On some functionals of the first passage times in jump models of stochastic volatility

Abstract

We compute some functionals related to the generalized joint Laplace transforms of the first times at which two-dimensional jump processes exit half strips. It is assumed that the state space components are driven by Cox processes with both independent and common (positive) exponential jump components. The method of proof is based on the solutions of the equivalent partial integro-differential boundary-value problems for the associated value functions. The results are illustrated on several two-dimensional jump models of stochastic volatility which are based on non-affine analogs of certain mean-reverting or diverting diffusion processes representing closed-form solutions of the appropriate stochastic differential equations.     

Convexity preserving jump-diffusion models for option pricing

We investigate which jump-diffusion models are convexity preserving. The study of convexity preserving models is motivated by monotonicity results for such models in the volatility and in the jump parameters. We give a necessary condition for convexity to be preserved in several-dimensional jump-diffusion models. This necessary condition is then used to show that, within a large class of possible models, the only convexity preserving models are the ones with linear coefficients.     

PRICING EUROPEAN OPTION IN A DOUBLE EXPONENTIAL JUMP-DIFFUSION MODEL WITH TWO MARKET STRUCTURE RISKS AND ITS COMPARISONS

Using Fourier inversion transform, P.D.E. and Feynman-Kac formula, the closedform solution for price on European call option is given in a double exponential jump-diffusion model with two different market structure risks that there exist CIR stochastic volatility of stock return and Vasicek or CIR stochastic interest rate in the market. In the end, the result of the model in the paper is compared with those in other models, including BS model with numerical experiment. These results show that the double exponential jump-diffusion model with CIR-market structure risks is suitable for modelling the real-market changes and very useful.     

On the approximation of stochastic differential equations

The stability properties of stochastic differential equations with respetct to the perturbation of the coefficients and of the driving processes are investigated in the topology of uniform convergence in probability     

Threshold dynamics of the stochastic epidemic model with jump-diffusion infection force

This paper formulates a stochastic SIR epidemic model by supposing that the infection force is perturbed by Brown motion and L\''{e}vy jumps. The globally positive and bounded solution is proved firstly by constructing the suitable Lyapunov function. Then, a stochastic basic reproduction number $R_0^{L}$ is derived, which is less than that for the deterministic model and the stochastic model driven by Brown motion. Analytical results show that the disease will die out if $R_0^{L}<1$, and $R_0^{L}>1$ is the necessary and sufficient condition for persistence of the disease. Theoretical results and numerical simulations indicate that the effects of L\''{e}vy jumps may lead to extinction of the disease while the deterministic model and the stochastic model driven by Brown motion both predict persistence. Additionally, the method developed in this paper can be used to investigate a class of related stochastic models driven by L\''{e}vy noise.     

Reweighted Nadaraya-Watson estimation of jump-diffusion models

In this paper,we study the nonparametric estimation of the second infinitesimal moment by using the reweighted Nadaraya-Watson (RNW) approach of the underlying jump diffusion model.We establish strong consistency and asymptotic normality for the estimate of the second infinitesimal moment of continuous time models using the reweighted Nadaraya-Watson estimator to the true function.     

The coding complexity of diffusion processes under -norm distortion

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under L^p[0,1]-norm distortion. We find explicit high resolution formulas in terms of the average diffusion coefficient seen by the process. The proof is based on a decoupling method introduced in a former article by the author. Given that link it remains to analyze the coding problem for a concatenation of Wiener processes and to solve the corresponding rate allocation problem.     

On Wong-Zakai approximation of stochastic differential equations

An approximation theorem of stochastic differential equations driven by semimartingales is proved, based on approximation of semimartingales by a sequence of processes with piecewise monotonic sample functions.     

Existence and uniqueness of the solutions of stochastic differential equations

The standard existence and uniqueness theorem for stochastic differential equations requires Lipschitz condition of the coefficients. In this paper, we extend these results to the case in which the coefficients are not required to be Lipschitz continuous, instead they only satisfy a ‘weak’ type of Lipschitz condition.     

On the rate of convergence of weak Euler approximation for nondegenerate SDEs driven by Lévy processes

The paper studies the rate of convergence of the weak Euler approximation for solutions to SDEs driven by Lévy processes, with Hölder-continuous coefficients. It investigates the dependence of the rate on the regularity of coefficients and driving processes. The equation considered has a nondegenerate main part driven by a spherically symmetric stable process.     

The coding complexity of diffusion processes under supremum norm distortion

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under supremum norm distortion. Tight asymptotic formulas are found under mild regularity assumptions. The main technical tool is a decoupling method which allows us to relate the complexity of the diffusion process to that of the Wiener process. The technique is also applicable when considering the L^p[0,1]$L^p[0,1]$-norm distortion.     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

Strong convergence rate of principle of averaging for jump-diffusion processes

We study jump-diffusion processes with two well-separated time scales. It is proved that the rate of strong convergence to the averaged effective dynamics is of order O(ɛ ^1/2), where ɛ ≪ 1 is the parameter measuring the disparity of the time scales in the system. The convergence rate is shown to be optimal through examples. The result sheds light on the designing of efficient numerical methods for multiscale stochastic dynamics.     

Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

In this article, we find the transition densities of the basic affine jump-diffusion (BAJD), which has been introduced by Duffie and Gârleanu as an extension of the CIR model with jumps. We prove the positive Harris recurrence and exponential ergodicity of the BAJD. Furthermore, we prove that the unique invariant probability measure π of the BAJD is absolutely continuous with respect to the Lebesgue measure and we also derive a closed-form formula for the density function of π.     

On Lp -Theory of stochastic partialdifferential equations of divergence form with continuous coefficients

We develop an L_p -theory of stochastic PDEs of divergence form. Under natural assumptions on the coefficients and the data, we show that the solutions belong to some modified stochastic Sobolev spaces. As a consequence of this result and certain embedding theorem, we also show that the solutions are Holder continuous in space and time a.s. for sufficiently large p     

On the Expectations of Multiple Stratonovich Integrals

Abstract

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectations of products of multiple stochastic integrals. In the Itô case, explicit formulae for the expectation of a multiple integral with integrand identically equal to 1 and for the product of two such integrals are known. In this paper formulae for the expectation of any multiple Stratonovich integral as well as for the product of a broad class of two Stratonovich integrals have been derived.     

On Hölder solutions of the integro-differential Zakai equation

The existence and uniqueness of solutions of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. The equation considered arises in a nonlinear filtering problem with a jump signal process and continuous observation.     

On a problem of stochastic differential games

The process of bargaining between management and union during a strike is modelled by a nonlinear stochastic differential game. It is assumed that the two sides bargain in the mood of a cooperative game. A pair of Pareto-optimal strategies is obtained.