Convergence of Solutions and Their Exit Times in Diffusion Models with Jumps
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CONVERGENCE OF SOLUTIONS AND THEIR EXIT TIMES IN DIFFUSION MODELS WITH JUMPS A. G. Moroza† and V. V. Tomashyka‡
UDC 519.21
Abstract. We consider a diffusion model with jumps given by a stochastic differential equation with a finite Poisson measure and coefficients depending on a parameter. It is shown that, in the case of convergence of the coefficients, both the solution and its exit times converge. Keywords: stochastic differential equation, exit time, process convergence, Yamada condition, Poisson measure. INTRODUCTION Diffusion models with jumps are used rather extensively and play an especially important role in modeling price processes in securities markets. For this reason, the investigations connected with such models, in particular, those specified by stochastic differential equations (SDEs) are rather urgent. In diffusion models with jumps, a situation often occurs when an optimal strategy for an investor is specified by exit times of a price process that exceed some level (see, for example, [1, 2]). In practice, the coefficients of an equation modeling a certain process are often unknown and are estimated with the help of some approximate methods. In this case, we obtain equations whose coefficients are close to the coefficients of the equation specifying the process being modeled. Thus, it would be desirable to guarantee, first, the closeness (in a sense) of solutions of approximate equations to the solution of the initial equation and, second, the closeness between optimal strategies that are specified by exit times of processes from a band. Taking into account the aforesaid, this work forms the conditions guaranteeing the convergence of solutions to equations with jumps and exit times of these solutions that go beyond a band in the case of convergence of coefficients of equations. As an auxiliary result, the stability of exit times with respect to a band is obtained. This result is of particular interest since the equations specifying the band edges whose overrun is an optimal strategy for an investor are seldom or never exactly solved. The question of convergence of solutions of equations with jumps was considered in [3, 4] and in many monographs, for example, in [5, 6]. The novelty of this work is that, instead of the Lipschitz condition, a weaker Yamada condition is imposed on the diffusion coefficient. This, in particular, allows one to consider Cox–Ingersoll–Ross equations that are often used in financial modeling. This article is organized as follows. In Sec. 1, the main object of this investigation is defined and all the assumptions made are described. Section 2 presents the results for the equations without jumps that are used for the proof of main results. In Secs. 3 and 4, the convergences of solutions of SDEs with jumps and exit times from a band, respectively, are established. 1. MAIN OBJECT OF INVESTIGATION Let (W , F , {Ft , t ³ 0}, P ) be a complete probability space with filtration satisfying the usual conditions. Let us consider the following sequence of stochastic differential equa
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