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Title. Optimal consumption under loss-averse multiplicative habit-formation preferences
Abstract. We consider a loss-averse version of the multiplicative habit formation preference and the corresponding optimal investment and consumption strategies over an infinite horizon. The agent's consumption preference is depicted by a general S-shaped utility function of her consumption-to-habit ratio. By considering the concave envelope of the S-shaped utility and the associated dual value function, we provide a thorough analysis of the HJB equation for the concavified problem via studying a related nonlinear free boundary problem. Based on established properties of the solution to this free boundary problem, we obtain the optimal consumption and investment policies in feedback form. Some new and technical verification arguments are developed to cope with generality of the utility function. The equivalence between the original problem and the concavified problem readily follows from the structure of the feedback policies. We also discuss some quantitative properties of the optimal policies under several commonly used S-shaped utilities, complemented by illustrative numerical examples and their financial implications.
Title. Sequential optimal contracting
Abstract. We study a principal-agent model in continuous time with multiple lump-sum payments (contracts) paid at different times. We study three types of agreements: (a) The contracts are negotiated at the initial time and are (possibly) path-dependent; (b) The contracts are negotiated at the initial time, and the payments depend on the increment of the output process; (c) The contracts are renegotiated. Following the approach introduced by Sannikov (2008), and Cvitanic-Possamai-Touzi (2018), we write the Stackelberg game between the principal and agent as a standard stochastic optimal control problem, which can be solved by verification arguments.
Title. A Resource Sharing Model with Local Time Interactions
Abstract. We consider a resource sharing model for $N$ financial firms which takes the form of a semimartingale reflected Brownian motion in the positive orthant. When the asset value of any firm hits zero, the other firms contribute to a local time reflection term to keep the distressed firm's assets non-negative. Depending on a critical parameter $\alpha$, which reflects either friction or subsidy, the interval of existence for this scheme is shown to be either infinite, or almost surely finite. In the mean-field limit, the behaviour of the Fokker--Planck equation also depends on $\alpha$: either solutions exists for all time or the equation exhibits finite-time blowup. The passage from finite to mean-field model is investigated. We also analyze connections between our model and systemic risk models involving hitting times, the up-the-river problem of Aldous, various free boundary PDEs, and Atlas models from Stochastic Portfolio Theory.
Title. From rank-based models with common noise to pathwise entropy solutions of SPDEs
Abstract. We study the mean field limit of a rank-based model with common noise, which arises as an extension to models for the market capitalization of firms in stochastic portfolio theory. We show that, under certain conditions on the drift and diffusion coefficients, the empirical cumulative distribution function converges to the solution of a stochastic PDE. A key step in the proof, which is of independent interest, is to show that any solution to an associated martingale problem is also a pathwise entropy solution to the stochastic PDE, a notion introduced in a recent series of papers by Gess, Souganidis, Lions, and Perthame).
Title. Optimal Loss Reporting in Continuous Time with Full Insurance
Abstract. Experience-rating systems, such as bonus-malus systems, offer discounts on future premiums to insureds who maintain a claim-free record, thereby generating incentives for insureds to not report certain losses. This paper formulates an optimal loss-reporting problem in a continuous-time framework for an insured with full insurance. The insured follows a barrier strategy for reporting losses and aims to maximize the expected exponential utility of her terminal wealth over a random horizon t. In the special case when t has a constant hazard rate, we obtained the optimal barrier strategy in closed form, which is a strictly positive constant. In the general case of a time-varying hazard rate for t, we obtain the optimal barrier strategy in semi-closed form, subject to solving a system of ordinary differential equations. Additionally, we uncover several noteworthy qualitative insights into both the optimal barrier strategy and the associated value functions.
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Title. Utility maximization under endogenous pricing
Abstract. We study the expected utility maximization problem of a large investor who is allowed to make transactions on tradable assets in an incomplete financial market with endogenous permanent market impacts. The asset prices are assumed to follow a nonlinear price curve quoted in the market as the utility indifference curve of a representative liquidity supplier. We show that optimality can be fully characterized via a system of coupled forward-backward stochastic differential equations (FBSDEs) which corresponds to a non-linear backward stochastic partial differential equation (BSPDE). We show existence of solutions to the optimal investment problem and the FBSDEs in the case where the driver function of the representative market maker grows at least quadratically or the utility function of the large investor falls faster than quadratically or is exponential. Furthermore, we derive smoothness results for the existence of solutions of BSPDEs. Examples are provided when the market is complete or the utility function is exponential.
Title. A mean field optimal stopping problem with common noise
Abstract. This paper focuses on a mean-field optimal stopping problem with common noise, inspired by Talbi, Touzi, and Zhang 2023. The goal is to establish the dynamic programming equation along with its comparison principle and convergence rate of the value function of the large population optimal stopping problem towards those of the mean field limit. The proof of comparison principle relies on the Ishii's Lemma proved in Bayraktar, Ekren, and Zhang 2023.
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Title. Neural Operators Can Play Dynamic Stackelberg Games
Abstract. Dynamic Stackelberg games are a broad class of two-player games in which the leader acts first, and the follower chooses a response strategy to the leader's strategy. Unfortunately, only stylized Stackelberg games are explicitly solvable since the follower's best-response operator (as a function of the control of the leader) is typically analytically intractable. This paper addresses this issue by showing that the \textit{follower's best-response operator} can be approximately implemented, uniformly on compact sets of controls of the leader, by an \textit{attention-based neural operator}, uniformly on compact subsets of adapted open-loop controls for the leader. We further show that the value of the Stackelberg game where the follower uses the approximate best-response operator approximates the value of the original Stackelberg game. Our main result is obtained using our universal approximation theorem for attention-based neural operators between spaces of square-integrable adapted stochastic processes, as well as stability results for a general class of Stackelberg games.
Title. Risk bounds for the marginal expected shortfall under dependence uncertainty
Abstract. Measuring the contribution of a bank or an insurance company to the overall systemic risk of the market is an important issue, especially in the aftermath of the 2007-2009 Financial Crisis and the financial downturn of 2020. Specifically, the Office of the Superintendent of Financial Institutions (OSFI) in Canada recommends a multifaceted approach to address this problem, with two major components being the size of the company and its interconnectedness within the market.
One appropriate quantitative measure of systemic risk contribution is the marginal expected shortfall (MES), introduced by Acharya et al. (2017) and known to actuarial scholars as the risk capital allocation rule based on the expected shortfall risk measure since at least Panjer and Jing (2001). The MES risk measure has attracted considerable attention and has been evaluated under various assumptions on the joint probability distribution of companies’ risks in the market of interest. However, in reality, the univariate probability distributions of distinct companies in the market can be quite distinct, and the dependence structure between the companies is rarely known.
In this paper, we derive the upper and lower bounds of MES under the assumption of full information on the marginal distributions of the companies’ risks and an unknown dependence structure. Furthermore, we also derive improved bounds for the MES risk measure when partial information on companies’ risk exposures–and hence their dependence–is available. We employ two well-known factor models, the multiplicative and additive background risk models, to describe such partial information.
Reference
Acharya, V. V., L. H. Pedersen, T. Philippon, and M. Richardson (2017). Measuring systemic risk. The review of financial studies 30 (1), 2–47.
Panjer, H. H. and J. Jing (2001). Solvency and capital allocation. University of Waterloo, Institute of Insurance and Pension Research.
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