Julien Dubedat
Various features of the two-parameter family of Schramm-Loewner evolutions SLE(κ,ρ) are studied. In particular, we derive certain restriction properties that lead to a ``strong duality'' conjecture, which is an identity in law between the outer boundary of a variant of the SLE(κ) process for κ\ge 4 and a variant of the SLE(16/κ) process.
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We present an explicit relation between representations of the Virasoro algebra and polynomial martingales in stochastic Loewner evolutions (SLE). We show that the Virasoro algebra is the spectrum generating algebra of SLE martingales. This is based on a new representation of the Virasoro algebra, inspired by the Borel-Weil construction, acting on functions depending on coordinates parametrizing conformal maps.
UST branches, martingales, and multiple SLE(2)
本文研究了均匀生成树中多个边界到边界分支的局部标度极限，发现其为局部多重SLE(2)过程，并通过鞅可观测性和分区函数收敛性进行了证明。
We implement a version of radial conformal field theory in a family of statistical fields generated by central charge modification of the Gaussian free field and show that the correlation functions of such fields under the insertion of one-leg operator form a collection of radial SLE martingale-observables. We apply the renormalization procedure to the multi-point vertex fields with the neutrality condition to expand this collection and study its basic properties.
Irregular SLE(4) martingales and isomonodromic deformations
研究非傅里叶型单调变形的参数演化，构造SLE(4)过程在双极点下的鞅可观测量，结合预Schwarzian和Schwarzian，通过连贯BPZ方程刻画其唯一性。
We implement a version of conformal field theory in a doubly connected domain to connect it to the theory of annulus SLE of various types, including the standard annulus SLE, the reversible annulus SLE, and the annulus SLE with several force points. This implementation considers the statistical fields generated under the OPE multiplication by the Gaussian free field and its central/background charge modifications with a weighted combination of Dirichlet and excursion-reflected boundary conditions. We derive the Eguchi-Ooguri version of Ward's equations and Belavin-Polyakov-Zamolodchikov equati…
We consider a coupling of the Gaussian free field with slit holomorphic stochastic flows, called ($δ,σ$)-SLE, which contains known SLE processes (chordal, radial, and dipolar) as particular cases. In physical terms, we study a free boundary conformal field theory with one scalar bosonic field, where Green's function is assumed to have some general regular harmonic part. We establish which of these models allow coupling with ($δ,σ$)-SLE, or equivalently, when the correlation functions induce local ($δ,σ$)-SLE martingales (martingale observables).
Scharmm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT we consider CFT correlation functions which are martingales with respect to SLE. We establish connection between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset…
We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick's theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half plane. This is then applied to stochastic…
An annulus SLE$_κ$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus SLE$_4$, SLE$_8$ and SLE$_{8/3}$. From the local martingale for annulus SLE$_4$ we find a candidate of discrete lattice model that may have annulus SLE$_4$ as its scaling limit. The local martingale for annulus SLE$_{8/3}$ is similar to those for chordal and radial SLE$_{8/3}$. But it seems that annulus SLE$_{8/3}$ does not satisfy the restriction property.
We present a relation between conformal field theories (CFT) and radial stochastic Schramm-Loewner evolutions (SLE) similar to that we previously developed for the chordal SLEs. We construct an important local martingale using degenerate representations of the Virasoro algebra. We sketch how to compute derivative exponants and the restriction martingales in this framework. In its CFT formulation, the SLE dual Fokker-Planck operator acts as the two-particle Calogero hamiltonian on boundary primary fields and as the dilatation operator on bulk primary fields localized at the fixed point of the S…
We construct martingale observables for systems of multiple SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov, extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit expressions for the Green's function and Schramm's formula in the case of two curves growing towards infinity. In the special case when the curves are `fused' and start at the same point, some of the formulas we prove were predicted in the physics literature.
A Multi-Marginal C-Convex Duality Theorem for Martingale Optimal Transport
本文将 Beiglböck 等人关于两边际鞅最优传输的凸对偶定理推广到多边际、多期设定。
Duality for a Martingale Transport Problem with Moment Constraints
该研究针对第二边缘分布由有限矩约束指定的弱鞅最优运输问题，推导有限维对偶公式并证明无对偶间隙，还明确了最优鞅耦合的结构及其与Bass鞅的关系。
Based on the multidimensional irreducible paving of De March & Touzi, we provide a multi-dimensional version of the quasi sure duality for the martingale optimal transport problem, thus extending the result of Beiglböck, Nutz & Touzi. Similar, we also prove a disintegration result which states a natural decomposition of the martingale optimal transport problem on the irreducible components, with pointwise duality verified on each component. As another contribution, we extend the martingale monotonicity principle to the present multi-dimensional setting. Our results hold in dimensions 1, 2, and…
In this paper, we introduce the notion of martingale Hardy-amalgam spaces: $ H^s_{p,q},\,\,\mathcal{Q}_{p,q}$ and $\mathcal{P}_{p,q}$. We present two atomic decompositions for these spaces. The dual space of $H^s_{p,q}$ for $0<p\le q\le 1$ is shown to be a Campanato-type space.
We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.
The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of cadlag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming princ…
We improve the geometric properties of SLE$(κ;\vecρ)$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $κ\in (4,8)$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from a random point. Using this fact together with a similar proposition in the case that $κ\ge 8$, we obtain a description of the boundary of a standard chordal SLE$(κ)$ hull for $κ>4$, at a finite stopping time. Finally, we prove that for $κ>4$, in many c…
We present in this paper some embeddings of various dyadic martingale Hardy-amalgam spaces $H^S_{p,q},\,\, H^s_{p,q},\,\,H^*_{p,q},\,\,\mathcal{Q}_{p,q}$ and $\mathcal{P}_{p,q}$ of the real line. In the same settings, we characterize the dual of $H^s_{p,q}$ for large $p$ and $q$. We also introduce a Garsia-type space $\mathcal{G}_{p,q}$ and characterize its dual space.
Martingale Optimal Transport and Martingale Schrödinger Bridges for Calibration of Stochastic Volatility Models
本文研究了连续时间鞅最优传输与鞅Schrödinger桥问题，建立了对偶公式并给出替代证明，应用于SVM至SPX期权及SPX与VIX期权的联合校准。
The objective of this paper is to develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In (A) we follow the approach taken in the Entropy Optimal Transport (EOT) primal problem by Liero et al. "Optimal entropy-transport problems and a new Hellinger-Kantorovic distance between positive measures", Invent. math. 2018, but we add the constraint, typical of Martingale Optimal Transport (MOT) theory, that the infimum of the cost functional is taken over martingale probability measures, instead of finite positive measures, as i…
We derive some geometric properties of chordal SLE$(κ;\vecρ)$ processes. Using these results and the method of coupling two SLE processes, we prove that the outer boundary of the final hull of a chordal SLE$(κ;\vecρ)$ process has the same distribution as the image of a chordal SLE$(κ';\vec{ρ'})$ trace, where $κ>4$, $κ'=16/κ$, and the forces $\vecρ$ and $\vec{ρ'}$ are suitably chosen. We find that for $κ\ge 8$, the boundary of a standard chordal SLE$(κ)$ hull stopped on swallowing a fixed $x\in\R\sem\{0\}$ is the image of some SLE$(16/κ;\vecρ)$ trace started from $x$. Then we obtain a new proof…
In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited ``duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and…
Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_κ$ curves is well defined for all $κ<8$. Our proof uses a two-interior-point local martingale.
In this note, we prove a version of the conjectured duality of Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal $\SLE_κ$, $κ>4$, and appropriate versions of $\SLE_{\hatκ}$, $\hatκ=16/κ$.
The space of H1 martingales is interesting because of its duality with the space of BMO martingales. It is straightforward to show that every H1 martingale is a uniformly integrable martingale. However, the converse is not true. That is to say, some uniformly integrable martingales are not H1 martingales. This brief note provides a template for systematically constructing such processes.
The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge-Kantorovich type martingale transport problem of maximizing the expe…
Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems f…
We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [BNT16] established such existence for weak (quasi-sure) duality, [BHP13] showed existence for the natural stronger pointwise duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when $y\mapsto c(x,y)$ is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost $c(x,y)$ is twice continuously differentiable in $y$. Further, this may not be improved as we give examples with $…
$q_x$-Bass Martingales: Optimality, Existence and Convergence
本研究扩展$q$-Bass鞅为$q_x$-Bass鞅，通过弱最优传输问题定义其为转移核最接近参考核族的鞅耦合，建立强对偶后提出计算算法并证其收敛与存在性，一维情形下简化为不动点迭代。
We give a definitive treatment of duality for optimal consumption over the infinite horizon, in a semimartingale incomplete market satisfying no unbounded profit with bounded risk (NUPBR). Rather than base the dual domain on (local) martingale deflators, we use a class of supermartingale deflators such that deflated wealth plus cumulative deflated consumption is a supermartingale for all admissible consumption plans. This yields a strong duality, because the enlarged dual domain of processes dominated by deflators is naturally closed, without invoking its closure. In this way we automatically …
Duality and the well-posedness of a martingale problem
本文证明了在存在有界连续对偶函数和对偶过程的条件下，鞅问题具有存在性与唯一性，无需依赖不同类型过程的近似序列，并通过Fleming-Viot等测度值扩散模型验证了该方法。
We propose variants of Schramm-Loewner evolution (SLE) that are related to superconformal algebras following the group theoretical formulation of SLE, in which the relevant stochastic differential equation is derived from a random process on an infinite dimensional Lie group. In this paper, we consider random processes on certain kind of groups of superconformal transformations generated by exponentiated elements of the Grassmann envelop of superconformal algebras. We also provide a prescription of obtaining local martingales from a representation of the superconformal algebra after integratio…
Geometry of vectorial martingale optimal transportations and duality
本文提出面向多资产期权定价与对冲的VMOT问题，证明其无限维线性规划的对偶最优解存在，并据此揭示原最优传输的几何及极端条件相关性结构。
In a two-period financial market where a stock is traded dynamically and European options at maturity are traded statically, we study the so-called martingale Schrödinger bridge Q*; that is, the minimal-entropy martingale measure among all models calibrated to option prices. This minimization is shown to be in duality with an exponential utility maximization over semistatic portfolios. Under a technical condition on the physical measure P, we show that an optimal portfolio exists and provides an explicit solution for Q*. This result overcomes the remarkable issue of non-closedness of semistati…
We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decompositio…
SLE Loop Measure and Liouville Quantum Gravity
本文利用刘维尔量子引力（LQG）研究SLE环路测度，证明了共形环路系综与SLE环路测度的等价性，建立了κ与16/κ的对偶性，并导出了电厚度矩的精确公式。
In this paper we consider random planar maps weighted by the self-dual Fortuin--Kasteleyn model with parameter $q \in (0,4)$. Using a bijection due to Sheffield and a connection to planar Brownian motion in a cone we obtain rigorously the value of the critical exponent associated with the length of cluster interfaces, which is shown to be $$ \frac{4}π \arccos \left( \frac{\sqrt{2 - \sqrt{q}}}{2} \right)=\frac{κ'}{8}. $$ where $κ' $ is the SLE parameter associated with this model. We also derive the exponent corresponding to the area enclosed by a loop which is shown to be 1 for all values of $…
The Fleming-Viot (FV) process is a measure-valued diffusion that models the evolution of type frequencies in a countable population which evolves under resampling (genetic drift), mutation, and selection. In the classic FV model the fitness (strength) of types is given by a measurable function. In this paper, we introduce and study the Fleming-Viot process in random environment (FVRE), when by random environment we mean the fitness of types is a stochastic process with càdlàg paths. We identify FVRE as the unique solution to a so called quenched martingale problem and derive some of its proper…
This paper focuses on the time-changed Q-Wiener process, a Hilbert space-valued sub-diffusion. It is a martingale with respect to an appropriate filtration, hence a stochastic integral with respect to it is definable. For the resulting integral, two change of variables formulas are derived. Via a duality theorem for integrals, existence and uniqueness theorems for stochastic differential equations (SDEs) driven by the time-changed Q-Wiener process are discussed. Associated fractional Fokker-Planck-Kolmogorov equations are derived using either a time-changed Itô formula or duality. Connections …
Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology
本文研究生物与公共卫生领域的随机双侧退化聚集-扩散方程，结合辅助非退化系统与Faedo--Galerkin方法证明鞅解的存在唯一性，并通过数值模拟揭示环境噪声对系统行为的影响，为公共卫生相关研究提供参考。
Duality and DeepMartingale for High-Dimensional Optimal Switching: Computable Upper Bounds and Approximation-Expressivity Guarantees
本文研究有限时间最优切换问题，提出基于深度学习的对偶框架，通过马尔可夫惩罚项推导对偶表示，并建立收敛性保证，同时提供神经网络的表达性分析，证明在高维情况下可避免维度灾难。
We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock ("process increasing in convex order", by Hirsch, Profeta, Roynette and Yor). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordere, Obloj, Spoida and Touzi. Under technical conditions, some explicit characteristi…
DeepMartingale: Duality of the Optimal Stopping Problem with Expressivity and High-Dimensional Hedging
DeepMartingale通过深度学习框架解决高维最优停止问题，提供紧致的对偶上界和可扩展的delta对冲策略。
Robust Hedging of American Options via Aggregated Snell Envelopes
该研究在非支配框架下构建Snell包络聚合器，用于一般半鞅环境下美式期权的鲁棒对冲，建立对偶关系并证明最小对冲策略存在，结果可应用于通过半鞅特征量化不确定性的金融市场模型。
Existence, uniqueness and ergodicity for the centered Fleming-Viot process
本文通过鞅问题刻画中心化 Fleming–Viot 过程，利用渐近展开与弱化对偶方法证明有限矩初始条件下解的存在唯一性，并确立其全变差指数遍历性及不变测度。
We consider the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.
Positive Markov processes in Laplace duality
本文建立了一个关于正马尔可夫过程拉普拉斯对偶的一般框架，证明了过程具有拉普拉斯对偶当且仅当满足某种完全单调性条件，并探讨了在边界0和∞处的弱连续性和吸收性性质。
In this paper we study a two-level branching model for virus populations under cell division. We assume that the cells are carrying virus populations which evolve as a branching particle system with competition, while the cells split according to a Yule process thereby dividing their virus populations into two independently evolving sub-populations. We then assume that sizes of the virus populations are huge and characterize the fast branching rate and huge population density limit as the solution of a well-posed martingale problem. While verifying tightness is quite standard, we provide a Fey…
Interfaces of the two-dimensional voter model in the context of SLE
本文从SLE视角研究二维选民模型界面的几何性质，发现其弦界面在局部具有分形维数3/2，但在宏观尺度上趋于变直，证实了Holmes等人的猜想。
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