# Registered Data

## [00382] Stochastic control and stochastic analysis in finance and insurance

**Session Date & Time**:- 00382 (1/3) : 2C (Aug.22, 13:20-15:00)
- 00382 (2/3) : 2D (Aug.22, 15:30-17:10)
- 00382 (3/3) : 2E (Aug.22, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: Stochastic control and stochastic analysis have played core roles in quantitative finance and insurance. Newly emerging financial and risk models, trading constraints, behavioral decision making and time inconsistency issues have brought many new mathematical challenges. Some novel PDE techniques, mean field game formulation, optimal transport, deep learning and reinforcement learning have been rapidly developed in addressing these problems. The goal of this minisymposium is to provide a forum for some experts to exchange ideas and explore possible collaborations in modelling and methodologies in financial and insurance applications.**Organizer(s)**: Xiaolu Tan, Kazutoshi Yamazaki, Xiang Yu**Classification**:__91Gxx__,__93Exx__,__60-XX__,__90Bxx__**Speakers Info**:- Shuoqing Deng (The Hong Kong University of Science and Technology)
- Guanxing Fu (The Hong Kong Polytechnic University)
- Kazutoshi Yamazaki (The University of Queensland)
- Xun Li (The Hong Kong Polytechnic University)
- Yating Liu (University of Paris Dauphine)
- Thibaut Mastrolia (University of California, Berkeley)
- Xiaolu Tan (The Chinese University of Hong Kong)
- Hiroshi Tsukada (Kagoshima University)
**Xiang Yu**(The Hong Kong Polytechnic University)- Jiacheng Zhang (University of California, Berkeley)
- Xiaowen Zhou (University of Concordia)
- Zhou Zhou (University of Sydney)

**Talks in Minisymposium**:**[02144] Optimal Consumption with Loss Aversion and Reference to Past Spending Maximum****Author(s)**:**Xun LI**(The Hong Kong Polytechnic University)- Xiang Yu (The Hong Kong Polytechnic University)
- Zhang Qinyi (The Hong Kong Polytechnic University)

**Abstract**: This talk studies an optimal consumption problem for a loss-averse agent with reference to past consumption maximum. To account for loss aversion on relative consumption, an S-shaped utility is adopted that measures the difference between the non-negative consumption rate and a fraction of the historical spending peak. We consider the concave envelope of the realization utility with respect to consumption, allowing us to focus on an auxiliary HJB variational in- equality on the strength of concavification principle and dynamic programming arguments. By applying the dual transform and smooth-fit conditions, the auxiliary HJB variational inequality is solved in closed-form piecewisely and some thresholds of the wealth variable are obtained. The optimal consumption and investment control of the original problem can be derived analytically in the piecewise feedback form. The rigorous verification proofs on optimality and concavification principle are provided. Some numerical sensitivity analysis and financial implications are also presented.

**[02210] Lévy bandits under Poissonian decision times****Author(s)**:**Kazutoshi Yamazaki**(University of Queensland)- Jose Luis Perez (CIMAT)

**Abstract**: We consider a version of the continuous-time multi-armed bandit problem where decision opportunities arrive at Poisson arrival times, and study its Gittins index policy. When driven by spectrally one-sided Lévy processes, the Gittins index can be written explicitly in terms of the scale function, and is shown to converge to that in the classical Lévy bandit of Kaspi and Mandelbaum (1995).

**[02775] CONVERGENCE OF POLICY IMPROVEMENT FOR ENTROPY-REGULARIZED STOCHASTIC CONTROL PROBLEMS****Author(s)**:- Yu-Jui Huang (University of Colorado, Boulder)
- Zhenhua Wang (University of Michigan)
**Zhou Zhou**(University of Sydney)

**Abstract**: For a general entropy-regularized stochastic control problem on an infinite horizon, we prove that a policy improvement algorithm (PIA) converges to an optimal relaxed control. Contrary to the standard stochastic control literature, classical Hölder estimates of value functions do not ensure the convergence of the PIA, due to the added entropy-regularizing term. To circumvent this, we carry out a delicate estimation by moving back and forth between appropriate Hölder and Sobolev spaces. This requires new Sobolev estimates designed specifically for the purpose of policy improvement and a nontrivial technique to contain the entropy growth. Ultimately, we obtain a uniform Hölder bound for the sequence of value functions generated by the PIA, thereby achieving the desired convergence result. Characterization of the optimal value function as the unique solution to an exploratory Hamilton– Jacobi–Bellman equation comes as a by-product.

**[02785] Functional convex order for the McKean-Vlasov equation****Author(s)**:**Yating Liu**(Paris Dauphine University)- Gilles Pagès (Sorbonne University)

**Abstract**: We introduce the functional convex order for two McKean-Vlasov processes X and Y with respective marginal distributions $(\mu_t)_{t\in[0, T]}$ and $(\nu_t)_{t\in[0, T]}$. For a convex functional $G$ defined on the product space involving the path space and its marginal distribution space, we obtain $\mathbb{E}\,G(X, (\mu_t)_{t\in[0, T]}) \leq \mathbb{E}\,G(Y, (\nu_t)_{t\in[0, T]})$ under appropriate conditions. This presentation also includes two applications of the convex order result to mean-field control and mean-field games.

**[02873] On the Entropy martingale optimal transport****Author(s)**:**Shuoqing Deng**(The Hong Kong University of Science and Technology)- Erhan Bayraktar (University of Michigan)
- Dominykas Norgilas (University of Michigan)

**Abstract**: We study the Entropy Martingale Optimal Transport of Doldi and Frittelli (Finance&Stochastics, 2023). Compared with classical MOT, marginal constraints and linear pricing rules are respectively replaced by penalisation on deviations from the reference measures and utility-induced nonlinear rules. Inspired by techniques from classical MOT, we prove the duality with different arguments and weaker conditions. In particular, combining minimax arguments and the optional decomposition theorem, we generalise their duality without continuity requirement for the dynamic strategy.

**[02892] Skew Brownian Motion with Two-Valued Drift****Author(s)**:**Xiaowen Zhou**(Concordia University)

**Abstract**: We consider a skew Brownian motion with two-valued drift as the unique solution to the following SDE dX_t=\big(\mu_- 1_{\{X_ta\}}\big)dt +dB_t+\beta dL^a_t(X), where $\mu_-$ and $\mu_+ $ are constants, $-1<\beta<1$, $B$ is a Brownian motion and $L^a_t(X)$ denotes the symmetric local time for $X$ at level $a$. Such a process can be identified as a regime switching model depending on whether the process $X$ takes values above or below level $a$. In this talk we present some properties for such processes and discuss an optimization problem that is motivated by the optimal dividend problem for risk processes. This talk is based on joint work with Zhongqin Gao.

**[02928] Pathwise uniqueness of SDEs driven by stable processes****Author(s)**:**Hiroshi Tsukada**(Kagoshima University)

**Abstract**: We consider one-dimensional stochastic differential equations (SDEs) driven by strictly stable processes. In this talk, we give some non-Lipschitz conditions on diffusion and drift coefficients under which the pathwise uniqueness of solutions to the SDEs is established. Moreover, we provide sufficient conditions for the non-contact property of strong solutions to the SDEs with different initial values.

**[03167] Incentive to shape equilibria in double auction markets****Author(s)**:**Thibaut Mastrolia**(UC Berkeley)- Mathieu Rosenbaum (Ecole Polytechnique)
- Joffrey Derchu (Ecole Polytechnique)

**Abstract**: We study a toy two-player game for periodic double auction markets with imperfect information between the players. It allows us to link market spreads with signal strength. We first derive some market statistics related to the model studied. Then, we characterize Nash equilibria in cases with or without incentives from the exchange. This enables us to derive new insights about price formation and incentives design.

**[03347] Continuous time q-learning for McKean-Vlasov control problems****Author(s)**:- Xiaoli Wei (Tsinghua Shenzhen International Graduate School)
**Xiang Yu**(The Hong Kong Polytechnic University)

**Abstract**: For continuous time McKean-Vlasov control problems, we study the continuous time version of Q-learning for reinforcement learning under entropy regularization. Due to the complexity of distribution dependence, the counterpart of the martingale characterization of q-function in the single-agent control problem fails in our framework. To resolve the challenge, we introduce two distinct q-functions, which share the same integral under all test stochastic policies. The first q-function is associated to the optimal policy and policy improvement, and the second q-function can be used to develop the weak martingale characterization of some related processes under all test stochastic policies. Based on the weak martingale characterization and the relationship between two q-functions, we can design some q-learning algorithms for the learning McKean-Vlasov control problems and present several financial applications.

**[03375] A mean-field version of Bank--El Karoui’s representation of stochastic processes****Author(s)**:**Xiaolu Tan**(The Chinese University of Hong Kong)

**Abstract**: We investigate a mean-field version of Bank--El Karoui's representation theorem of stochastic processes. Under different technical conditions, we established some existence and uniqueness results. As motivation and first applications, the results of mean-field representation provide a unified approach for studying various mean-field games (MFGs) in the setting with common noise and multiple populations, including the MFG of timing and the MFG with singular control, etc. As a crucial technical step, a stability result was provided on the classical Bank--El Karoui’s representation theorem. It has its own interests and other applications, such as deriving the stability results of optimizers (in the strong sense) for a class of optimal stopping and singular control problems.

**[04566] On time-consistent equilibrium stopping under aggregation of diverse discount rates****Author(s)**:**Jiacheng Zhang**(UC Berkeley)- Shuoqing Deng (The Hong Kong University of Science and Technology)
- Xiang Yu (The Hong Kong Polytechnic University)

**Abstract**: This paper studies the central planner's decision making on behalf of a group of members with diverse discount rates. In the context of optimal stopping, we work with a smooth aggregation preference to incorporate all heterogeneous discount rates with an attitude function that reflects the aggregation rule in the same spirit of ambiguity aversion in the smooth ambiguity preference proposed in Klibanoff et al.(2005). The optimal stopping problem renders to be time inconsistent, for which we develop an iterative approach using consistent planning and characterize all time-consistent equilibria as fixed points of an operator in the setting of one-dimensional diffusion processes. We provide some sufficient conditions on both the underlying models and the attitude function such that the smallest equilibrium attains the optimal equilibrium in which the attitude function becomes equivalent to the linear aggregation rule as of diversity neutral. When the sufficient condition of the attitude function is violated, we can illustrate by various examples that the characterization of the optimal equilibrium may differ significantly from some existing results for an individual agent, which now sensitively depends on the attitude function and the diversity distribution of discount rates.