An $f$-divergence approach for optimal portfolios in exponential Levy models

No-Free-Lunch equivalences for exponential Levy models

We provide equivalence of numerous no-free-lunch type conditions for financial markets where the asset prices are modeled as exponential Levy processes, under possible convex constraints in the use of investment strategies. The general message is the following: if any kind of free lunch exists in these models it has to be of the most egregious type, generating an increasing ealth. Furthermore, we connect the previous to the existence of the numeraire portfolio, both for its particular expositional clarity in exponential Levy models and as a first step in obtaining analogues of the no-free-lunc…

On portfolios generated by optimal transport

First introduced by Fernholz in stochastic portfolio theory, functionally generated portfolio allows its investment performance to be attributed to directly observable and easily interpretable market quantities. In previous works we showed that Fernholz's multiplicatively generated portfolio has deep connections with optimal transport and the information geometry of exponentially concave functions. Recently, Karatzas and Ruf introduced a new additive portfolio generation whose relation with optimal transport was studied by Vervuurt. We show that additively generated portfolio can be interprete…

Minimal Kullback-Leibler Divergence for Constrained Lévy-Itô Processes

Given an n-dimensional stochastic process X driven by P-Brownian motions and Poisson random measures, we seek the probability measure Q, with minimal relative entropy to P, such that the Q-expectations of some terminal and running costs are constrained. We prove existence and uniqueness of the optimal probability measure, derive the explicit form of the measure change, and characterise the optimal drift and compensator adjustments under the optimal measure. We provide an analytical solution for Value-at-Risk (quantile) constraints, discuss how to perturb a Brownian motion to have arbitrary var…

Minimal $f^q$-martingale measures for exponential Lévy processes

Let $L$ be a multidimensional Lévy process under $P$ in its own filtration. The $f^q$-minimal martingale measure $Q_q$ is defined as that equivalent local martingale measure for $\mathcal {E}(L)$ which minimizes the $f^q$-divergence $E[(dQ/dP)^q]$ for fixed $q\in(-\infty,0)\cup(1,\infty)$. We give necessary and sufficient conditions for the existence of $Q_q$ and an explicit formula for its density. For $q=2$, we relate the sufficient conditions to the structure condition and discuss when the former are also necessary. Moreover, we show that $Q_q$ converges for $q\searrow1$ in entropy to the m…

基于效用的回收风险测度：投资组合优化的冗余性与近似方法

Utility-Based Recovery Risk Measures: Redundancy and Approximation for Portfolio Optimization

该研究针对投资组合优化，刻画了基于效用的回收风险测度的冗余性，推导了相关筛选条件，开发了非线性OCE效用的分段线性近似框架并验证了其误差与收敛性。

Lévy噪声驱动的非线性SPDE：适定性、最优控制与不变测度

Nonlinear SPDE driven by Levy noise: Well-posedness, optimal control and invariant measure

针对乘性Lévy噪声驱动的非线性SPDE，论文结合紧性、Skorokhod表示与压缩法证明强解唯一存在，并通过变分法获得最优控制，同时证明无控制方程的不变测度存在。

半鞅的简化演算：乘法补偿子和测度变换

Simplified calculus for semimartingales: Multiplicative compensators and changes of measure

本文提出复值半鞅的乘法补偿方法，证明其随机指数可成为真鞅，推广了Lévy–Khintchin公式，并应用于均值-方差投资组合和测度变换，简化了绝对连续测度变换的处理。

Logarithmic divergences from optimal transport and Rényi geometry

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using the duality in optimal transport, we introduce and study the one-parameter family of $L^{(\pm α)}$-divergences. It includes the Bregman divergence corresponding to the Euclidean quadratic cost, and the $L$-divergence introduced by Pal and the author in connection with portfolio theory and a logarithmic cost function. They admit natural generalizations of exponential family that are closely related to the $α$-family and $q$-exponential family. In partic…

一种 Musielak-Orlicz 方法用于长期记忆过程中的不确定性建模

A Musielak-Orlicz approach for modeling uncertainties in long-memory processes

本文提出了一种新的数学框架，用于建模超OU过程中的不确定性，通过状态依赖的分歧函数在Musielak-Orlicz空间中同时评估重构和莱维测度的扭曲，通过求解优化问题确定累积量的上下界，展示了该方法在水环境应用中的实用性。

在具有跳跃的多变量仿射Volterra模型下Merton问题的最优解

Optimal Merton's Problem under Multivariate Affine Volterra Models with Jumps

本文研究了在允许跳跃的多资产金融市场中，具有指数、幂和对数效用的投资者的资产选择问题，采用martingale optimality principle解决非马尔可夫和非半鞅环境下的Merton优化问题，并通过数值实验展示路径粗糙性和跳跃对价值函数和最优策略的影响。

Exponentially concave functions and a new information geometry

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consi…