Stochastic One-Sided Osgood Condition

• The stochastic one-sided Osgood condition is a generalization of monotonicity that controls non-Lipschitz dynamics in stochastic differential systems.
• It guarantees the existence and uniqueness of L1 solutions in multidimensional BSDEs by leveraging integrability and stochastic inequalities.
• The framework underpins practical applications in finance, risk measures, and nonlinear PDEs by accommodating random, non-uniform coefficients.

The stochastic one-sided Osgood condition is a structural generalization of monotonicity and Lipschitz-type regularity appearing in the analysis of stochastic differential equations (SDEs), backward SDEs (BSDEs), and related stochastic systems. It permits non-Lipschitz, possibly nonlinear, but integrably controlled drift or generator dynamics and ensures well-posedness—particularly existence and uniqueness—under minimal regularity. This condition has become central in establishing robust theory for multidimensional BSDEs and SDEs under weak integrability and non-uniform stochastic coefficients, as exemplified by (Lai et al., 15 Sep 2025).

1. Mathematical Formulation of the Stochastic One-Sided Osgood Condition

The condition concerns generator functions $$g: \Omega \times [0,T] \times \mathbb{R}^k \times \mathbb{R}^{k\times d}\to\mathbb{R}^k$$ in multidimensional BSDEs: $$y_t = \xi + \int_t^T g(s, y_s, z_s)\, ds - \int_t^T z_s \, dB_s, \qquad t\in[0,T].$$ The stochastic one-sided Osgood condition (assumption (H1)) posits that, for any $y_1, y_2 \in \mathbb{R}^k$ and $z \in \mathbb{R}^{k\times d}$,

$\Bigl\langle \frac{y_1-y_2}{|y_1-y_2|} \mathbbm{1}_{\{|y_1-y_2|\neq 0\}},\, g(\omega,t,y_1,z)-g(\omega,t,y_2,z) \Bigr\rangle \le u_t(\omega) \, \rho(|y_1-y_2|),$

for all $(\omega, t)$, where $u \in L^\infty(\Omega; L^1([0,T]))$ is a nonnegative stochastic process and $\rho$ is a nondecreasing, concave function with $\rho(0) = 0$, $\rho(u)>0$ for $u>0$, and

$\int_{0^+} \frac{du}{\rho(u)} = +\infty.$

This generalizes both standard monotonicity (Osgood with linear $\rho$) and deterministic Osgood conditions by permitting both explicit time-random dependence and nonlinearity.

The condition is not two-sided (does not enforce symmetry under exchange of $y_1$, $y_2$), which is crucial in stochastic models with inherent directionality (e.g., jumps, dissipative flows, or directional risk in finance).

2. Existence and Uniqueness of $L^1$ Solutions for Multidimensional BSDEs

The main result [(Lai et al., 15 Sep 2025), Theorem 3] establishes that under assumptions (H1)-(H5), including the stochastic one-sided Osgood condition, a general growth control in $y$, a stochastic Lipschitz (with random coefficient) and (possibly) sublinear growth in $z$ (assumptions (H3)-(H4)), and an integrability requirement on both $\xi$ and $g(\cdot,0,0)$, there exists a unique $L^1$ solution $(y,z)$ such that $y$ is of class (D): $E\left[\int_0^T u_t |y_t| \,dt\right] < +\infty.$ The proof relies on constructing a Picard iteration sequence and a priori estimates. Convergence is shown by controlling successive differences via stochastic Gronwall-type and Bihari-type inequalities adapted to the random framework. The uniqueness statement demonstrates that, provided the additional integrability holds, any two solutions must coincide—again leveraging the Osgood structure to prevent explosive deviation.

3. Stochastic Lipschitz and General Growth Assumptions

The stochastic one-sided Osgood condition is combined with a stochastic Lipschitz bound in $z$: $$|g(\omega,t,y,z_1) - g(\omega,t,y,z_2)| \le v_t(\omega) |z_1-z_2|,$$ for a $\mathbb{F}$-predictable, nonnegative $v\in L^\infty(\Omega; L^2([0,T]))$, and a general (not necessarily linear) integrability growth condition in $y$: $$\psi_r(\omega,t) := \sup_{|y|\le r} |g(\omega,t,y,0)|$$ must be integrable in expectation for all $r$.

The integrability condition imposed on the "first component" of $y$,

$E\left[\int_0^T u_t |y_t| dt\right] < +\infty,$

ensures that gains due to stochastic fluctuations do not accumulate uncontrollably—thus allowing the Osgood mechanism to "tame" the nonlinearity.

4. Stochastic Gronwall and Bihari Type Inequalities

To handle non-uniform, random coefficients and weaker solution integrability, the analysis develops stochastic analogues of classical deterministic inequalities:

• Stochastic Gronwall Inequality: For an adapted nonnegative process $\mu_t$ satisfying

$$\mu_t \leq E\bigl[ \eta + \int_t^T (\beta_s \mu_s + f_s) ds \big| \mathcal{F}_t \bigr]$$

with $L^1$ integrability, one can deduce exponential-type bounds for $\mu_t$.

• Stochastic Bihari Inequality: For

$$\mu_t \leq c + E\left[ \int_t^T \beta_s \rho(\mu_s) ds \middle| \mathcal{F}_t \right],$$

and the same Osgood condition on $\rho$, an explicit bound is achieved:

$$\mu_t \leq \Theta^{-1}\left( \Theta(c) + \int_t^T \| \beta_s \|_{L^1} ds \right)$$

where $\Theta(x) = \int_1^x du/\rho(u)$. Both inequalities allow the stochastic modulating coefficients (e.g., $u_t$, $v_t$) to vary unpredictably, reflecting a genuine stochastic flexibility central to applications.

These techniques provide the a priori machinery for controlling Picard differences and local increments of the solution, allowing generalization to flexible and weakly integrable settings.

5. Broader Implications and Applications

The stochastic one-sided Osgood condition allows the theory of multidimensional BSDEs to be developed in $L^1$—significant for applications where solutions may not be square integrable.

Practical Consequences:

• Stochastic Coefficient Models: Coefficient processes are permitted to be random and sample-path dependent, reflecting model uncertainty, pathwise controls, or environmental randomness.
• Nonlinear Filtering and Risk: Osgood nuclei can be embedded in risk measure computations, g-expectations, or in financial pricing under risk-sensitive or nonlinear growth (e.g., superlinear penalties or "dissipative" effects).
• Nonlinear PDEs: Probabilistic representations of nonlinear PDEs via BSDEs now admit wider classes (e.g., non-globally Lipschitz, Osgood only) of nonlinearities, especially in high-dimensional, degenerate, or singular settings.

The extension of the Osgood paradigm to the stochastic and multidimensional setting significantly broadens the tractable class of BSDEs, providing a framework for both theoretical investigation and applications in areas such as stochastic control, finance, and interacting particle systems.

6. Technical Summary Table

Aspect	Deterministic Osgood	Stochastic One-Sided Osgood
Drift/Generator control	$|b(x)-b(y)| \leq \rho(|x-y|)$	$$\langle u, g(y_1)-g(y_2)\rangle \leq u_t(\omega) \rho(|y_1-y_2|)$$
Osgood requirement	$\int_0^+ du/\rho(u) = +\infty$	Same; $\rho$: concave/nondecreasing, Osgood divergence
Coefficient regularity	Deterministic, uniform	Stochastic, random, non-uniform
Solution space	$L^p$, $p \geq 1$	$L^1$, class (D), under random weights
Uniqueness mechanism	Bihari's inequality	Stochastic Bihari, Gronwall, integrability of $u_t$

The stochastic one-sided Osgood condition, with random or time-dependent coefficients and general nonlinearities, stands as a minimal—yet sharp—requirement for well-posedness of multidimensional BSDEs with only $L^1$-level data. Its integration with stochastic analysis tools (martingale inequalities, stochastic fixed point theory, and a posteriori estimates) provides a flexible yet rigorous theoretical underpinning for current and future directions in the mathematical theory of stochastic dynamical systems.