R¨ udiger,Exponential ergodicity of an aﬃne two-factor model based on the α -root process, Adv. in Appl. Probab. 49 (2017), no. 4, 1144–1169. MR 3732190

2017
[28]

, Existence of limiting distribution for aﬃne processes , arXiv:1812.05402 [math.PR] (2018)

2018 arXiv
[29]

Peng Jin, Jonas Kremer, and Barbara R¨ udiger,Moments and ergodicity of the jump-diﬀusion CIR process, to appear in Stochastics (2019+)

2019
[30]

Martin Keller-Ressel and Aleksandar Mijatovi´ c, On the limit distributions of continuous-state branching processes with immigration , Stochastic Process. Appl. 122 (2012), no. 6, 2329–2345. MR 2922631

2012
[31]

Victoria Knopova and Alexei Kulik, Parametrix construction of the transition probability den sity of the solution to an SDE driven by α -stable noise , Ann. Inst. Henri Poincar´ e Probab. Stat. 54 (2018), no. 1, 100–140. MR 3765882

2018
[32]

Tadeusz Kulczycki and Micha/suppress l Ryznar,Transition density estimates for diagonal systems of SDEs driven by cylindrical α -stable processes, ALEA Lat. Am. J. Probab. Math. Stat. 15 (2018), no. 2, 1335–1375. MR 3877025

2018
[33]

, Semigroup properties of solutions of sdes driven by l´ evy pr ocesses with independent coor- dinates, arXiv:1906.07173 [math.PR] (2019)

2019 arXiv
[34]

Tadeusz Kulczycki, Micha/suppress l Ryznar, and Pawe/suppress l Sztonyk,Strong feller property for sdes driven by multiplicative cylindrical stable noise , arXiv:1811.05960v1 [math.PR] (2018)

2018 arXiv
[35]

67, De Gruyter, Berlin, 2018, With applications to limit theore ms

Alexei Kulik, Ergodic behavior of Markov processes , De Gruyter Studies in Mathematics, vol. 67, De Gruyter, Berlin, 2018, With applications to limit theore ms. MR 3791835

2018
[36]

MR 2760602

Zenghu Li, Measure-valued branching Markov processes , Probability and its Applications (New York), Springer, Heidelberg, 2011. MR 2760602

2011
[37]

Zenghu Li and Chunhua Ma, Asymptotic properties of estimators in a stable Cox-Ingers oll-Ross model, Stochastic Process. Appl. 125 (2015), no. 8, 3196–3233. MR 3343292

2015
[38]

Eberhard Mayerhofer, Robert Stelzer, and Johanna Vestw eber, Geometric Ergodicity of Aﬃne Processes on Cones , arXiv e-prints (2018), arXiv:1811.10542

2018 arXiv
[39]

Tweedie, Markov chains and stochastic stability , second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W

Sean Meyn and Richard L. Tweedie, Markov chains and stochastic stability , second ed., Cambridge University Press, Cambridge, 2009, With a prologue by Peter W. Glynn. MR 2509253

2009
[40]

Xuhui Peng and Rangrang Zhang, Exponential ergodicity for SDEs under the total variation , J. Evol. Equ. 18 (2018), no. 3, 1051–1067. MR 3859440

2018
[41]

Theory Related Fields 105 (1996), no

Jean Picard, On the existence of smooth densities for jump processes , Probab. Theory Related Fields 105 (1996), no. 4, 481–511. MR 1402654

1996
[42]

Marco Romito, A simple method for the existence of a density for stochastic evolutions with rough coeﬃcients , Electron. J. Probab. 23 (2018), Paper no. 113, 43. MR 3885546

2018
[43]

III , Monographs in Mathematics, vol

Hans Triebel, Theory of function spaces. III , Monographs in Mathematics, vol. 100, Birkh¨ auser Verlag, Basel, 2006. MR 2250142

2006
[44]

Jian Wang, On the exponential ergodicity of L´ evy-driven Ornstein-Uh lenbeck processes, J. Appl. Probab. 49 (2012), no. 4, 990–1004. MR 3058984

2012
[45]

Glynn, Aﬃne Jump-Diﬀusions: Stochastic Stability and Limit The- orems, arXiv e-prints (2018), arXiv:1811.00122

Xiaowei Zhang and Peter W. Glynn, Aﬃne Jump-Diﬀusions: Stochastic Stability and Limit The- orems, arXiv e-prints (2018), arXiv:1811.00122. (Martin Friesen) F aculty for Mathematics and Natural Sciences, University o f Wupper- tal, Germany E-mail address : friesen@math.uni-wupp...

2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [20]

, Existence of densities for stochastic diﬀerential equation s driven by L´ evy processes with anisotropic jumps, arXiv:1810.07504 [math.PR] (2018)

work page Pith review arXiv 2018

[2] [21]

, Boundary behavior of multi-type continuous-state branchi ng processes with immigration , arXiv:1902.01162 [math.PR] (2019)

work page Pith review arXiv 2019

[3] [1]

6, Springer, Cham; Bocconi University Press, Milan, 2015

Aur´ elien Alfonsi,Aﬃne diﬀusions and related processes: simulation, theory an d applications, Boc- coni & Springer Series, vol. 6, Springer, Cham; Bocconi University Press, Milan, 2015. MR 3363174

work page 2015

[4] [2]

M´ aty´ as Barczy, Leif D¨ oring, Zenghu Li, and Gyula Pap,Stationarity and ergodicity for an aﬃne two-factor model, Adv. in Appl. Probab. 46 (2014), no. 3, 878–898. MR 3254346

work page 2014

[5] [3]

M´ aty´ as Barczy, Zenghu Li, and Gyula Pap,Stochastic diﬀerential equation with jumps for multi- type continuous state and continuous time branching proces ses with immigration , ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), no. 1, 129–169. MR 3340375

work page 2015

[6] [4]

R. F. Bass and M. Cranston, The Malliavin calculus for pure jump processes and applicat ions to local time, Ann. Probab. 14 (1986), no. 2, 490–532. MR 832021

work page 1986

[7] [5]

Krzysztof Bogdan, Pawe/suppress l Sztonyk, and Victoria Knopova,Heat kernel of anisotropic nonlocal operators, arXiv:1704.03705v1 [math.AP] (2017)

work page Pith review arXiv 2017

[8] [6]

The martingale problem for a class of nonlocal operators of diagonal type

Jamil Chaker, The martingale problem for anisotropic nonlocal operators , arXiv:1802.05888v1 [math.PR], to appear in: Math. Nachrichten (2018)

work page Pith review arXiv 2018