On Path-dependent Volterra Integral Equations: Strong Well-posedness and Stochastic Numerics

Abstract: The aim of this paper is to provide a comprehensive analysis of the path-dependent Stochastic Volterra Integral Equations (SVIEs), in which both the drift and the diffusion coefficients are allowed to depend on the whole trajectory of the process up to the current time. We investigate the existence and uniqueness (aka the strong well-posedness) of solutions to such equations in the $L<sup>p$ setting, $p&gt;0$, locally in time and their properties specifically their path regularity and flows. Then, we introduce a numerical approximation method based on an interpolated $K-$integrated Euler-Maruyama scheme to simulate numerically the process, and we prove the convergence, with an explicit rate, of this scheme towards the strong solution in the $L<sup>p$ norm.

• The paper establishes strong existence and uniqueness for path-dependent stochastic Volterra equations with potentially singular kernels by using Bochner-space fixed-point arguments rather than semimartingale methods.
• The analysis derives continuous modifications, Hölder regularity, moment and flow estimates, and extends bounds from sufficiently large moments to all positive moments through a functional-representation splitting argument.
• The paper introduces an implementable interpolated K-integrated Euler scheme with O(n²) pathwise complexity and strong rates governed by kernel singularity and coefficient regularity, reducing to h^(α−1/2) for fractional kernels under suitable Lipschitz conditions.

Setting and contribution

The paper studies path-dependent stochastic Volterra integral equations (SVIEs) of the form

$$X_t = x_0(t) + \int_0^t K_1(t,s)\, b(s, X_\cdot^s)\, ds + \int_0^t K_2(t,s)\, \sigma(s, X_\cdot^s)\, dW_s,$$

on a finite horizon $[0,T]$, where both coefficients depend on the entire stopped trajectory $X_{\cdot}^{s}$, the kernels $K_1, K_2$ may be singular (e.g., fractional kernels $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$), and the initial condition is an $\mathcal{F}_0$-measurable random function $x_0(t) = X_0\varphi(t)$ independent of $W$. The authors' stated contributions are threefold: (i) strong existence and uniqueness in an $L^p$ setting for any sufficiently large $p$, proved via a fixed-point argument in Bochner spaces rather than the semimartingale techniques of earlier work; (ii) qualitative properties — continuous modification, Hölder path regularity, moment bounds, flow estimates, and a Blagoveščenskii–Freidlin-type functional representation; and (iii) an interpolated $[0,T]$0-integrated Euler–Maruyama scheme with an explicit strong convergence rate in $[0,T]$1.

A key structural observation is that solutions are generally neither semimartingales nor Markov processes, so classical tools such as BDG applied directly to the solution fail. The paper circumvents this by introducing, for fixed $[0,T]$2, the auxiliary local martingale $[0,T]$3, which restores access to BDG at the cost of working with Bochner-space norms $[0,T]$4 weighted by an exponential factor $[0,T]$5.

Strong well-posedness

Under Assumption 2.2 — kernel integrability conditions $[0,T]$6 or $[0,T]$7, kernel continuity $[0,T]$8, Lipschitz continuity of $[0,T]$9 on path space uniformly in time, and integrability/Hölder assumptions on $X_{\cdot}^{s}$0 — Proposition 3.4 shows that the Picard map $X_{\cdot}^{s}$1 is Lipschitz on the Banach space $X_{\cdot}^{s}$2 with a Lipschitz constant that vanishes as $X_{\cdot}^{s}$3. Theorem 3.5 then yields a unique strong solution satisfying

$X_{\cdot}^{s}$4

The proof strategy follows Dobrushin–Sznitman-style contraction arguments adapted to the Volterra structure. For regular kernels, where the solution is a semimartingale, the authors note that a simpler supremum-norm Picard argument via stochastic Fubini suffices. A notable feature is that existence and uniqueness are first established only in the Bochner space, without pathwise continuity; continuity is recovered a posteriori through Kolmogorov's criterion or the Garsia–Rodemich–Rumsey inequality, in contrast to most prior treatments.

Path regularity, moments, flows, and representation

Theorem 3.9 establishes increment estimates of order $X_{\cdot}^{s}$5, hence a Hölder-continuous modification of every order below $X_{\cdot}^{s}$6, together with $X_{\cdot}^{s}$7-bounds on the Hölder seminorm and on $X_{\cdot}^{s}$8. These bounds initially require $X_{\cdot}^{s}$9 — a restriction the authors explicitly flag as unusual. It is removed in Theorem 3.13 by a splitting lemma (Lemma 3.16): because the solution admits a measurable functional representation $K_1, K_2$0 (Theorem 3.18, extending Blagoveščenskii–Freidlin to the path-dependent Volterra setting), any bound valid for large $K_1, K_2$1 extends to all $K_1, K_2$2 by conditioning on truncated initial values. The same machinery delivers flow estimates: Lipschitz dependence of the solution on the initial function with constant $K_1, K_2$3, and Hölder continuity of the flow $K_1, K_2$4 in the starting point, using Kolmogorov's criterion jointly in time and space.

Numerical scheme and convergence rate

The proposed discretization is an interpolated $K_1, K_2$5-integrated Euler scheme: drift and diffusion are frozen at grid points but evaluated along the piecewise-affine interpolator $K_1, K_2$6 of past values, and the kernels are integrated exactly over each subinterval:

$K_1, K_2$7

The scheme is simulable whenever closed forms exist for the deterministic weight matrix $K_1, K_2$8 and the Gaussian vectors $K_1, K_2$9; the appendix details their joint simulation via a stable $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$0 Cholesky decomposition, exploiting the telescopic submatrix structure so that a single factorization serves all columns. Computational complexity is $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$1 per path, reflecting the inherent non-Markovianity.

The main numerical result, Theorem 4.8, gives, under the joint Hölder–Lipschitz condition $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$2 on the coefficients,

$K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$3

for every $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$4, with fixed-time rate $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$5 without the $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$6 loss. The proof combines a generalized Grönwall lemma of Volterra type (Corollary 4.4), which handles the mixed linear and square-root integral terms arising from drift and diffusion respectively, with the GRR-based sup-norm transfer and the splitting lemma to cover all $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$7. The technical core is Lemma 4.3, a Young-inequality decomposition bounding weighted Volterra integrals of non-decreasing functions in terms of $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$8 and $K_\alpha(t)=t^{\alpha-1}/\Gamma(\alpha)$9.

For the fractional kernel $\mathcal{F}_0$0 with $\mathcal{F}_0$1, the effective rate reduces to $\mathcal{F}_0$2, i.e., the Hurst exponent $\mathcal{F}_0$3 when coefficients are Lipschitz in time. In the long-memory regime ($\mathcal{F}_0$4) this exceeds the classical $\mathcal{F}_0$5 rate, while in the rough regime it degrades accordingly — consistent with the low Hölder regularity ($\mathcal{F}_0$6) empirically observed in volatility data.

Numerical illustration

Simulations use a scaled convolutive equation with path-dependent coefficients of delay/volatility-functional form, driven by a pseudo-fractional Brownian motion. Sample paths for $\mathcal{F}_0$7 ($\mathcal{F}_0$8) and $\mathcal{F}_0$9 ($x_0(t) = X_0\varphi(t)$0) illustrate the roughening as $x_0(t) = X_0\varphi(t)$1 decreases. Convergence is verified empirically by comparing schemes on nested grids ($x_0(t) = X_0\varphi(t)$2 refinement) driven by the same Brownian motion: since $x_0(t) = X_0\varphi(t)$3 in $x_0(t) = X_0\varphi(t)$4, the telescoping identity bounds the true error by a geometric series in the pairwise grid-refinement error, yielding $x_0(t) = X_0\varphi(t)$5. Log-log plots of mean and endpoint errors over 1000 Monte Carlo iterations show slopes matching, and occasionally exceeding, the predicted rate $x_0(t) = X_0\varphi(t)$6 for both $x_0(t) = X_0\varphi(t)$7 and $x_0(t) = X_0\varphi(t)$8.

Limitations and open questions

Several restrictions are acknowledged or implicit. The well-posedness theory requires global Lipschitz continuity of $x_0(t) = X_0\varphi(t)$9 and $W$0 on path space; singular-kernel regimes relevant to rough volatility often involve locally Lipschitz or polynomial-growth coefficients (e.g., quadratic volatilities), which fall outside the present framework except through special recastings such as the co-kernel SDE reduction of Example 2.2. The sup-norm convergence rate carries an unavoidable $W$1 loss relative to the fixed-time rate, and closing this gap is not addressed. The empirical verification relies on self-consistency between nested grids rather than comparison against an exact solution, which is unavailable in closed form. Finally, weak convergence rates, variance-reduction, and multilevel extensions of the interpolated $W$2-integrated scheme are not treated here and remain open.

Conclusion

The paper provides a self-contained treatment of path-dependent SVIEs covering strong well-posedness in Bochner spaces, sharp path-regularity and flow estimates valid for all moments via a functional-representation splitting argument, and a fully implementable Euler-type scheme with explicit strong rates governed by the kernel singularity exponent and coefficient time-regularity. The combination of the generalized Volterra Grönwall lemma and the Blagoveščenskii–Freidlin representation constitutes the methodological backbone, and the numerical experiments confirm the theoretical rates for fractional kernels across both rough and long-memory regimes.