We introduce and study a new stochastic SARS model (based on stochastic SIR model with semi-Markov regime-switchings) and investigate its behavior in averaging, merging and diffusion approximation schemes. The main method is based on general theory of random differential equations in semi-Markov...

In this paper, we further study various new Hawkes processes, namely, so-called general compound and regime-switching general compound Hawkes processes to model the price processes in the limit order books. We prove Law of Large Numbers (LLN) and Functional Central Limit Theorems (FCLT) for...

In this paper, we introduce a new model for the risk process based on general compound Hawkes process (GCHP) for the arrival of claims. We call it risk model based on general compound Hawkes process (RMGCHP). The Law of Large Numbers (LLN) and the Functional Central Limit Theorem (FCLT) are...

In this paper we introduce two new Hawkes processes, namely, compound and regime-switching compound Hawkes processes, to model the price processes in limit order books. We prove Law of Large Numbers and Functional Central Limit Theorems (FCLT) for both processes. The two FCLTs are applied to...

In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regards to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit...

In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. Law of...

In this paper, we price covariance and correlation swaps for financial markets with Markov-modulated volatilities. As an example, we consider stochastic volatility driven by two-state continuous Markov chain. In this case, numerical example is presented for VIX and VXN volatility indeces (S&P...

The paper considers a general semi-Markov model for Limit Order Books with two states, which incorporates price changes that are not fixed to one tick. Furthermore, we introduce an even more general case of the semi-Markov model for Limit Order Books that incorporates an arbitrary number of...

We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance, i.e., for a wealth portfolio X(t) consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital R(t) of an insurance company...

The complexity of pricing variance, volatility, covariance, correlation swaps involves how to determine the dynamics of stochastic processes for underlying assets and their volatilities. In this way, sometimes it is simpler to consider of swaps pricing involving the so-called pseudo- statistics,...