In this paper, we study financial markets with stochastic volatilities driven by fractional Brownian motion with Hurst index H1/2. Our models include fractional versions of Ornstein-Uhlenbeck, Vasicek, geometric Brownian motion and continuous-time GARCH models. We price variance and volatility...

The valuation of the variance swaps for local Levy based stochastic volatility with delay (LLBSVD) is discussed in this paper. We provide some analytical closed forms for the expectation of the realized variance for the LLBSVD. As applications of our analytical solutions, we fit our model to 10...

In this paper, we model financial markets with semi-Markov volatilities and price covarinace and correlation swaps for this markets. Numerical evaluations of varinace, volatility, covarinace and correlations swaps with semi-Markov volatility are presented as well. The novelty of the paper lies...

We consider a merging principle for the endemic SIR model in a semi-Markov random media. Under merging conditions of a semi-Markov media we show that the perturbed endemic SIR model converges to the merged Markov endemic SIR model with coe cients which depend on merged Markov process. Novelty of...

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...

Some commodity prices, like oil and gas, exhibit the mean reversion, unlike stock price. It means that they tend over time to return to some long-term mean.In this paper we consider a risky asset S_t following the mean-reverting stochastic process. The aim of this paper is to obtain an explicit...

The valuation of the variance swaps for local stochastic volatility with delay and jumps is discussed in this paper. We provide some analytical closed forms for the expectation of the realized variance for the stochastic volatility with delay and jumps. Besides, we also present a lower bound for...

The jumps in stock market volatility are found to be so active that this discredits many recently proposed stochastic volatility models without jumps (Bollerslev et al (2008)). The most convincing evidence comes from recent nonparametric work using high-frequency data as in Barndorff-Nielsen and...

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...