Conference Agenda

One of the fundamental results in optimal stopping theory states that for Markovian problems, optimal stopping times exist in the class of (Markovian) pure first-entry times. On the other hand, when it comes to Markovian games of optimal stopping, equilibria in the class of (Markovian) pure first-entry times do only exist under restrictive assumptions, while the most general existence results do only provide equilibria in the class of randomized stopping times. Due to the vastness of the class of randomized stopping times such equilibria can hardly be pinpointed and their inherent path-dependency compromises subgame perfection. Therefore, it is natural to restrict the scope to the inbetween-class of Markovian randomized stopping times. We outline a general scheme to existence of equilibria in that class based on the example of a war-of-attrition type Dynkin game.

Consider a Brownian motion $(B_t)_{t \geq 0}$ on a given complete probability space $(\Omega, \mathcal{F}, \mathbb P)$ with the filtration $\mathbb{F} := (\mathcal{F}_t)_{t \geq 0}$ being the standard augmentation of $(\mathcal F^B_t)_{t\ge 0}$. We fix $T > 0$ and let $\mathcal{T}$ denote the set of all $[0,T]$ valued $\mathbb{F}$-stopping times. Suppose that $L$ and $U$ are $\text{càdlàg}$, $\mathbb{F}$-adapted processes of class (D), satisfying \[ L_t \leq U_t, \quad t \in [0, T]. \] We study Dynkin games governed by a nonlinear $\mathbb{E}^f$-expectation with payoff processes $L$ and $U$ which do not necessarily satisfy the Mokobodzki's condition—the existence of a $\text{càdlàg}$ semimartingale between the barriers. To address this, we introduce the notion of Mokobodzki's stochastic intervals $\mathcal{M}(\tau)$, i.e. maximal stochastic intervals on which Mokobodzki's condition is satisfied for the pair $(L,U)$ when starting from $\tau\in\mathcal T$. Based on this concept, we prove that the Dynkin game on stopping times in $\mathcal T$ under the nonlinear expectation $\mathbb{E}^f$ has a value, i.e. \begin{equation}\label{1} \mathop{\mathrm{ess\,inf}}_{\sigma \ge\theta} \mathop{\mathrm{ess\,sup}}_{\tau \ge \theta} \mathbb{E}^f_{\theta, \tau \wedge \sigma}(L_{\tau} \mathbf{1}_{\{\tau \leq \sigma\}} + U_{\sigma} \mathbf{1}_{\{\sigma < \tau\}}) = \mathop{\mathrm{ess\,sup}}_{\tau \ge \theta} \mathop{\mathrm{ess\,inf}}_{\sigma \ge\theta} \mathbb{E}^f_{\theta, \tau \wedge \sigma}(L_{\tau} \mathbf{1}_{\{\tau \leq \sigma\}} + U_{\sigma} \mathbf{1}_{\{\sigma < \tau\}}) \end{equation} for any stopping time $\theta \in \mathcal{T}$. Moreover, we show that the family $\{V^f(\theta), \, \theta \in \mathcal{T}\}$ of said values is aggregable to a process $Y$ that is a semimartingale on each Mokobodzki's interval $\mathcal{M}(\theta)$, $\theta \in \mathcal{T}$, and that there exist saddle points for this game in case $L_{t-}\le L_t$, $U_{t-}\ge U_t$, $t\in [0,T]$. A complete novelty, even in the case of the standard expectation, is that we provide a minimal saddle point.

The method we propose is based on the theory of Reflected BSDEs that we suitable extend to the case when the barriers $L,U$ are not supposed to satisfy Mokobodzki's condition. As a by-product we show that $Y$ is a solution to some Reflected BSDEs and moreover $Y^n_t\to Y_t$, where \[ Y^n_t=L_T+\int_t^Tf(s,Y^n_s,Z^n_s)\,ds+n\int_t^T(Y^n_s-L_s)^-\,ds-n\int_t^T(Y^n_s-U_s)^+\,ds-\int_t^TZ^n_s\,dB_s, \] i.e. $Y^n$ is the first component of a solution to nonlinear BSDE with the penalty terms. The additional advantage of applying BSDEs is that \(f: \Omega \times [0, T] \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}\) may be assumed to be merely monotone and continuous in \(y\) (with no restrictions on the growth), Lipschitz continuous in \(z\), and integrable at zero, i.e. $\mathbb E\int_0^T|f(s,0,0)|\, ds<\infty$. Furthermore, as we mentioned before $L,U$ are merely of class (D) which means that the families $(L_\tau)_{\tau\in\mathcal T}$, $(U_\tau)_{\tau\in\mathcal T}$ are uniformly integrable.

The presented results were obtained in cooperation with Tomasz Klimsiak.

Bibliography:

Klimsiak, T., Rzymowski, M.: Mokobodzki's intervals: an approach to Dynkin games when value process is not a semimartingale. arXiv:2407.15601