We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents' rewards are rank-dependent. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence

We propose a new optimal consumption model in which the degree of addictiveness of habit formation is directly controlled through a constraint on admissible consumption. In particular, we assume that the individual is unwilling to consume at a rate below a certain proportion 0α≤1 of her...

We consider a zero-sum optimal stopping game in which the value of the reward is revealed when the second player stops, instead of it being revealed after the first player's stopping time. Such problems appear in the context of financial mathematics when one sells and buys two different American...

In one dimension, the theory of the G-normal distribution is well-developed, and many results from the classical setting have a nonlinear counterpart. Significant challenges remain in multiple dimensions, and some of what has already been discovered is quite nonintuitive. By answering several...

In this paper, we develop a new mathematical technique which allows us to express the joint distribution of a Markov process and its running maximum (or minimum) through the marginal distribution of the process itself. This technique is an extension of the classical reflection principle for...

We consider the optimal problem $\sup_{\tau\in\mathcal{T}_{\eps,T}}\mathbb{E}\left[\sum_{i=1}^n \phi_{(\tau-\eps^i)^ }^i\right]$, where $T0$ is a fixed time horizon, $(\phi_t^i)_{0\leq t\leq T}$ is progressively measurable with respect to the Brownian filtration, $\eps^i\in[0,T]$ is a constant,...

We consider the optimal dividend problem under a habit formation constraint that prevents the dividend rate to fall below a certain proportion of its historical maximum, the so-called drawdown constraint. This is an extension of the optimal Duesenberry's ratcheting consumption problem, studied...

Let $\Omega$ be one of $\X^{N 1},C[0,1],D[0,1]$: product of Polish spaces, space of continuous functions from $[0,1]$ to $\mathbb{R}^d$, and space of RCLL (right-continuous with left limits) functions from $[0,1]$ to $\mathbb{R}^d$, respectively. We first consider the existence of a probability...

We revisit the dividend payment problem in the dual model of Avanzi et al. Using the fluctuation theory of spectrally positive Levy processes, we give a short exposition in which we show the optimality of barrier strategies for all such Levy processes. Moreover, we characterize the optimal...

Regularity of the impulse control problem for a non-degenerate n-dimensional jump diffusion with infinite activity and finite variation jumps was recently examined in [4]. Here we extend the analysis to include infinite activity and infinite variation jumps. More specifically, we show that the...