Maximum-minimum bidirectional options are a kind of exotic path dependent options. In the constant elasticity of variance (CEV) model a combining trinomial tree was structured to approximate the nonconstant volatility that is a function of the underlying asset. On this basis a simple and efficient recursive algorithm was developed to compute the risk-neutral probability of each different node for the underlying asset reaching a maximum or minimum price and the total number of maxima (minima) in the trinomial tree. With help of it the computational problems can be effectively solved arising from the inherent complexities of different types of maximum-minimum bidirectional options when the underlying asset evolves as the trinomial CEV model. Numerical results demonstrate the validity and the convergence of the approach mentioned above for the different parameter values set in the trinomial...

Maintaining the stability of bipedal walking remains a major challenge in humanoid robotics, primarily due to the large number of hyperparameters involved and the need to adapt to dynamic environments and external disturbances. Traditional methods for determining these hyperparameters, such as heuristic approaches, are often time- consuming and potentially suboptimal. In this thesis, we present an integrated approach combining advanced control and reinforcement learning techniques to improve the stability of bipedal walking, particularly in the face of ground disturbances and speed variations. Our main contribution lies in the integration of two complementary approaches: (1) an intrinsically stable model predictive control (IS-MPC) combined with whole-body admittance control, and (2) a reinforcement learning module implemented in the mc_rtc framework. This system allows for continuous mo...