Mathematics and Financial Economics最新文献

We consider a class of optimal advertising problems under uncertainty for the introduction of a new product into the market, on the line of the seminal papers of Vidale and Wolfe (Oper Res 5:370–381, 1957) and Nerlove and Arrow (Economica 29:129–142, 1962). The main features of our model are that, on one side, we assume a carryover effect (i.e. the advertisement spending affects the goodwill with some delay); on the other side we introduce, in the state equation and in the objective, some mean field terms that take into account the presence of other agents. We take the point of view of a planner who optimizes the average profit of all agents, hence we fall into the family of the so-called “Mean Field Control” problems. The simultaneous presence of the carryover effect makes the problem infinite dimensional hence belonging to a family of problems which are very difficult in general and whose study started only very recently, see Cosso et al. [Ann Appl Probab 33(4):2863–2918, 2023]. Here we consider, as a first step, a simple version of the problem providing the solutions in a simple case through a suitable auxiliary problem.

This paper studies an optimal investment–consumption problem for competitive agents with exponential or power utilities and a common finite time horizon. Each agent regards the average of habit formation and wealth from all peers as benchmarks to evaluate the performance of her decision. We formulate the n-agent game problems and the corresponding mean field game problems under the two utilities. One mean field equilibrium is derived in a closed form in each problem. In each problem with n agents, an approximate Nash equilibrium is then constructed using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained. In addition, we provide some numerical illustrations of our results.

In this paper we further extend the optimal bubble riding model proposed in Tangpi and Wang (Optimal bubble riding: a mean field game with varying entry times, 2022) by allowing for price-dependent entry times. Agents are characterized by their individual entry threshold that represents their belief in the strength of the bubble. Conversely, the growth dynamics of the bubble is fueled by the influx of players. Price-dependent entry naturally leads to a mean field game of controls with common noise and random entry time, for which we provide an existence result. The equilibrium is obtained by first solving discretized versions of the game in the weak formulation and then examining the measurability property in the limit. In this paper, the common noise comes from two sources: the price of the asset which all agents trade, and also the the exogenous bubble burst time, which we also discretize and incorporate into the model via progressive enlargement of filtration.

We consider a stochastic differential game, where each player continuously controls the diffusion intensity of her own state process. The players must all choose from the same diffusion rate interval ([sigma _1, sigma _2]), and have individual random time horizons that are independently drawn from the same distribution. The players whose states at their respective time horizons are among the best (p in (0,1)) of all terminal states receive a fixed prize. We show that in the mean field version of the game there exists an equilibrium, where the representative player chooses the maximal diffusion rate when the state is below a given threshold, and the minimal rate else. The symmetric n-fold tuple of this threshold strategy is an approximate Nash equilibrium of the n-player game. Finally, we show that the more time a player has at her disposal, the higher her chances of winning.

We develop a dynamic model economy where self-employed entrepreneurs allocate their net worth to their firm capital and risk-less government bonds, facing borrowing constraints, uninsurable labour endowment and capital depreciation risk. We derive a numerical approximation of the model’s equilibrium and compare it with a benchmark economy with no capital risk. Unlike labour endowment risk, capital risk reduces aggregate capital accumulation and wages and generates a positive risk premium. Low- (high-) net-worth entrepreneurs, whose consumption depends primarily on labour (financial) income, hold higher (lower) capital risk exposure. These patterns exacerbate inequality by increasing the share of financially constrained individuals and fattening the tails of the net worth distribution. Fiscal policy affects these outcomes by redistributing resources and affecting the risk premium. Capital tax cuts benefit more low- or high-net-worth entrepreneurs, depending on whether taxes on bonds or labour income finance them.

We study binary opinion formation in a large population where individuals are influenced by the opinions of other individuals. The population is characterised by the existence of (i) communities where individuals share some similar features, (ii) opinion leaders that may trigger unpredictable opinion shifts in the short term (iii) some degree of incomplete information in the observation of the individual or public opinion processes. In this setting, we study three different approximate mechanisms: common sampling approximation, independent sampling approximation, and, what will be our main focus in this paper, McKean–Vlasov (or mean-field) approximation. We show that all three approximations perform well in terms of different metrics that we introduce for measuring population level and individual level errors. In the presence of a common noise represented by the major influencers opinions processes, and despite the absence of idiosyncratic noises, we derive a propagation of chaos type result. For the particular case of a linear model and particular specifications of the major influencers opinion dynamics, we provide additional analysis, including long term behavior and fluctuations of the public opinion. The theoretical results are complemented by some concrete examples and numerical analysis, illustrating the formation of echo-chambers, the propagation of chaos, and phenomena such as snowball effect and social inertia.

The present paper studies a kind of robust optimization problems with constraint. The problem is formulated through Backward Stochastic Differential Equations (BSDEs) with quadratic generators. A necessary condition is established for the optimal solution using a terminal perturbation method and properties of Bounded Mean Oscillation (BMO) martingales. The necessary condition is further proved to be sufficient for the existence of an optimal solution under an additional convexity assumption. Finally, the optimality condition is applied to discuss problems of partial hedging with ambiguity, fundraising under ambiguity and randomized testing problems for a quadratic g-expectation.

This paper considers the non-zero-sum stochastic differential game problem between two ambiguity-averse insurers (AAIs) with common shock. Each AAI’s surplus process consists of a proportional reinsurance protection and an investment in a money account, a stock and a credit default swap (CDS) with the objective of maximizing the expected utility of her relative terminal surplus with respect to that of her competitors. We consider default contagion risk of CDSs through a Markovian model with interacting default intensities. It is worthwhile to consider the uncertainty of the model on both the insurer herself and her competitors. In our model, we describe the surplus processes of two insurers by two jump-diffusion models with a common shock. Under jump-diffusion models, the robust Nash equilibrium strategies and the value functions for the all-default, one-default and all-alive case are derived under a worst-case scenario, respectively. Finally, through some numerical examples, we found some interesting results about the effects of some model parameters on the robust Nash equilibrium strategies, such as, the common shocks and the individual claims have the opposite effect on reinsurance investment.

We consider the strategic interaction of n investors who are able to influence a stock price process and at the same time measure their utilities relative to the other investors. Our main aim is to find Nash equilibrium investment strategies in this setting in a financial market driven by a Brownian motion and investigate the influence the price impact has on the equilibrium. We consider both CRRA and CARA utility functions. Our findings show that the problem is well-posed as long as the price impact is at most linear. Moreover, numerical results reveal that the investors behave very aggressively when the price impact is close to a critical parameter.