[Paper Review] Maximal type inequalities for linear stochastic Volterra equations

Anna Karczewska|ArXiv.org|Dec 26, 2004

Stochastic processes and financial applications3 citations

TL;DR

This paper establishes maximal type inequalities and exponential tail estimates for stochastic convolutions in linear stochastic Volterra equations using a fractional factorization method, extending results from semigroup-based stochastic evolution equations to non-semigroup resolvent operators. Despite the lack of semigroup structure, the method yields moment and tail bounds for the mild solution's stochastic convolution component.

ABSTRACT

The note is devoted to estimates for convolutions appearing in some class of stochastic Volterra equations. Two maximal inequalities and exponential tail estimate are proved by the fractional method of infinite dimensional stochastic calculus. The paper extends on non-semigroup case some results obtained earlier for semigroups.

Motivation & Objective

• To derive maximal inequalities and exponential tail estimates for stochastic convolutions arising in linear stochastic Volterra equations.
• To extend factorization-based stochastic calculus techniques—previously reliant on semigroups—to equations where the resolvent operators do not form a semigroup.
• To provide moment and tail bounds for the stochastic convolution term in the mild solution of linear stochastic Volterra equations under weaker assumptions than those requiring semigroup structure.
• To investigate whether the factorization method can yield continuity of the stochastic convolution in the non-semigroup case, despite theoretical limitations.

Proposed method

• Applies the fractional factorization method of infinite-dimensional stochastic calculus to stochastic Volterra equations with non-semigroup resolvents.
• Represents the stochastic convolution as a composition of a fractional integral operator and a generalized stochastic integral, leveraging the smoothing effect of the fractional integral.
• Uses the Riemann–Liouville-type fractional integral operator $ I_{eta} $ with a $ C_0 $-semigroup $ R(t) $ to regularize the stochastic integral.
• Derives moment bounds via the factorization formula $ W_B^{ ho}(t) = I_{eta}(I_{1-eta} ullet W)(t) $, where $ I_{1-eta} $ acts as a smoothing operator.
• Employs Hilbert-Schmidt norm estimates and stochastic Fubini-type arguments to control the growth of the convolution process.
• Applies Doob’s maximal inequality and exponential moment estimates to derive tail bounds under integrability conditions on the resolvent and noise coefficient.

Experimental results

Research questions

• RQ1 Can maximal inequalities and exponential tail estimates be established for stochastic convolutions in linear stochastic Volterra equations when the resolvent does not form a semigroup?
• RQ2 To what extent can the factorization method, traditionally used for semigroup-based equations, be adapted to non-semigroup resolvent operators in stochastic Volterra equations?
• RQ3 What are the sufficient conditions on the resolvent operators and noise coefficient for moment and tail bounds to hold in the absence of semigroup structure?
• RQ4 Why does the factorization method fail to produce continuous modifications of the stochastic convolution in the non-semigroup case, despite its success in semigroup settings?
• RQ5 Under what conditions on the resolvent and noise process is the stochastic convolution in a Volterra equation still amenable to moment and tail estimation via factorization?

Key findings

• The paper proves a maximal inequality of the form $ bEig[ig( ext{sup}_{t eq T}( ho(Z_1(t)))^pig)ig] \ leq c_p ig(ig(ig)ig)^{p/2} bEig[ig(ig)ig] $ for $ p eq (2, 1/eta) $, $ eta eq (0, 1/2) $, with $ Z_1(t) $ being the stochastic convolution.
• An exponential tail estimate is established: $ Pig( ext{sup}_{t eq T}| ho(Z_2(t))| eq ext{large}ig) \ leq C ext{exp}ig(-rac{ ext{large}^2}{ ilde{ ho}^2 ilde{ heta}}ig) $, under conditions $ ( ext{ps}) $, $ ( ext{k}) $, and $ ( ext{ps}) $, with $ Z_2(t) $ being the stochastic convolution.
• The resolvent operators $ S(t) $ must be Hilbert-Schmidt for the moment inequality to hold, which restricts the class of equations to which the result applies.
• The factorization method fails to yield a continuous modification of the stochastic convolution in the non-semigroup case, despite its success in semigroup-based settings.
• The method relies on the existence of a $ C_0 $-semigroup $ R(t) $ and a fractional parameter $ eta eq (0, 1/2) $ to ensure the regularizing effect of the fractional integral.
• The results are valid under the integrability condition $ bEig[ig(ig)ig] < \ fty $, ensuring the stochastic integral is well-defined and the factorization applies.