Some applications of quantum probability to stochastic differential equations in Hilbert space

1989 ◽  
pp. 77-90 ◽  
Author(s):  
Alberto Frigerio
Keyword(s):  
Hilbert Space ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Quantum Probability

Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay

2019 ◽  
Vol 20 (01) ◽  
pp. 2050003
Author(s):  
Xiao Ma ◽  
Xiao-Bao Shu ◽  
Jianzhong Mao
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Calculus ◽  
Stochastic Differential Equations ◽  
Periodic Solutions ◽  
Infinite Delay ◽  
Almost Periodic Solutions ◽  
Almost Periodic

In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results.

scholarly journals Stochastic differential equations in infinite dimensional Hilbert space and its optimal control problem with Lévy processes

2021 ◽  
Vol 7 (2) ◽  
pp. 2427-2455
Author(s):  
Meijiao Wang ◽  
◽  
Qiuhong Shi ◽  
Maoning Tang ◽  
Qingxin Meng ◽  
...  
Keyword(s):  
Optimal Control ◽  
Hilbert Space ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Teugels Martingales

<abstract><p>The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens <sup>[<xref ref-type="bibr" rid="b21">21</xref>]</sup>). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.</p></abstract>

The passage from random walk to diffusion in quantum probability

1988 ◽  
Vol 25 (A) ◽  
pp. 151-166 ◽  
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Random Walk ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discrete Time ◽  
Continuous Time ◽  
Quantum Probability ◽  
Diffusion Limit ◽  
Quantum Random Walk

The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.

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