Distributions of Functionals of Switching Diffusions with Jumps
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DISTRIBUTIONS OF FUNCTIONALS OF SWITCHING DIFFUSIONS WITH JUMPS A. N. Borodin∗
UDC 519.2
The paper deals with methods for calculating distributions of functionals of switching diffusions with jumps. The switching between two collections of diffusion coefficients occurs at the Poisson time moments which are independent of the initial diffusions. At the same moments, the diffusion can have jumps. The process controlling the switching is determined by a Markov chain. Bibliography: 6 titles.
This investigation combines the approaches of papers [1] and [2]. A class of diffusions with switchings and jumps carried out in Poisson time points is considered. There are many papers devoted to switching diffusions, see, for example, the monograph [3]. There are several collections of diffusion coefficients corresponding to the classical diffusions. Switching from one collection to another occurs at random times corresponding to the moments of jumps of a Poisson process independent of the initial diffusions. In addition, the switching diffusions have jumps at the indicated moments. We are interested in results that allow us to calculate the distributions of various functionals of a diffusion with switchings and jumps. For classical diffusions, in particular for a Brownian motion, the paper of M. Kac [4] is of fundamental importance for development of the theory of distributions of integral functionals. 1. Switching diffusion with jumps The Poisson process N (t), t ≥ 0, with intensity λ1 > 0, which is responsible for switching can be represented as follows: l τk ≤ t 1[0,t] (τ1 ), N (t) := max l : k=1
where τk , k = 1, 2, . . . , are independent exponentially distributed with parameter λ1 random variables, d P(τk < t) = λ1 e−λ1 t 1[0,∞) (t). dt
The diffusion-switching process L(t), t ≥ 0, is defined as follows. There is a homogeneous Markov chain d(n), n = 0, 1, . . . , on the set {1, . . . , r} and with transition probabilities {pl,k }, l = 1, . . . , r, k = 1, . . . , r. Then L(t) := d(N (t)). j = (Yj (1), Yj (2), . . . , Yj (r)), j = 1, 2, . . . , be i.i.d. random vectors independent of the Let Y 0 = (0, . . . , 0). process N . Let Y Let W (t), t ≥ 0, be a Brownian motion independent of the Poisson process N and the j , j = 1, 2, . . . . variables Y Consider homogeneous diffusions X(l, t), t ≥ 0, l = 1, . . . , r. For each l, such a diffusion is a solution of the stochastic differential equation: a.s. for any t ≥ 0, t t (1.1) X(l, t) = x + μ(l, X(l, u)) du + σ(l, X(l, u)) dW (u). 0
0
∗
St.Petersburg Department of Steklov Mathematical Institute, St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 28–45. Original article submitted October 10, 2018. 1072-3374/20/2511-0015 ©2020 Springer Science+Business Media, LLC 15
Let μ(l, x) and σ(l, x), x ∈ R, l = 1, . . . , r, be continuously differentiable functions satisfying the linear growth condition |μ(l, x)| + |σ(l, x)| ≤ C(1 + |x|)
for all x ∈ R. μ(l, x) are bounded. Assume th
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