By Archil Gulisashvili
Asymptotic research of stochastic inventory fee versions is the critical subject of the current quantity. designated examples of such types are stochastic volatility versions, which were constructed as a solution to definite imperfections in a celebrated Black-Scholes version of alternative pricing. In a inventory rate version with stochastic volatility, the random habit of the volatility is defined by means of a stochastic strategy. for example, within the Hull-White version the volatility strategy is a geometrical Brownian movement, the Stein-Stein version makes use of an Ornstein-Uhlenbeck procedure because the stochastic volatility, and within the Heston version a Cox-Ingersoll-Ross strategy governs the habit of the volatility. one of many author's major ambitions is to supply sharp asymptotic formulation with blunders estimates for distribution densities of inventory costs, choice pricing features, and implied volatilities in numerous stochastic volatility types. the writer additionally establishes sharp asymptotic formulation for the implied volatility at severe moves quite often stochastic inventory expense versions. the current quantity is addressed to researchers and graduate scholars operating within the sector of monetary arithmetic, research, or chance conception. The reader is predicted to be conversant in parts of classical research, stochastic research and likelihood idea.
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Additional info for Analytically Tractable Stochastic Stock Price Models
The same conclusion is valid in the weak sense. 79). 46 It is clear that the results obtained in Sect. 25)). It suffices to replace a by qm and b by q. 77) becomes ρt (y) = 4qm 4qeqt 4qy0 4qeqt y 2 χ ; 2 , 2 qt , 2 qt 2 qt c (e − 1) c (e − 1) c c (e − 1) y > 0. 25) is used to model a random behavior of the interest rate (see [CIR85]). In mathematics, the CIR process is called the Feller process after W. Feller who studied the diffusion equation associated with this process. 77) was formulated in [CIR85] without proof.
4) holds for W and Z. 4) holds for W and Z. We will next prove that the converse statements are also true. 4). 6) hold. 5) holds. 6) is similar. Let us start with the case where ρ ∈ (−1, 1). Put W= 1 1 − ρ2 W− ρ 1 − ρ2 Z. 5) holds. It remains to prove that the process W is a standard Brownian motion. First we notice that W , W t = t. s. Applying the Lévy characterization theorem, we see that the process W is a standard Brownian motion. Next, let ρ = 1 (the case where ρ = −1 is similar). 4) with ρ = 1 implies W − Z, W − Z t = 0 for all t ≥ 0.
8), we have b1 (x1 , x2 ) = μx1 , σ11 (x1 , x2 ) = x1 f (x2 ), σ21 (x1 , x2 ) = ρσ (x2 ) b2 (x1 , x2 ) = b(x2 ), σ12 (x1 , x2 ) = 0, and σ22 (x1 , x2 ) = 1 − ρ 2 σ (x2 ). 7) does not satisfy the linear growth condition. , [IW81]), we cannot apply these theorems to establish the unique strong solvability of Eq. 7). 4 below). For the information on the concepts and facts used in the formulations and proofs of these lemmas, we refer the reader to the books on stochastic differential equations, mentioned at the end of Sect.