Stochastic Gronwall Inequality
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Stochastic Gronwall inequality is a generalization of Gronwall's inequality and has been used for proving the
well-posedness The mathematical term well-posed problem stems from a definition given by 20th-century French mathematician Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that: # a solution exists, # the ...
of path-dependent
stochastic differential equations A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
with local monotonicity and coercivity assumption with respect to supremum norm.


Statement

Let X(t),\, t\geq 0 be a non-negative right-continuous (\mathcal_t)_-
adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every rea ...
. Assume that A: ,\infty)\to[0,\infty) is a deterministic non-decreasing càdlàg Function (mathematics)">function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
with A(0)=0 and let H(t),\,t\geq 0 be a non-decreasing and
càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subse ...
adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every rea ...
starting from H(0)\geq 0. Further, let M(t),\,t\geq 0 be an (\mathcal_t)_-
local martingale In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local m ...
with M(0)=0 and
càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subse ...
paths. Assume that for all t\geq 0, X(t)\leq \int_0^t X^*(u^-)\,d A(u)+M(t)+H(t), where X^*(u):=\sup_X(r). and define c_p=\frac. Then the following estimates hold for p\in (0,1) and T>0: *If \mathbb \big(H(T)^p\big)<\infty and H is predictable, then \mathbb\left left(X^*(T)\right)^p\Big\vert\mathcal_0\rightleq \frac\mathbb\left H(T))^p\big\vert\mathcal_0\right\exp \left\lbrace c_p^A(T)\right\rbrace; *If \mathbb \big(H(T)^p\big)<\infty and M has no negative jumps, then \mathbb\left left(X^*(T)\right)^p\Big\vert\mathcal_0\rightleq \frac\mathbb\left H(T))^p\big\vert\mathcal_0\right\exp \left\lbrace (c_p+1)^A(T)\right\rbrace; *If \mathbb H(T)<\infty, then \displaystyle;


Proof

It has been proven by
Lenglart's inequality In the mathematical theory of probability, Lenglart's inequality was proved by Èrik Lenglart in 1977. Later slight modifications are also called Lenglart's inequality. Statement Let be a non-negative right-continuous \mathcal_t- adapted proce ...
.


References

{{DEFAULTSORT:Stochastic Gronwall inequality Stochastic differential equations Articles containing proofs Probabilistic inequalities