Maruyama Representation in SDEs

• Maruyama Representation is a framework that uses discrete-time Euler–Maruyama approximations to construct solutions for McKean–Vlasov SDEs with non-Lipschitz coefficients.
• It rigorously demonstrates weak existence and pathwise uniqueness through moment bounds, tightness arguments, and generalized Grönwall methods.
• The method achieves an optimal strong convergence rate of O(Δt), making it a robust tool for both theoretical analysis and practical applications.

The Maruyama representation refers to the construction of solutions to stochastic differential equations (SDEs)—in particular, McKean–Vlasov SDEs with non-Lipschitz coefficients—via discrete-time Euler–Maruyama (EM) approximations. This framework is rigorously analyzed in the context of weak and strong existence, pathwise uniqueness, and convergence rates, as explicated by Ding–Qiao (2019) (Ding et al., 2019).

1. Euler–Maruyama Scheme for McKean–Vlasov SDEs

The McKean–Vlasov SDE under consideration is posed on $\R^d$ with dynamics: $dX_t = b(t, X_t, \mu_t)\,dt + \sigma(t, X_t, \mu_t)\,dW_t, \quad \mu_t = \Law(X_t),$ where $b$ is the drift, $\sigma$ the diffusion, and $\mu_t$ the law of $X_t$. The EM time-discretization employs mesh points $t_k = k\Delta t$, $\Delta t = T/2^n$, yielding discrete iterates: $\begin{aligned} X^{(n)}_0 &= X_0, \quad \Law(X_0) = \mu_0,\ X^{(n)}_{t_{k+1}} &= X^{(n)}_{t_k} + b(t_k, X^{(n)}_{t_k}, \mu^{(n)}_{t_k})\,\Delta t + \sigma(t_k, X^{(n)}_{t_k}, \mu^{(n)}_{t_k})\,(W_{t_{k+1}} - W_{t_k}), \end{aligned}$ where $\mu^{(n)}_{t_k} = \Law(X^{(n)}_{t_k})$ denotes the empirical law at each step. The EM process $X^{(n)}$ can be interpolated to all $t \in [0,T]$ by holding values constant in drift between mesh points and integrating the Brownian increment for the martingale part.

2. Non-Lipschitz Conditions on Coefficients

Two key conditions underpin the analysis:

• (H1) Linear Growth: For all $(x, \mu) \in \R^d \times \Mc_2(\R^d)$,

$$|b(x, \mu)|^2 + \|\sigma(x, \mu)\|^2 \le L_1 \left(1 + |x|^2 + \int |y|^2\,\mu(dy)\right),$$

controlling moment growth of the process.

• (H2) One-Sided Non-Lipschitz Continuity: For $(x_i, \mu_i)$, exists concave, strictly increasing functions $K_1,K_2$ vanishing at 0 and $L_2>0$ such that

$$\begin{split} 2\langle x_1 - x_2, b(x_1, \mu_1) - b(x_2, \mu_2)\rangle + \|\sigma(x_1, \mu_1) - \sigma(x_2, \mu_2)\|^2\ \le L_2 \left( K_1(|x_1-x_2|^2) + K_2(\rho^2(\mu_1, \mu_2)) \right), \end{split}$$

with $\rho(\mu, \nu)$ a Wasserstein-type distance based on quadratic-growth test functions. An equivalent “log-Lipschitz” alternative (H2′) is noted.

3. Existence of Weak Solutions Using EM Approximation

A three-step procedure establishes the existence of weak solutions:

1. Moment Bounds (Lemma 3.4): Uniform in $n$ and time,

$$\sup_{0 \le t \le T} \mathbb{E}\bigl|X^{(n)}_t\bigr|^{2p} < \infty, \quad \mathbb{E}\bigl|X^{(n)}_t - X^{(n)}_s\bigr|^{2p} \le C(t-s)^p,$$

derived via discrete Itô/Burkholder–Davis–Gundy and Grönwall arguments.

2. Tightness and Identification of Limits (Proposition 3.5): The laws $\mathbb{P}^{(n)} = \Law(X^{(n)}_\cdot)$ are tight in $C([0,T];\R^d)$, and any weak limit $\mathbb{P}^*$ solves the McKean–Vlasov martingale problem:

$$M^f_t = f(X_t) - f(X_0) - \int_0^t [\mathcal{L}_{\mu_s} f](X_s)\,ds$$

for $f \in C^2$, where $$\mathcal{L}_\mu f(x) = \frac12 \mathrm{tr}[\sigma\sigma^*(x, \mu) D^2 f(x)] + \langle b(x, \mu), \nabla f(x)\rangle$$.

3. Completion: Existence of a filtered probability space and adapted process $X$ that solves the SDE in the weak sense is secured.

4. Pathwise Uniqueness and Generalized Grönwall Arguments

Pathwise uniqueness is ensured by analysis of two solutions $(X^1, X^2)$ driven by the same $W$. Defining $Z_t = X^1_t - X^2_t$, Itô’s formula and condition (H2) imply

$\mathbb{E}|Z_t|^2 \le L_2 \int_0^t \left( K_1(\mathbb{E}|Z_s|^2) + K_2(\rho^2(\Law(X^1_s), \Law(X^2_s))) \right)\,ds.$

This reduces, when $\Law(X^1_s) = \Law(X^2_s)$ if $Z_s=0$, to an Osgood-type integral inequality. Lemma 3.6 yields $\mathbb{E}|Z_t|^2 = 0$ identically, enforcing $X^1_t \equiv X^2_t$ almost surely. By Yamada–Watanabe’s principle for McKean–Vlasov equations, weak existence and pathwise uniqueness guarantee a strong solution.

5. Strong Convergence Rate of the Euler–Maruyama Approximation

The EM scheme achieves an optimal strong convergence rate as formalized in Theorem 4.1. Under (H1), (H2), and $\mathbb{E}|X_0|^{2p}<\infty$ for all $p$, for some small $T_0>0$,

$$\mathbb{E}\left[\sup_{0 \le t \le T_0} |X^{(n)}_t - X_t|^2\right] = O(\Delta t),$$

and by patching in time, this rate is extended over $[0,T]$. In the classical case, the standard $\mathcal{L}^2$-rate $O(\Delta t)$ is recovered. The proof leverages a decomposition of the error $H_t = X^{(n)}_t - X_t$ into drift, diffusion, and martingale components, application of (H2)-type estimates, and a Grönwall-type lemma for mixed Osgood inequalities.

6. Continuum Limit: Representation of Solutions

The continuous solution admits a representation as the limit of discrete EM approximations: $$X_t = \lim_{n \to \infty} \left( X^{(n)}_{\kappa_n(t)} + b(\kappa_n(t), X^{(n)}_{\kappa_n(t)}, \mu^{(n)}_{\kappa_n(t)}) (t - \kappa_n(t)) + \sigma(\kappa_n(t), X^{(n)}_{\kappa_n(t)}, \mu^{(n)}_{\kappa_n(t)})(W_t - W_{\kappa_n(t)}) \right),$$ which converges to the integral form

$$X_t = X_0 + \int_0^t b(X_s, \mu_s)\,ds + \int_0^t \sigma(X_s, \mu_s)\,dW_s,$$

with convergence in law (weak existence) and mean-square uniform sense (strong convergence). Thus, the EM approximation constructs an explicit representation for solutions to the McKean–Vlasov SDE.

7. Summary Table: Key Properties of Maruyama Representation (Ding–Qiao, 2019)

Property	Condition/Hypothesis	Result
Moment bounds	(H1) Linear growth	Uniform in $n$, time
Weak existence	(H1), (H2)	Limit of discrete martingale schemes
Pathwise uniqueness	(H2), Osgood lemma	Holds; implies strong solution
Strong convergence rate	(H1), (H2), $\mathbb{E}|X_0|^{2p}$	$O(\Delta t)$ globally

This framework establishes the EM-based Maruyama representation as a central analytic and constructive tool for McKean–Vlasov SDEs with non-Lipschitz coefficients, ensuring existence, uniqueness, and optimal convergence properties as proved in Ding–Qiao (2019) (Ding et al., 2019).