The study investigates the necessary maximum principle for robust optimal control problems associated with quadratic backward stochastic differential equations (BSDEs). The system coefficients depend on parameter 𝜃, while the generator of BSDEs exhibits quadratic growth with respect to 𝑧. To address the uncertainty present in the model, the variational inequality is derived using weak convergence techniques. Additionally, due to the generator being quadratic with respect to 𝑧, the forward adjoint equations are stochastic differential equations with unbounded coefficients, involving mean oscillation martingales. By using the reverse Hölder inequality and John−Nirenberg inequality, we demonstrate that the solutions are continuous with respect to parameter 𝜃. Moreover, the necessary and sufficient conditions for robust optimal control are established using the linearization method.

We study a robust utility maximization problem in the case of incomplete market and logarithmic utility with general stochastic constraints. Our problem is equivalent to the maximization of nonlinear expected logarithmic utility. We characterize the optimal solution using quadratic backward stochastic differential equations.

The Shilkret integral or idempotent expectation is a sublinear functional which is very close to being a sublinear expectation since it satisfies all the required properties but its domain is not a linear space. In this paper, we prove that it admits a law of large numbers which is structurally similar to Peng’s LLN for sublinear expectations although significant differences exist. As regards the central limit theorem, the situation is radically different as the $\sqrt{n}$ normalization can lead to a trivial limit and other normalizations are possible for variables with a finite second moment or even bounded.

In this paper, we study the asymptotic behaviors of implied volatility in an affine jump-diffusion model. By assuming that log stock prices under the risk-neutral measure follow an affine jump-diffusion model, we show that an explicit form of the moment-generating function for log stock price can be obtained by solving a set of ordinary differential equations. A large-time large deviation principle for log stock prices is derived by applying the Gärtner-Ellis theorem. We characterize the asymptotic behaviors of implied volatility in the large-maturity and large-strike regimes using the rate function in the large deviation principle. The asymptotics of the implied volatility for fixed-maturity, large-strike and small-strike regimes are also studied. Numerical results are provided to validate the theoretical work.

In this paper, we study the existence and uniqueness of the solution to a reflected backward stochastic differential equation (RBSDE) with the generator $g(t,y,z)={G}_{f}^{F}(t,y,z)+f(y)|z{|}^{2}$, where $f(y)$ is a locally integrable function defined on an open interval $D$, and ${G}_{f}^{F}(t,y,z)$ is induced by 𝑓 and a Lipschitz continuous function 𝐹. Both the solution ${Y}_{t}$ and the obstacle ${L}_{t}$ of this RBSDE take values in $D$. As applications, we provide a probabilistic interpretation of an obstacle problem for a quadratic PDE with a singular term, whose solution takes values in $D$, and study an optimal stopping problem for the payoff of American options under general utilities.

We consider the optimal stopping time problem under model uncertainty, $R(v)=\underset{\mathbb{P}\in \mathcal{P}}{\text{ess sup}}\underset{\tau \in {\mathcal{S}}_{v}}{\text{ess sup}}{E}^{\mathbb{P}}[Y(\tau )|{\mathcal{F}}_{v}]$, for every stopping time $v$, within the framework of families of random variables indexed by stopping times. This setting is more general than the classical setup of stochastic processes, notably allowing for general payoff processes that are not necessarily right-continuous. Under weaker integrability, with regularity assumptions for the reward family $Y=(Y(v),v\in S)$, the existence of an optimal stopping time is demonstrated. Sufficient conditions for the existence of an optimal model are then determined. For this purpose, we present a universal optional decomposition for the generalized Snell envelope family associated with $Y$. This decomposition is then employed to prove the existence of an optimal probability model and to study its properties.