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Gaussian-type density estimates for mixed SDEs driven by correlated fractional Brownian motions

Maximilian Buthenhoff, Ercan Sönmez

TL;DR

The paper develops a framework to obtain Gaussian-type density bounds for weak solutions of multidimensional SDEs driven by a linear combination of completely correlated fractional Brownian motions with Hurst indices $H_1,H_2$. It avoids Malliavin calculus and stochastic dynamical systems, employing a Girsanov transform for two fBMs and exponential Orlicz spaces, together with a conditionally Gaussian process to establish density existence. Under bounded drift and Hölder regularity (when $\min(H_1,H_2)>1/2$), it proves the existence of a Lebesgue density for $X_{T_0}$ and provides two-sided Gaussian bounds $\frac{C_1'}{\sigma^d(T_0)} e^{-\frac{C_2'|x-x_0|^2}{\sigma^2(T_0)}} \le p_{T_0}(x) \le \frac{C_1}{\sigma^d(T_0)} e^{-\frac{C_2|x-x_0|^2}{\sigma^2(T_0)}}$, with $\sigma^2(T_0) = \int_0^{T_0} [a_1 K_{H_1}(T_0,t) + a_2 K_{H_2}(T_0,t)]^2 dt$. The results are sharp in the short-range regime and extend Gaussian-type density bounds to mixed fBM SDEs without resorting to Malliavin calculus, broadening applicability to systems with memory and long-range dependence.

Abstract

In this work, we investigate the existence and properties of Gaussian-like densities for weak solutions of multidimensional stochastic differential equations driven by a mixture of completely correlated fractional Brownian motions. We consider both the short-range and long-range dependent regimes, imposing a singular drift in the short-range dependent case and a Hölder continuous drift in the long-range dependent setting. Our approach avoids the use of Malliavin calculus and stochastic dynamical systems, relying instead on the Girsanov theorem and the framework of exponential Orlicz spaces. By considering a conditionally Gaussian process, we establish the existence of a density with respect to the Lebesgue measure. Furthermore, we derive Gaussian-type upper and lower bounds for this density, illustrating the optimality of our results in the short-range dependent case.

Gaussian-type density estimates for mixed SDEs driven by correlated fractional Brownian motions

TL;DR

The paper develops a framework to obtain Gaussian-type density bounds for weak solutions of multidimensional SDEs driven by a linear combination of completely correlated fractional Brownian motions with Hurst indices . It avoids Malliavin calculus and stochastic dynamical systems, employing a Girsanov transform for two fBMs and exponential Orlicz spaces, together with a conditionally Gaussian process to establish density existence. Under bounded drift and Hölder regularity (when ), it proves the existence of a Lebesgue density for and provides two-sided Gaussian bounds , with . The results are sharp in the short-range regime and extend Gaussian-type density bounds to mixed fBM SDEs without resorting to Malliavin calculus, broadening applicability to systems with memory and long-range dependence.

Abstract

In this work, we investigate the existence and properties of Gaussian-like densities for weak solutions of multidimensional stochastic differential equations driven by a mixture of completely correlated fractional Brownian motions. We consider both the short-range and long-range dependent regimes, imposing a singular drift in the short-range dependent case and a Hölder continuous drift in the long-range dependent setting. Our approach avoids the use of Malliavin calculus and stochastic dynamical systems, relying instead on the Girsanov theorem and the framework of exponential Orlicz spaces. By considering a conditionally Gaussian process, we establish the existence of a density with respect to the Lebesgue measure. Furthermore, we derive Gaussian-type upper and lower bounds for this density, illustrating the optimality of our results in the short-range dependent case.

Paper Structure

This paper contains 19 sections, 19 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main result
  3. Fractional Brownian motion
  4. Girsanov theorem
  5. Main result
  6. Weak existence and uniqueness of solutions
  7. Existence of weak solution
  8. Case 1: $H_1 = 1/2$.
  9. Case 2: $H_1,H_2 \in (0,1/2)$.
  10. Case 3: $H_1 \in (0,1/2)$, $H_2 \in (1/2,1)$.
  11. Case 4: $H_1,H_2 \in (1/2,1)$.
  12. Uniqueness in law
  13. Existence of a density with respect to the Lebesgue measure
  14. The conditionally Gaussian process
  15. Existence of density
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $B^{H_1}, B^{H_2}$ be two completely correlated fBMs with Hurst parameters $H_1$ and $H_2$ as above. Moreover, let $(u_t)_{t \in [0,T]}$ and $(v_t)_{t \in [0,T]}$ be $(\mathcal{A}_t)_{t\in [0,T]}$-adapted processes with integrable paths such that $\int_0^{\cdot} u_s \mathrm{d}s \in I_{0+}^{H_1+1 for every $t \in (0,T]$. If satisfies $\mathbb{E}[L_T] = 1$, then, under the probability measure $

Theorems & Definitions (38)

  • Theorem 2.1: Girsanov theorem for two fBMs
  • Theorem 2.2
  • Definition 3.1: Weak solution
  • Proposition 3.2: cf. nualart2021regularization
  • proof
  • Theorem 3.3: Existence of weak solution
  • proof
  • Theorem 3.4
  • proof
  • Proposition 4.1: Hölder continuity
  • ...and 28 more