Stochastic equations of non-negative processes with jumps

By Zongfei Fu, Zenghu Li. 2008

230 citations recorded by the source corpus.

We study stochastic equations of non-negative processes with jumps. The existence and uniqueness of strong solutions are established under Lipschitz and non-Lipschitz conditions. Under suitable conditions, the comparison properties of solutions are proved. Those results are applied to construct continuous state branching processes with immigration as strong solutions of stochastic equations.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2e6f435403b5abd553f4e8e675555fa62fcad610

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In Finance

1. Zenghu Li
2. Chao Zhu
3. Chunhua Ma
4. Wei Xu
5. Fei Pu
6. Fubao Xi
7. Vincent Bansaye
8. S. M'el'eard

In Mathematical Physics

1. Zenghu Li
2. Chunhua Ma
3. Gyula Pap
4. Mátyás Barczy
5. Vincent Bansaye
6. Fei Pu
7. Wei Xu
8. S. M'el'eard

Citations in the corpus

• Strong solutions of jump-type stochastic equations (2012, background): “The results improve those in [11, 15] and can be applied to equations like (1.”
• Asymptotic properties of estimators in a stable Cox–Ingersoll–Ross model (2013, methodology): “By Theorem 6.2 of Fu and Li (2010), for any initial value X0, which is a positive F0-measurable random variable, there is a unique positive strong solution {Xt : t ≥ 0} to (1.2).”
• Stochastic equations, flows and measure-valued processes (2010, background): “We refer the reader to Le Jan and Raimond (2004), Ma and Xiang (2001) and Xiang (2009) for the study of stochastic flows of mappings and measures in abstract settings. Let {Bs,t : −∞ < s ≤ t < ∞} be the stochastic flow of bridges associated to a Λ-coalescent in the sense of Bertoin and Le Gall (2003”
• Strong solutions for stochastic differential equations with jumps (2009, background): “We extend the results of [4] in two directions.”
• Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence, Feller and strong Feller properties, and exponential ergodicity (2017, background): “4 is in the same spirit of Yamada and Watanabe’s argument for pathwise uniqueness in Yamada and Watanabe (1971) and Fu and Li (2010), Li and Mytnik (2011). The key idea is to construct a sequence of monotone C(2) functions {ψn} satisfying certain conditions so that one can bound the growth of the L1”
• Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior (2015, methodology): “We refer to Fu and Li [39] for a general framework on the existence and uniqueness of such equations.”
• Asymptotic results for exponential functionals of Levy processes (2016, background): “(1.3) By Theorem 6.2 in Fu and Li (2010), there exists a unique positive strong solution {X(t) : t ≥ 0} to (1.3).”
• On the extinction of Continuous State Branching Processes with catastrophes (2013, methodology): “More precisely, from Fu and Li [16] we have”
• Continuous-State Branching Processes with Immigration (2019, methodology): “For the stable branching CBI-process, a strong stochastic differential equation driven by Lévy processes was established in Fu and Li (2010).”
• Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration (2014, background): “3 in Chapter II in Ikeda and Watanabe (1989), since the intensity measure dsm(dr) of p is deterministic.”