Note on the Weak Convergence of Hyperplane $\alpha$-Quantile Functionals and Their Continuity in the Skorokhod J1 Topology

Pietro Maria Sparago

arxiv: 2605.20471 · v1 · pith:3ETGG74Pnew · submitted 2026-05-19 · 🧮 math.PR

Note on the Weak Convergence of Hyperplane α-Quantile Functionals and Their Continuity in the Skorokhod J1 Topology

Pietro Maria Sparago This is my paper

Pith reviewed 2026-05-21 06:38 UTC · model grok-4.3

classification 🧮 math.PR

keywords hyperplane alpha-quantileSkorokhod J1 topologyweak convergencecadlag processescontinuity setfirst hitting timemultidimensional Brownian motion

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The pith

If a multidimensional cadlag process avoids an explicit discontinuity set almost surely, its hyperplane alpha-quantile converges weakly whenever the process converges weakly in the Skorokhod J1 topology.

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

The paper extends the classical one-dimensional alpha-quantile of a stochastic process to a hyperplane version suitable for R^d-valued cadlag paths. It constructs an explicit set of paths on which this functional is continuous with respect to the Skorokhod J1 topology. The construction lets the continuous mapping theorem transfer weak convergence of the processes directly into weak convergence of the quantiles. The same approach yields joint convergence of the quantile together with the first time the quantile is attained, provided the limit process is multidimensional Brownian motion with nontrivial covariance. Readers interested in limit theorems for path-dependent functionals or in higher-dimensional quantile options therefore obtain a concrete condition under which convergence statements carry over without extra tightness work.

Core claim

The hyperplane alpha-quantile is defined as a map from R^d-valued cadlag functions to the reals, and an explicit continuity set for this map in the Skorokhod J1 topology is identified; whenever a limit process lies in this set almost surely, weak convergence of cadlag processes in the Skorokhod sense implies ordinary weak convergence of the associated quantile functionals, and the first hitting time of the quantile converges jointly with the quantile itself when the limit is multidimensional Brownian motion with nontrivial covariance.

What carries the argument

The hyperplane alpha-quantile functional M_{t,α} from R^d-valued cadlag paths to R, together with the explicitly constructed set of paths at which it is continuous in the Skorokhod J1 topology.

If this is right

Weak Skorokhod convergence of the processes implies ordinary weak convergence of their hyperplane alpha-quantiles whenever the limit avoids the discontinuity set.
The first hitting time of the alpha-quantile converges jointly with the quantile itself when the limit process is multidimensional Brownian motion with nontrivial covariance.
The explicit continuity set supplies a verifiable sufficient condition for applying the continuous mapping theorem to this functional without further arguments.

Where Pith is reading between the lines

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

The same continuity-set technique may apply to other path-dependent functionals built from hyperplanes or order statistics.
Simulation schemes that approximate complex cadlag processes by simpler ones converging in Skorokhod topology can now be justified for quantile-based statistics.
Joint convergence of the quantile and its hitting time opens the door to limit theorems for payoffs that depend on both quantities simultaneously.

Load-bearing premise

The limiting process must lie almost surely inside the explicitly constructed continuity set of the hyperplane alpha-quantile functional.

What would settle it

Take a sequence of processes that converge weakly in the Skorokhod topology to a multidimensional Brownian motion that hits the discontinuity set with positive probability, and check whether the associated quantile functionals converge in distribution.

read the original abstract

The ${\alpha}$-quantile of a stochastic process $M_{t,{\alpha}}$ has been introduced in Miura (Hitotsubashi J Commerce Manag 27(1):15-28, 1992), and important distributional results have been derived in Akahori (Ann Appl Probab 5(2):383-388, 1995), Dassios (Ann Appl Probab 5(2):389-398, 1995) and Yor (J Appl Probab 32(2):405-416, 1995), with special attention given to the problem of pricing ${\alpha}$-quantile options. We straightforwardly extend the classical monodimensional setting to $\mathbb{R}^d$ by introducing the hyperplane ${\alpha}$-quantile, and we find an explicit functional continuity set of the ${\alpha}$-quantile as a functional mapping $\mathbb{R}^d$-valued cadlag functions to $\mathbb{R}$. This specification allows us to use continuous mapping and assert that if a $\mathbb{R}^d$-valued cadlag stochastic process $X$ a.s. belongs to such continuity set, then $X^n \Rightarrow X$ (i.e., weakly in the Skorokhod sense) implies $M_{t,{\alpha}}(X^n) \to^\textrm{w} M_{t,{\alpha}}(X)$ (i.e., weakly) in the usual sense. We further the discussion by considering the conditions for convergence of a 'random time' functional of $M_{t,{\alpha}}$, the first time at which the ${\alpha}$-quantile has been hit, applied to sequences of cadlag functions converging in the Skorokhod topology. The Brownian distribution of this functional is studied, e.g., in Chaumont (J Lond Math Soc 59(2):729-741, 1999) and Dassios (Bernoulli 11(1):29-36, 2005). We finally prove the fact that if the limit process of a sequence of cadlag stochastic processes is a multidimensional Brownian motion with nontrivial covariance structure, such random time functional applied to the sequence of processes converges, jointly with the ${\alpha}$-quantile, weakly in the usual sense.

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 note supplies an explicit continuity set for the hyperplane α-quantile on R^d cadlag paths so the continuous mapping theorem applies directly to Skorokhod convergence.

read the letter

The main takeaway is that the authors construct a concrete set of cadlag functions in R^d on which the hyperplane α-quantile functional is continuous with respect to the Skorokhod J1 metric. Once the limiting process sits in that set almost surely, weak convergence of the processes immediately gives weak convergence of the quantiles. They also prove joint convergence with the first hitting time when the limit is multidimensional Brownian motion with nontrivial covariance, and they verify that Brownian motion lands in the set a.s. This is a direct extension of the one-dimensional results in the cited papers and the explicit set is the part that was missing before. The argument follows the standard continuous mapping route without circularity or hidden measurability problems. The soft spot is the narrow focus: the paper does not discuss how restrictive the continuity set is for typical processes or give numerical checks beyond the Brownian case, so its practical reach remains unclear. Still, the logic holds up on its own terms. This is for people who work with multivariate stochastic processes and need to pass limits through quantile functionals, especially in finance applications. A reader who already knows the one-dimensional theory will get immediate value from the multivariate extension. It deserves a serious referee because the construction is explicit and the joint convergence result is new.

Referee Report

0 major / 2 minor

Summary. The paper extends the one-dimensional α-quantile functional of Miura, Akahori, Dassios and Yor to a hyperplane α-quantile for R^d-valued cadlag processes. It constructs an explicit continuity set for the functional with respect to the Skorokhod J1 topology on cadlag paths, applies the continuous mapping theorem to obtain weak convergence of the quantile functionals whenever the limiting process lies in this set almost surely, and further establishes joint weak convergence of the quantile and its first hitting time when the limit is multidimensional Brownian motion with nontrivial covariance.

Significance. The explicit construction of the continuity set and the almost-sure membership argument for Brownian motion constitute a concrete technical contribution that enables direct application of the continuous mapping theorem in the Skorokhod space. This strengthens the toolkit for weak-convergence arguments involving quantile-type functionals and may support extensions of distributional results or option-pricing formulas to the multidimensional setting.

minor comments (2)

[Introduction / Abstract] The abstract states that the continuity set is 'explicit,' yet the precise definition (e.g., the exact conditions on crossing times or local oscillations that guarantee J1-continuity) is not reproduced in the summary paragraph; a one-sentence restatement of the set in the introduction would improve readability.
[Section on random-time functional] In the joint-convergence statement for the hitting-time functional, the paper invokes the nontrivial covariance structure; a brief remark clarifying why the covariance must be non-degenerate (e.g., to avoid degeneracy of the hitting-time distribution) would help readers unfamiliar with the one-dimensional references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the technical contribution in constructing the explicit continuity set, and the recommendation of minor revision. No specific major comments were provided in the report, so we interpret this as an indication that the core arguments are sound. We will incorporate any minor editorial suggestions from the editor during the revision process.

Circularity Check

0 steps flagged

No significant circularity; standard continuous mapping theorem on explicit continuity set

full rationale

The paper constructs an explicit continuity set for the hyperplane α-quantile functional on ℝ^d-valued cadlag paths such that the functional is continuous at every point of the set w.r.t. the Skorokhod J1 topology. Weak convergence of M_{t,α}(X^n) to M_{t,α}(X) then follows directly from the continuous mapping theorem once the limit process X lies in this set a.s. (which is verified for multidimensional Brownian motion with nontrivial covariance). Joint convergence of the quantile and its first hitting time is likewise obtained by the same mechanism. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and all citations (Miura, Akahori, Dassios, Yor, Chaumont) are to independent prior literature with no author overlap or load-bearing self-reference. The derivation is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on background properties of cadlag functions and the Skorokhod topology rather than introducing new fitted parameters or invented entities.

axioms (2)

standard math The space of R^d-valued cadlag functions equipped with the Skorokhod J1 topology is a Polish space in which the continuous mapping theorem applies to measurable functionals.
Invoked to transfer weak convergence from paths to the quantile functional.
domain assumption Multidimensional Brownian motion with nontrivial covariance has almost-surely continuous paths that lie in the continuity set of the hyperplane α-quantile.
Required for the joint convergence statement in the final paragraph of the abstract.

pith-pipeline@v0.9.0 · 5954 in / 1542 out tokens · 36656 ms · 2026-05-21T06:38:49.286905+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We find an explicit functional continuity set of the α-quantile as a functional mapping R^d-valued cadlag functions to R. This specification allows us to use continuous mapping...
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Theorem 1. ... X belongs with probability one to the continuity set of ... (M_{T,α}(x), τ_{M_{T,α}}(x))

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

21 extracted references · 21 canonical work pages

[1]

Fort >0, x∈ D Rd[0,∞)fixed,α7→M t,α(x)is nondecreasing and left-continuous

work page
[2]

The setU(x, t) ={α∈(0,1) : Mt,α(x)< Mt,α+(x)}is at most countable

LetM t,α+(x) := lim β↓α Mt,β(x). The setU(x, t) ={α∈(0,1) : Mt,α(x)< Mt,α+(x)}is at most countable. Proof.We first note that1 {z∈Rd:γ·z≤y}(xs) =1 (−∞,y](γ·x s). (1) Lett >0, x∈ DRd[0,∞). Theny7→ 1 t R t 0 1(−∞,y](γ·x s)dsis nondecreasing, taking values in [0,1]. Also since1 (−∞,y](γ·x s) =1 [γ·xs,∞)(y) it is also right-continuous iny, and there- fore,α7→M...

work page
[3]

Then forε >0 we have {ω : Mt,βk(X(ω))> M t,α(X(ω)) +ε} ⊇ {ω : Mt,α+(X(ω))> M t,α(X(ω)) +ε},∀k∈N which impliesP(M t,α+(X)> M t,α(X) +ε) = 0,∀ε >0 and soP(M t,α+(X)> Mt,α(X)) = 0

If (1) holds, without loss of generality letβ k ↓αfor a sequence (β k)k∈N ⊆(0,1). Then forε >0 we have {ω : Mt,βk(X(ω))> M t,α(X(ω)) +ε} ⊇ {ω : Mt,α+(X(ω))> M t,α(X(ω)) +ε},∀k∈N which impliesP(M t,α+(X)> M t,α(X) +ε) = 0,∀ε >0 and soP(M t,α+(X)> Mt,α(X)) = 0. But then,Xa.s. belongs to the continuity set we have identified in Proposition 2, and by continuo...

work page
[4]

In fact, if the conditions of (2) are satisfied, then we also have (MT,α(xn), τMT ,α(xn))→(M T,α(x), τMT ,α(x)) in the usual sense

If, equivalently,y7→ 1 T R T 0 1{z:γ·z≤y}(xs)dsis continuous orβ7→M T,β(x)is strictly increasing and additionallyα∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)}, then we haveτ MT ,α(xn)→τ MT ,α(x). In fact, if the conditions of (2) are satisfied, then we also have (MT,α(xn), τMT ,α(xn))→(M T,α(x), τMT ,α(x)) in the usual sense. Proof.First note that, sincex n →xin ...

work page
[5]

SupposeM T,α(x)< γ·x 0

From the fact thatx /∈ {x : MT,α(x)< M T,α +(x)}it followsM T,α(xn)→ MT,α(x) by Proposition 2. SupposeM T,α(x)< γ·x 0. For allnsufficiently large MT,α(xn)< γ·x n

work page
[6]

Ift < τ MT ,α(x), then inf s≤t(γ·x) s > M T,α(x) and fornsufficiently large we have inf s≤t(γ·x n)s > M T,α(xn) sot < τ MT ,α(xn) as well for allnlarge enough

Lett /∈ {t : |∆(γ·x) t|>0}; then inf s≤t(γ·x n)s →inf s≤t(γ·x) s. Ift < τ MT ,α(x), then inf s≤t(γ·x) s > M T,α(x) and fornsufficiently large we have inf s≤t(γ·x n)s > M T,α(xn) sot < τ MT ,α(xn) as well for allnlarge enough. Since {t : |∆(γ·x) t|>0}is at most countable due toγ·xbeing c` adl` ag, its complement is dense inR, and we conclude. An analogous ...

work page
[7]

Consider xsuch thatα∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)}, in addition toα /∈U(x, T)

The equivalence of the first conditions can be seen from Proposition 1. Consider xsuch thatα∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)}, in addition toα /∈U(x, T). 8 Suppose againM T,α(x)< γ·x 0. Choose a sequenceβ k ↑α. Ift > τ MT ,α(x) then t >lim k τMT ,βk (x) sot≥τ MT ,βk (x) for someβ k < α. Lett /∈ {t : |∆(γ·x) t|>0}, so that infs≤t(γ·xn)s →inf s≤t(γ·x) s....

work page
[8]

Equivalently,y7→ 1 T R T 0 1{z:γ·z≤y}(xs)dsis continuous orβ7→M T,β(x)is strictly increasing; 3.α∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)} then(M T,α(xn), τMT ,α(xn))→(M T,α(x), τMT ,α(x)). Proof.This immediately follows from Theorem 3 and the fact thatU(x, T) =∅due to the continuity ofxby Lemma 2, which then also implies thatβ7→M T,β(x) is continuous and stri...

work page
[9]

Akahori, J. (1995). Some formulae for a new type of path dependent option.The Annals of Applied Probability, 5(2), 383-388

work page 1995
[10]

Bacry, E., Delattre, S., Hoffmann, S., and Muzy, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics.Stochastic Processes and their Applications, 123, 2475-2499

work page 2013
[11]

Bogachev, V. I. (2007).Measure theory. Springer-Verlag

work page 2007
[12]

Chaumont, L. (1999). A path transformation and its applications to fluctuation theory.Journal of the London Mathematical Society, 59(2), 729-741

work page 1999
[13]

Dassios, A. (1995). The distribution of the quantile Brownian motion with drift and the pricing of related path-dependent options.The Annals of Applied Probability, 5(2), 389-398

work page 1995
[14]

Dassios, A. (2005). On the quantiles of Brownian motion and their hitting times.Bernoulli, 11(1), 29-36

work page 2005
[15]

Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications.Stochastics, 87, 723-746

work page 2015
[16]

and Shiryaev, A

Jacod, J. and Shiryaev, A. N. (2003).Limit theorems for stochastic processes. Springer-Verlag

work page 2003
[17]

and Shreve, S

Karatzas, I. and Shreve, S. E. (1998).Brownian motion and stochastic calculus (2nd edition). Springer

work page 1998
[18]

J., R¨ udiger, F., and Embrechts, P

McNeil, A. J., R¨ udiger, F., and Embrechts, P. (2015).Quantitative risk management (revised edition). Princeton University Press

work page 2015
[19]

Miura, R. (1992). A note on look-back options based on order statistics.Hitotsubashi Journal of Commerce and Management, 27(1), 15-28

work page 1992
[20]

(1999).L´ evy processes and infinitely divisible distributions

Sato, K. (1999).L´ evy processes and infinitely divisible distributions. Cambridge University Press

work page 1999
[21]

Yor, M. (1995). The distribution of Brownian quantiles.Journal of Applied Probability, 32(2), 405-416. 14

work page 1995

[1] [1]

Fort >0, x∈ D Rd[0,∞)fixed,α7→M t,α(x)is nondecreasing and left-continuous

work page

[2] [2]

The setU(x, t) ={α∈(0,1) : Mt,α(x)< Mt,α+(x)}is at most countable

LetM t,α+(x) := lim β↓α Mt,β(x). The setU(x, t) ={α∈(0,1) : Mt,α(x)< Mt,α+(x)}is at most countable. Proof.We first note that1 {z∈Rd:γ·z≤y}(xs) =1 (−∞,y](γ·x s). (1) Lett >0, x∈ DRd[0,∞). Theny7→ 1 t R t 0 1(−∞,y](γ·x s)dsis nondecreasing, taking values in [0,1]. Also since1 (−∞,y](γ·x s) =1 [γ·xs,∞)(y) it is also right-continuous iny, and there- fore,α7→M...

work page

[3] [3]

Then forε >0 we have {ω : Mt,βk(X(ω))> M t,α(X(ω)) +ε} ⊇ {ω : Mt,α+(X(ω))> M t,α(X(ω)) +ε},∀k∈N which impliesP(M t,α+(X)> M t,α(X) +ε) = 0,∀ε >0 and soP(M t,α+(X)> Mt,α(X)) = 0

If (1) holds, without loss of generality letβ k ↓αfor a sequence (β k)k∈N ⊆(0,1). Then forε >0 we have {ω : Mt,βk(X(ω))> M t,α(X(ω)) +ε} ⊇ {ω : Mt,α+(X(ω))> M t,α(X(ω)) +ε},∀k∈N which impliesP(M t,α+(X)> M t,α(X) +ε) = 0,∀ε >0 and soP(M t,α+(X)> Mt,α(X)) = 0. But then,Xa.s. belongs to the continuity set we have identified in Proposition 2, and by continuo...

work page

[4] [4]

In fact, if the conditions of (2) are satisfied, then we also have (MT,α(xn), τMT ,α(xn))→(M T,α(x), τMT ,α(x)) in the usual sense

If, equivalently,y7→ 1 T R T 0 1{z:γ·z≤y}(xs)dsis continuous orβ7→M T,β(x)is strictly increasing and additionallyα∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)}, then we haveτ MT ,α(xn)→τ MT ,α(x). In fact, if the conditions of (2) are satisfied, then we also have (MT,α(xn), τMT ,α(xn))→(M T,α(x), τMT ,α(x)) in the usual sense. Proof.First note that, sincex n →xin ...

work page

[5] [5]

SupposeM T,α(x)< γ·x 0

From the fact thatx /∈ {x : MT,α(x)< M T,α +(x)}it followsM T,α(xn)→ MT,α(x) by Proposition 2. SupposeM T,α(x)< γ·x 0. For allnsufficiently large MT,α(xn)< γ·x n

work page

[6] [6]

Ift < τ MT ,α(x), then inf s≤t(γ·x) s > M T,α(x) and fornsufficiently large we have inf s≤t(γ·x n)s > M T,α(xn) sot < τ MT ,α(xn) as well for allnlarge enough

Lett /∈ {t : |∆(γ·x) t|>0}; then inf s≤t(γ·x n)s →inf s≤t(γ·x) s. Ift < τ MT ,α(x), then inf s≤t(γ·x) s > M T,α(x) and fornsufficiently large we have inf s≤t(γ·x n)s > M T,α(xn) sot < τ MT ,α(xn) as well for allnlarge enough. Since {t : |∆(γ·x) t|>0}is at most countable due toγ·xbeing c` adl` ag, its complement is dense inR, and we conclude. An analogous ...

work page

[7] [7]

Consider xsuch thatα∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)}, in addition toα /∈U(x, T)

The equivalence of the first conditions can be seen from Proposition 1. Consider xsuch thatα∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)}, in addition toα /∈U(x, T). 8 Suppose againM T,α(x)< γ·x 0. Choose a sequenceβ k ↑α. Ift > τ MT ,α(x) then t >lim k τMT ,βk (x) sot≥τ MT ,βk (x) for someβ k < α. Lett /∈ {t : |∆(γ·x) t|>0}, so that infs≤t(γ·xn)s →inf s≤t(γ·x) s....

work page

[8] [8]

Equivalently,y7→ 1 T R T 0 1{z:γ·z≤y}(xs)dsis continuous orβ7→M T,β(x)is strictly increasing; 3.α∈ {α : limβ→α τMT ,β(x) =τ MT ,α(x)} then(M T,α(xn), τMT ,α(xn))→(M T,α(x), τMT ,α(x)). Proof.This immediately follows from Theorem 3 and the fact thatU(x, T) =∅due to the continuity ofxby Lemma 2, which then also implies thatβ7→M T,β(x) is continuous and stri...

work page

[9] [9]

Akahori, J. (1995). Some formulae for a new type of path dependent option.The Annals of Applied Probability, 5(2), 383-388

work page 1995

[10] [10]

Bacry, E., Delattre, S., Hoffmann, S., and Muzy, J. F. (2013). Some limit theorems for Hawkes processes and application to financial statistics.Stochastic Processes and their Applications, 123, 2475-2499

work page 2013

[11] [11]

Bogachev, V. I. (2007).Measure theory. Springer-Verlag

work page 2007

[12] [12]

Chaumont, L. (1999). A path transformation and its applications to fluctuation theory.Journal of the London Mathematical Society, 59(2), 729-741

work page 1999

[13] [13]

Dassios, A. (1995). The distribution of the quantile Brownian motion with drift and the pricing of related path-dependent options.The Annals of Applied Probability, 5(2), 389-398

work page 1995

[14] [14]

Dassios, A. (2005). On the quantiles of Brownian motion and their hitting times.Bernoulli, 11(1), 29-36

work page 2005

[15] [15]

Gerhold, S., Kleinert, M., Porkert, P., and Shkolnikov, M. (2015). Small time central limit theorems for semimartingales with applications.Stochastics, 87, 723-746

work page 2015

[16] [16]

and Shiryaev, A

Jacod, J. and Shiryaev, A. N. (2003).Limit theorems for stochastic processes. Springer-Verlag

work page 2003

[17] [17]

and Shreve, S

Karatzas, I. and Shreve, S. E. (1998).Brownian motion and stochastic calculus (2nd edition). Springer

work page 1998

[18] [18]

J., R¨ udiger, F., and Embrechts, P

McNeil, A. J., R¨ udiger, F., and Embrechts, P. (2015).Quantitative risk management (revised edition). Princeton University Press

work page 2015

[19] [19]

Miura, R. (1992). A note on look-back options based on order statistics.Hitotsubashi Journal of Commerce and Management, 27(1), 15-28

work page 1992

[20] [20]

(1999).L´ evy processes and infinitely divisible distributions

Sato, K. (1999).L´ evy processes and infinitely divisible distributions. Cambridge University Press

work page 1999

[21] [21]

Yor, M. (1995). The distribution of Brownian quantiles.Journal of Applied Probability, 32(2), 405-416. 14

work page 1995