Differentiability of Stochastic Differential Equation Driven by d -Dimensional G -Brownian Motion with Respect to the In
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Differentiability of Stochastic Differential Equation Driven by d-Dimensional G-Brownian Motion with Respect to the Initial Data Rania Bougherra1 · Hacène Boutabia1 · Manel Belksier1 Received: 31 December 2019 / Revised: 14 September 2020 / Accepted: 22 October 2020 © Iranian Mathematical Society 2020
Abstract The present paper is devoted to the study of the differentiability of solutions of stochastic differential equations driven by d-dimensional G-Brownian motion with respect to the initial data. Matricial stochastic differential equation of derivative and its inverse are given. This extends results obtained by Lin in 2013, in the one-dimensional case. Keywords G-Brownian motion · G-Expectations · G-Stochastic integrals · G-Stochastic differential equations · Random matrices Mathematics Subject Classification 60B20 · 60H10 · 60H05
1 Introduction Stochastic differential equations (SDEs in short) have been intensively investigated in many areas of sciences and industry. For example, in biology, economics, finance, chemistry, physics, electronics, mechanics, etc, for more details, see [8]. Among the important results, we cite the differentiability of solutions of SDEs which was intensively studied in 2014 by Buckdahn et al. [2] and in 2015 by Banõs [1].
Communicated by Majid Gazor.
B
Hacène Boutabia [email protected] Rania Bougherra [email protected] Manel Belksier [email protected]
1
Department of Mathematics, Faculty of Science, LaPS Laboratory, Badji-Mokhtar University, Annaba, Algeria
123
Bulletin of the Iranian Mathematical Society
In 2006, a new stochastic process called G-Brownian motion has been introduced by Peng [9–13] on sublinear expectation space, which is generated by the so-called nonlinear G-heat equation. The related stochastic calculus and SDEs driven by G-Brownian motion has been also established. As in the classical framework, the differentiability with respect to the initial data of SDEs driven by real G-Brownian motion has been studied in 2013 by Lin [7], which has some potential applications, e.g., to obtain the maximum principle for stochastic optimal control systems. Motivated by the above-mentioned works, we aim in this paper to study under suitable assumptions, the differentiability of solutions of G-SDEs in d-dimensional case via fixed point theorem and Picard approximation. Let us consider the following G-SDE: ⎧ d d ⎪ ⎨ dX = A (X ) dt + A (X ) dB k + A (X ) d B i , B j , t
0
⎪ ⎩
t
k
k=1
t
t
X0 = x ∈
i, j
i, j=1
t
t
Rd ,
fields on Rd , (Bt )t≥0 is a dwhere Ak , Ai, j ; k ∈ 0, d, i, j ∈ 1,
di arej vector dimensional G-Brownian motion and B , B t t≥0 its quadratic covariation process. Our approach, differs from that of Lin [7], is to introduce equivalent norms depending on a positive parameter λ. The key of our work is to prove that the derivative of nth Picard approximation of X t is the (n − 1)th of another approximation of the process Yt , which will be the derivative of the solution X t . The paper is organized as follows: in Sect. 2
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