---
title: Maruyama Representation in SDEs
url: https://www.emergentmind.com/topics/maruyama-representation
type: topic
---

# Maruyama Representation in SDEs

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) [1903.11754].

## 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) [1903.11754].

Source: https://www.emergentmind.com/topics/maruyama-representation