Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.

We study a discrete-time variant of the non-Markovian SEIS (Susceptible-Exposed-Infectious-Susceptible) model, occurring on complex networks. The model assumes for an arbitrary form of the Discrete Temporal Probability Functions (DTPFs), that govern the transitions from Exposed to Infectious state (incubation period) and Exposed/Infectious back to Susceptible state (recovery period); this enables the model to address a wide range of real-world spreading phenomena. Theoretical analysis, based on methods from systems theory, leads to an expression that defines the epidemic threshold, for the analyzed model, as a critical relation between the DTPF's, infection rate and the network topology (the largest eigenvalue of the networks adjacency matrix), in a form that extends the result for the Markovian case. Validity of the suggested model, and the obtained theoretical result are confirmed by the numerical analyzes. We argue that the approach used in the paper, may be further extended to describe a wide variety of model variants and sub-models, occurring both on natural, as well as technological (engineering) networks.

Contemporary numerical weather prediction schemes are based on ensemble forecasting. Ensemble members are obtained by taking different (perturbed) models started with different initial conditions. We introduce one type of improved model that represents interactive ensemble of individual models. The improved model's performance is tested with the Lorenz 96 toy model. One complex model is considered as reality, while its imperfect models are taken to be structurally simpler and with lower resolution. The improved model is defined as one with tendency that is weighted average of the tendencies of individual models. The weights are calculated from past observations by minimizing the average difference between the improved model's tendency and that of the reality. It is numerically verified that the improved model has better ability for short-term prediction than any of the individual models.

We study random walks on complex networks with transition probabilities which depend on the current and previously visited nodes. By using an absorbing Markov chain we derive an exact expression for the mean first passage time between pairs of nodes, for a random walk with a memory of one step. We have analyzed one particular model of random walk, where the transition probabilities depend on the number of paths to the second neighbors. The numerical experiments on paradigmatic complex networks verify the validity of the theoretical expressions, and also indicate that the flattening of the stationary occupation probability accompanies a nearly optimal random search.