Pith. sign in

REVIEW 2 major objections 4 minor 49 references

Drawdowns, Drawups, and Occupation Times under General Markov Models

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Five drawdown quantities become linear systems under any Markov model.

desk verdict Solid CTMC extension to five drawdown functionals, but the convergence theorem for repeated drawdowns assumes a path condition that fails for the diffusion models used in the numerics. read the letter →

arxiv 2506.00552 v1 pith:ZSYZRSSX submitted 2025-05-31 q-fin.MF

Pingping Zeng , Gongqiu Zhang , Weinan Zhang This is my paper
classification q-fin.MF MSC 60J2860J6060J7691G2091G3091G60
keywords continuous-timeMarkovchainapproximationdrawdowndrawupoccupationtimefirstpassageprocessLaplacetransformcomputationalfinance
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

A drawdown is the drop of an asset from its running maximum, and it is hard to price because the whole path matters. This paper claims that five standard drawdown quantities can be computed by replacing a general Markov process with a continuous-time Markov chain (CTMC) on a finite grid and solving explicit linear systems. It proves that, as the grid shrinks, the CTMC stopping times and occupation times converge in distribution to the original process's quantities under Assumption 1. The algorithms run in $O(N^3)$ time for general Markov models and $O(N)$ for diffusions, matching the complexity of path-independent pricing. If correct, this makes drawdown risk tractable for models with jumps, state-dependent coefficients, and L\'evy processes, not only Brownian motion or diffusions.

What carries the argument

The load-bearing object is the first-passage resolvent of the CTMC, $P_S(k;f)=E[e^{-\int_0^{T_S}k(Y_s)\,ds}f(Y_{T_S})]$, which solves the linear system $(k-G)P_S=0$ and is available in closed form via (3). Every target quantity is written as a recursion of such resolvents by decomposing the drawdown time as $\tau_a=T^-_{x-a}$ on the event $\{\tau_a<T^+_x\}$, appending the running maximum or minimum as an extra state when the quantity depends on drawups or recovery, as in the $R$ and $S$ resolvents of (8) and (11). The recursions are solved backward from the upper absorbing state, and Sherman-Morrison-Woodbury updates for the inverse matrices yield the claimed complexity: $O(N^3)$ for general Markov models, $O(N)$ for diffusions for most quantities, and closed-form ratios for L\'evy models.

What would settle it

Run the CTMC scheme and a high-precision Monte Carlo simulation on a Markov process of the form (1) whose jumps can land exactly on the drawdown boundary, then compare the distribution of the second drawdown time with recovery as the mesh shrinks; if the limits disagree, the condition $P(X\in U\cap V\cap W)=1$ fails and the theorem's conclusion does not hold for that model.

Watch

Extended reading notes

Core claim

The central claim is that each of the five quantities satisfies an exact finite linear system once the Markov process $X$ is replaced by a CTMC $Y$: equation (7) for the drawdown time preceding the drawup time, (9) for the occupation time of the underlying until the drawdown time, (10) for the occupation time of the drawdown process until the drawdown time, (13) for the $n$-th drawdown time without recovery, and (16) for the $n$-th drawdown time with recovery. These systems are built from the first-passage resolvent of the chain, so the only approximation is the chain itself. Theorem 1 then states that the CTMC stopping times and occupation times converge jointly in distribution to those of the original process as the mesh goes to zero, under Assumption 1 and path-regularity conditions on the sets $U$, $V$, $W$, and $\widetilde{W}$. Numerical tables for Black-Scholes, CEV, double-exponential jump-diffusion, and variance-gamma models show extrapolated relative errors below one percent within seconds for digital options and drawdown insurance products.

Load-bearing premise

The convergence guarantee depends on Assumption 1 and on path-regularity conditions for the $n$-th drawdown times, $P(X\in U\cap\widetilde{W})=1$ and $P(X\in U\cap V\cap W)=1$, which the paper states but does not verify for the four models it tests numerically.

Editorial extensions

If this is right

  • The five drawdown quantities can be computed for any Markov process of the form (1), removing the Brownian-motion or diffusion restrictions of earlier drawdown results.
  • For the occupation-time quantities and the $n$-th drawdown functionals, the complexity matches path-independent pricing: $O(N^3)$ for general Markov models, $O(N)$ for diffusions, and $O(N^2)$ or $O(N^4)$ for the more path-dependent drawdown-before-drawup quantity.
  • For L\'evy models, formulas (12), (15), and (18) turn several quantities into closed-form ratios, cutting runtime to under a second for some insurance and digital-option prices.
  • The joint convergence in Theorem 1 means that Laplace-inverted derivative prices inherit the approximation: both the stopping time and the accumulated occupation time converge together.

Reading between the lines

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

  • Going beyond the paper, the same recursion should directly implement the drawup occupation time until a drawdown and until an independent exponential time sketched in Remark 8, giving a natural stress test of the method on additional path-dependent functionals.
  • A Monte Carlo check of the path-regularity conditions $P(X\in U\cap\widetilde{W})=1$ and $P(X\in U\cap V\cap W)=1$ for Black-Scholes, CEV, jump-diffusion, and variance-gamma parameters would turn the conditional convergence guarantee into an unconditional one for the models actually tested.
  • Because the cost is linear in the number of CTMC states for diffusions, the same linear systems could be embedded in calibration or portfolio-optimization loops that evaluate drawdown quantities thousands of times, which the paper does not itself run.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a continuous-time Markov chain (CTMC) approximation framework for computing five drawdown-related quantities under general one-dimensional Markov models: the drawdown time preceding the drawup time, generalized occupation times of the underlying process and of the drawdown process until drawdown, and the n-th drawdown times without and with recovery. For each quantity the authors derive linear systems satisfied under the CTMC and propose recursive algorithms with complexity O(N^3) for general Markov models and O(N) or O(N^2) for diffusions. Section 4 states a convergence theorem, Theorem 1, asserting distributional convergence of the CTMC stopping times and occupation times to their continuous counterparts as the mesh shrinks. Numerical experiments for Black-Scholes, CEV, double exponential jump-diffusion, and variance gamma models report first-order convergence and small errors.

Significance. The algorithmic contribution is substantial: explicit linear systems, efficient recursions, and complexity bounds for five nontrivial path-dependent quantities are valuable and largely self-contained. The numerical experiments are extensive and demonstrate practical accuracy. However, the central convergence guarantee for the two n-th drawdown quantities rests on Theorem 1(4)-(5), whose hypotheses are not satisfied by the diffusion and jump-diffusion models used in the corresponding numerical tables. This gap means the paper's main theoretical claim is currently established only for the first three quantities, not for the last two under the stated model classes.

major comments (2)
  1. [Section 4, definition of U and Theorem 1(4)-(5)] The set U requires that for every u≥0 the two hitting times inf{t≥u:ω_{u,t}-ω_t≥a} and inf{t≥u:ω_{u,t}-ω_t>a} coincide. For a continuous path that ever reaches drawdown a and subsequently downcrosses level a, take u to be a downcrossing time (the first return to a from above after the initial hit). At that u the drawdown equals a and immediately afterwards is strictly less than a, so the first time ≥a equals u while the first time >a is strictly larger; hence the path is not in U. Nondegenerate diffusions downcross level a infinitely often after the first hit, so for the Black-Scholes, CEV, and DEJD models used in Tables 4 and 5 we have P(X∈U)=0 rather than 1. Therefore Theorem 1(4) and (5) do not apply to these central examples, and the paper does not establish convergence of the CTMC approximations for the n-th drawdown quantities without and with recovery. The authors need to replace U by a condition that is actually satisfied by the intended model classes (for instance, a condition expressed in terms of quasi-left-continuity and the absence of exact drawdown equality at the relevant stopping times) and rework the proof accordingly.
  2. [Section 4, Theorem 1(4)-(5) and Appendix B.6] Even aside from the specific failure of U, the proof of Theorem 1(4)-(5) uses the membership of X in U to convert inequalities of the form '≥a' into '>a' in the limsup argument, and this step is essential for the claimed continuity of the n-th drawdown time mappings. Since U almost surely fails for continuous-path models, the proof cannot be patched by a minor rewording; the argument itself needs to be modified, for example by showing that the CTMC stopping times are asymptotically unaffected by the difference between weak and strict threshold crossing. Until such a proof is supplied, the convergence claims for the quantities in Sections 3.4 and 3.5 should be stated only conditionally on a verified path property, or the numerical convergence for the diffusion and jump-diffusion examples should be presented as empirical rather than as a consequence of Theorem 1.
minor comments (4)
  1. [Appendix B.1] In the proof of Proposition 1, the expression 'Y_{Y_x^+}' appears twice and is clearly a typo for the running maximum at the stopping time T_x^+; this makes the displayed derivation hard to follow.
  2. [Table 5] In the DEJD row with Nx=40, the reported absolute error is 0.1020, while the benchmark and CTMC values differ by 0.01020; this appears to be a factor-of-ten typographical error.
  3. [References] The reference 'Cherny and Ob/suppress l´oj' contains corrupted text and should read 'Cherny, V. and Oblój, J.'; the same corruption appears in the introduction.
  4. [Section 5] The benchmark values are stated to be obtained by wide grids and extrapolation, but the precise grid ranges and extrapolation procedure are not specified in the tables or text, which makes independent reproduction of the reported results more difficult.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the five linear systems are derived by first-step decompositions, the numerical values are solved from those exact systems with no fitted parameters, and the convergence proof rests on stated prior theorems rather than on the quantities being predicted.

full rationale

The paper's derivation chain is self-contained in the relevant sense: each of the five quantities is defined as an expectation under the CTMC approximation, and Propositions 1-5 derive exact linear systems (7), (9), (10), (13), and (16) by first-step (strong Markov) decompositions. The algorithms then solve these linear systems exactly; no parameter is calibrated to the target outputs, and no 'prediction' is a rename of an input. The convergence theorem is not circular: Theorem 1 assumes weak convergence of the CTMC approximation (from Mijatovic and Pistorius 2013) and invokes the prior drawdown-time continuity result of Zhang and Li (2023c) for the first drawdown time, but that prior result does not include the five occupation-time and nth-drawdown functionals proved here. The self-citations to Zhang and Li (2023c) are load-bearing in the sense that Lemma 1, Lemma 2, Assumption 1, and parts of Theorem 1 rely on that paper, but they are prior published mathematical results with explicitly stated assumptions and are used as building blocks, not as a way of assuming the present conclusions. The numerical 'benchmarks' are obtained from the same CTMC method on finer grids with extrapolation, so they are convergence checks rather than independent exact prices; this is a validation convention, not a fitted-input circularity. The unverified path-regularity hypotheses P(X in U cap ~W)=1 and P(X in U cap V cap W)=1 in Theorem 1(4)-(5) are assumption-support concerns and would affect correctness if false, but they are not a circular reduction of the theorem's conclusion to its inputs. Overall, the central derivation has independent mathematical content, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard Markov-process assumptions and on the existence of a weakly convergent CTMC sequence; the paper does not introduce new parameters or entities. The main unverified burden is the path-regularity conditions U,V,W,~W and the inversion accuracy.

assumptions (4)
  • domain assumption Assumption 1: X has no monotone sample paths, is quasi-left continuous, and its Levy measure satisfies stated conditions (absolute continuity or diffusion component).
    Used to ensure tau_a^(n) converges in distribution to tau_a; invoked in Theorem 1 and taken from Zhang and Li (2023c).
  • domain assumption P(X in U cap ~W)=1 and P(X in U cap V cap W)=1 for the n-th drawdown times without and with recovery.
    Required for the continuity of the stopping-time mappings in Theorem 1 parts (4) and (5); not verified for the numerical models.
  • domain assumption There exists a sequence of CTMCs Y^(n) such that Y^(n) converges weakly to X as mesh goes to 0, with the CTMC generator G approximating the generator L in Appendix A.
    The weak convergence is a sufficient condition from Mijatovic and Pistorius (2013); the paper assumes such approximating chains exist for the models considered.
  • domain assumption The Abate-Whitt Fourier-series algorithm inverts the Laplace transforms accurately.
    Used in Section 5 to convert the computed Laplace transforms into option prices; numerical accuracy of inversion is assumed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Drawdowns, Drawups, and Occupation Times under General Markov Models." pith.science (2026). https://pith.science/paper/ZSYZRSSX

@misc{pith2026250600552,
  author       = {Pith},
  title        = {Pith review of: Drawdowns, Drawups, and Occupation Times under General Markov Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSYZRSSX}},
  note         = {Machine review of arXiv:2506.00552}
}
read the original abstract

Drawdown risk, an important metric in financial risk management, poses significant computational challenges due to its highly path-dependent nature. This paper proposes a unified framework for computing five important drawdown quantities introduced in Landriault et al. (2015) and Zhang (2015) under general Markov models. We first establish linear systems and develop efficient algorithms for such problems under continuous-time Markov chains (CTMCs), and then establish their theoretical convergence to target quantities under general Markov models. Notably, the proposed algorithms for most quantities achieve the same complexity order as those for path-independent problems: cubic in the number of CTMC states for general Markov models and linear when applied to diffusion models. Rigorous convergence analysis is conducted under weak regularity conditions, and extensive numerical experiments validate the accuracy and efficiency of the proposed algorithms.

Figures

Figures reproduced from arXiv: 2506.00552 by the authors.

Figure 1
Figure 1. log10(ǫ) vs. log10(Nx) for evaluating Px,y(τa < τbb ∧ T) under different models. 6 Conclusions In this paper, we propose a unified framework for computing five important drawdown quantities under general Markov models. Utilizing the CTMC approximation, we derive linear systems for the associated quantities defined under the CTMC and develop algorithms to solve these systems. In particular, by exploiting the inherent… view at source ↗
Figure 2
Figure 2. log10(ǫ) vs. log10(Nx) for evaluating Ex h e −rf τa 1{ R τa 0 1{Xs<ξ} ds<T} i under different models. CTMC approximation for rough stochastic local volatility models was constructed in Yang et al. (2025). Acknowledgments Pingping Zeng acknowledges the support from the National Natural Science Foundation of China (Grant No. 12171228). Gongqiu Zhang is supported by the National Natural Science Foundation of China (Gra… view at source ↗
Figure 3
Figure 3. log10(ǫ) vs. log10(Nx) for evaluating Ex h e −rf τa 1{ R τa 0 1{Xs−Xs>ξ} ds<T} i under different models. A Construction of the CTMC Approximation This section is devoted to constructing a CTMC Y with the state space S to approximate a Markov process X whose infinitesimal generator L is depicted in (1). It suffices to specify the transition rate matrix G with elements G(x, y) for x, y ∈ S, such that the infinitesimal… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: log10(ǫ) vs. log10(Nx) for evaluating P∞ k=1 Ex h e −rf τ˜a,k 1{τ˜a,k<T} i under different mod￾els. x − = sup{y ∈ S : y < x}, δ−x = x − x −, x ∈ S\{y0}, δx = δ +x + δ −x 2 , x ∈ S\{y0, yN }, ∇ ±g(x) = ± g(x ±) − g(x) δ±x , x ∈ S\{y0, yN }. On the other hand, the integr…
Figure 5
Figure 5. Figure 5: log10(ǫ) vs. log10(Nx) for evaluating E h e −rf τa,k 1{τa,k<T} [PITH_FULL_IMAGE:figures/full_fig_p027_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    Abate, J. and W. Whitt (1992). The F ourier-series method for inverting transforms of probability distributions. Queueing Systems\/ 10 , 5--87

    work page 1992

  2. [2]

    Avram, and M

    Asmussen, S., F. Avram, and M. R. Pistorius (2004). Russian and A merican put options under exponential phase-type L \'e vy models. Stochastic Processes and their Applications\/ 109\/ (1), 79--111

    work page 2004

  3. [3]

    Avram, F., A. E. Kyprianou, and M. R. Pistorius (2004). Exit problems for spectrally negative L \'e vy processes and applications to ( C anadized) R ussian options. The Annals of Applied Probability\/ 14\/ (1), 215--238

    work page 2004

  4. [4]

    Black, F. and M. Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy\/ 81\/ (3), 637--654

    work page 1973

  5. [5]

    Song, and S

    Cai, N., Y. Song, and S. Kou (2015). A general framework for pricing A sian options under M arkov processes. Operations Research\/ 63\/ (3), 540--554

    work page 2015

  6. [6]

    Uryasev, and M

    Chekhlov, A., S. Uryasev, and M. Zabarankin (2005). Drawdown measure in portfolio optimization. International Journal of Theoretical and Applied Finance\/ 8\/ (1), 13--58

    work page 2005

  7. [7]

    Cherny, V. and J. Ob \'o j (2013). Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model. Finance and Stochastics\/ 17 , 771--800

    work page 2013

  8. [8]

    Cox, J. C. (1996). The constant elasticity of variance option pricing model. Journal of Portfolio Management\/ 23\/ (5), 15--17

    work page 1996

Show all 49 references
  1. [9]

    Li, and G

    Cui, Y., L. Li, and G. Zhang (2024). Pricing and hedging autocallable products by M arkov chain approximation. Review of Derivatives Research\/ 27\/ (3), 259--303

    2024

  2. [10]

    Lee, and Y

    Cui, Z., C. Lee, and Y. Liu (2018). Single-transform formulas for pricing A sian options in a general approximation framework under M arkov processes. European Journal of Operational Research\/ 266\/ (3), 1134--1139

    2018

  3. [11]

    Cui, Z. and D. Nguyen (2016). Omega diffusion risk model with surplus-dependent tax and capital injections. Insurance: Mathematics and Economics\/ 68 , 150--161

    2016

  4. [12]

    Cui, Z. and D. Nguyen (2018). Magnitude and speed of consecutive market crashes in a diffusion model. Methodology and Computing in Applied Probability\/ 20 , 117--135

    2018

  5. [13]

    Cvitanic, J. and I. Karatzas (1995). On portfolio optimization under ``drawdown" constraints. Institute for Mathematics and Its Applications\/ 65 , 35

    1995

  6. [14]

    Douady, R., A. N. Shiryaev, and M. Yor (2000). On probability characteristics of ``downfalls" in a standard B rownian motion. Theory of Probability & Its Applications\/ 44\/ (1), 29--38

    2000

  7. [15]

    Eriksson, B. and M. R. Pistorius (2015). American option valuation under continuous-time M arkov chains. Advances in Applied Probability\/ 47\/ (2), 378--401

    2015

  8. [16]

    Grossman, S. J. and Z. Zhou (1993). Optimal investment strategies for controlling drawdowns. Mathematical Finance\/ 3\/ (3), 241--276

    1993

  9. [17]

    Hadjiliadis, O. and J. Ve c e r (2006). Drawdowns preceding rallies in the B rownian motion model. Quantitative Finance\/ 6\/ (5), 403--409

    2006

  10. [18]

    Jacod, J. and A. Shiryaev (2013). Limit Theorems for Stochastic Processes , Volume 288. Springer Science & Business Media

    2013

  11. [19]

    Kou, S. G. (2002). A jump-diffusion model for option pricing. Management Science\/ 48\/ (8), 1086--1101

    2002

  12. [20]

    Li, and H

    Landriault, D., B. Li, and H. Zhang (2015). On the frequency of drawdowns for B rownian motion processes. Journal of Applied Probability\/ 52\/ (1), 191--208

    2015

  13. [21]

    Li, and H

    Landriault, D., B. Li, and H. Zhang (2017a). On magnitude, asymptotics and duration of drawdowns for L \'e vy models. Bernoulli\/ 23\/ (1), 432--458

    2017

  14. [22]

    Li, and H

    Landriault, D., B. Li, and H. Zhang (2017b). A unified approach for drawdown (drawup) of time-homogeneous M arkov processes. Journal of Applied Probability\/ 54\/ (2), 603--626

    2017

  15. [23]

    Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. The Annals of Probability\/ 5\/ (4), 601--607

    1977

  16. [24]

    Zeng, and G

    Li, L., P. Zeng, and G. Zhang (2024). Speed and duration of drawdown under general M arkov models. Quantitative Finance\/ 24\/ (3-4), 367--386

    2024

  17. [25]

    Madan, D. B., P. P. Carr, and E. C. Chang (1998). The variance gamma process and option pricing. Review of Finance\/ 2\/ (1), 79--105

    1998

  18. [26]

    Magdon-Ismail, M. and A. F. Atiya (2004). Maximum drawdown. Risk Magazine\/ 17\/ (10), 99--102

    2004

  19. [27]

    Magdon-Ismail, M., A. F. Atiya, A. Pratap, and Y. S. Abu-Mostafa (2004). On the maximum drawdown of a B rownian motion. Journal of Applied Probability\/ 41\/ (1), 147--161

    2004

  20. [28]

    Mijatovi\' c , A. and M. Pistorius (2013). Continuously monitored barrier options under M arkov processes. Mathematical Finance\/ 23\/ (1), 1--38

    2013

  21. [29]

    Mijatovi \'c , A. and M. R. Pistorius (2012). On the drawdown of completely asymmetric L \'e vy processes. Stochastic Processes and their Applications\/ 122\/ (11), 3812--3836

    2012

  22. [30]

    Pospisil, L. and J. Vecer (2008). PDE methods for the maximum drawdown. Journal of Computational Finance\/ 12\/ (2), 59--76

    2008

  23. [31]

    Pospisil, L. and J. Vecer (2010). Portfolio sensitivity to changes in the maximum and the maximum drawdown. Quantitative Finance\/ 10\/ (6), 617--627

    2010

  24. [32]

    Vecer, and O

    Pospisil, L., J. Vecer, and O. Hadjiliadis (2009). Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups. Stochastic Processes and their Applications\/ 119\/ (8), 2563--2578

    2009

  25. [33]

    Schuhmacher, F. and M. Eling (2011). Sufficient conditions for expected utility to imply drawdown-based performance rankings. Journal of Banking & Finance\/ 35\/ (9), 2311--2318

    2011

  26. [34]

    Shepp, L. and A. N. Shiryaev (1993). The R ussian option: reduced regret. The Annals of Applied Probability\/ 3\/ (3), 631--640

    1993

  27. [35]

    Yin, and Q

    Song, Q., G. Yin, and Q. Zhang (2013). Weak convergence methods for approximation of the evaluation of path-dependent functionals. SIAM Journal on Control and Optimization\/ 51\/ (5), 4189--4210

    2013

  28. [36]

    Taylor, H. M. (1975). A stopped B rownian motion formula. The Annals of Probability\/ 3\/ (2), 234--246

    1975

  29. [37]

    Vecer, J. (2006). Option pricing: Maximum drawdown and directional trading. Risk\/ 19\/ (12), 88

    2006

  30. [38]

    Vecer, J. (2007). Preventing portfolio losses by hedging maximum drawdown. Wilmott\/ 5\/ (4), 1--8

    2007

  31. [39]

    Woodbury, M. A. (1950). Inverting Modified Matrices . Department of Statistics, Princeton University

    1950

  32. [40]

    Ma, and Z

    Yang, W., J. Ma, and Z. Cui (2025). A general valuation framework for rough stochastic local volatility models and applications. European Journal of Operational Research\/ 322\/ (1), 307--324

    2025

  33. [41]

    Zhang, G. and L. Li (2022). Analysis of M arkov chain approximation for diffusion models with nonsmooth coefficients. SIAM Journal on Financial Mathematics\/ 13\/ (3), 1144--1190

    2022

  34. [42]

    Zhang, G. and L. Li (2023a). A general approach for lookback option pricing under M arkov models. Quantitative Finance\/ 23\/ (9), 1305--1324

    2023

  35. [43]

    Zhang, G. and L. Li (2023b). A general approach for P arisian stopping times under M arkov processes. Finance and Stochastics\/ 27\/ (3), 769--829

    2023

  36. [44]

    Zhang, G. and L. Li (2023c). A general method for analysis and valuation of drawdown risk. Journal of Economic Dynamics and Control\/ 152 , 104669

    2023

  37. [45]

    Zhang, H. (2015). Occupation times, drawdowns, and drawups for one-dimensional regular diffusions. Advances in Applied Probability\/ 47\/ (1), 210--230

    2015

  38. [46]

    Zhang, H. and O. Hadjiliadis (2010). Drawdowns and rallies in a finite time-horizon: drawdowns and rallies. Methodology and Computing in Applied Probability\/ 12 , 293--308

    2010

  39. [47]

    Zhang, H. and O. Hadjiliadis (2012). Drawdowns and the speed of market crash. Methodology and Computing in Applied Probability\/ 14 , 739--752

    2012

  40. [48]

    Leung, and O

    Zhang, H., T. Leung, and O. Hadjiliadis (2013). Stochastic modeling and fair valuation of drawdown insurance. Insurance: Mathematics and Economics\/ 53\/ (3), 840--850

    2013

  41. [49]

    Li, and G

    Zhang, X., L. Li, and G. Zhang (2021). Pricing A merican drawdown options under M arkov models. European Journal of Operational Research\/ 293\/ (3), 1188--1205

    2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.