Euler-Maruyama Updates for Scalar SDEs

From an Ito SDE to a Discrete Update

Numerical simulation replaces a continuous-time stochastic process by values on a finite time grid. For a scalar SDE, Euler–Maruyama uses the simplest local approximation: over a short interval, evaluate the drift and diffusion at the left endpoint and hold those values fixed for that step. The Brownian increment is not replaced by a deterministic quantity; it remains a random input whose variance equals the step length.

Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). Consider the scalar Ito SDE \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\), with initial value \(X_0\). Here \(\mu\) is the drift coefficient and \(\sigma\) is the diffusion coefficient. Unless another measure is explicitly specified, all expectations and Brownian increments in this tutorial are under \(\mathbb{P}\).

For definiteness, assume \(X_0\) is \(\mathcal{F}_0\)-measurable with \(\mathbb{E}[X_0^2]<\infty\), and that \(\mu,\sigma\) are measurable in time and globally Lipschitz in the state, uniformly in time, with linear growth. Thus, for some finite constant \(C\) and all \(t,x,y\),

These are standard sufficient conditions for a square-integrable solution of the SDE and for the finite-grid recursion below to be well defined. They are not necessary for every possible simulation problem. A numerical recursion can also be written for other coefficients, but its integrability and stability then need to be checked separately.

Choose a deterministic grid \(0=t_0<t_1<\cdots<t_N=T\), set \(h_n=t_{n+1}-t_n\), and write \(\Delta B_n=B_{t_{n+1}}-B_{t_n}\). Brownian motion gives \(\Delta B_n\sim N(0,h_n)\), independent of \(\mathcal{F}_{t_n}\). Euler–Maruyama starts at \(X_0^{\mathrm{EM}}=X_0\) and updates by

Equivalently, one can generate independent standard normal variables \(Z_n\sim N(0,1)\), independent of the information at the start of the step, and set \(\Delta B_n=\sqrt{h_n}Z_n\). The factor is \(\sqrt{h_n}\), not \(h_n\), because \(\operatorname{Var}(\sqrt{h_n}Z_n)=h_n\).

Why the Recursion Has This Form

On the interval \([t_n,t_{n+1}]\), freeze the coefficients at the known value \(X_n^{\mathrm{EM}}\). Integrating those frozen coefficients gives the drift contribution \(\mu(t_n,X_n^{\mathrm{EM}})h_n\) and the stochastic contribution \(\sigma(t_n,X_n^{\mathrm{EM}})\Delta B_n\). The second identity is an Ito-integral identity, so its integrability hypothesis matters.

Proof. The random variable \(\sigma(t_n,X_n^{\mathrm{EM}})\) is \(\mathcal{F}_{t_n}\)-measurable. A process constant on this interval with that value is adapted and progressively measurable there. Its square-integrability is verified directly: \(\mathbb{E}\int_{t_n}^{t_{n+1}}H_s^2\,ds=h_n\mathbb{E}[\sigma(t_n,X_n^{\mathrm{EM}})^2]<\infty\). For a simple predictable integrand constant on \((t_n,t_{n+1}]\), the Ito integral is defined as its value times the Brownian increment. Applying that definition gives the stated identity. Equality in \(L^2\) means the squared difference has expectation zero, so the random variables agree almost surely. \(\square\)

The square-integrability condition cannot be replaced by mere finiteness almost surely: an almost surely finite coefficient can have infinite second moment and then fail to belong to the \(\mathcal{L}^2\) class used to define the Ito integral. Under the Lipschitz and linear-growth assumptions above, the condition holds at every step. Indeed, if \(\mathbb{E}[(X_n^{\mathrm{EM}})^2]<\infty\), linear growth implies \(\mathbb{E}[\mu(t_n,X_n^{\mathrm{EM}})^2]<\infty\) and \(\mathbb{E}[\sigma(t_n,X_n^{\mathrm{EM}})^2]<\infty\). The update then has a finite second moment: use \(|a+b+c|^2\le3(a^2+b^2+c^2)\), \(\mathbb{E}[(\Delta B_n)^2]=h_n\), and independence of \(\Delta B_n\) from \(\mathcal{F}_{t_n}\). Starting from the assumed square-integrable \(X_0\), induction proves the claim for every grid index.

In continuous-time notation, the associated piecewise-frozen approximation is \(Y_t=X_0+\int_0^t\mu(t_n,X_n^{\mathrm{EM}})\,ds+\int_0^t\sigma(t_n,X_n^{\mathrm{EM}})\,dB_s\), where on each subinterval the index \(n\) is chosen with \(t_n\le s<t_{n+1}\). The stochastic integrand is progressively measurable and has finite \(\mathbb{E}\int_0^T H_s^2\,ds\) under the assumptions just stated. At grid points, this representation gives exactly the recursion. It is an approximation to the SDE, not generally its exact solution.

Worked Scalar Updates

Interpretation and Common Mistakes

The update uses Ito increments. It is not a Stratonovich update: under sufficient smoothness, the scalar conversion is \(\int_0^t \sigma(s,X_s)\circ dB_s=\int_0^t\sigma(s,X_s)\,dB_s+\tfrac12\int_0^t\sigma_x(s,X_s)\sigma(s,X_s)\,ds\). Thus a Stratonovich SDE has a drift correction when rewritten in Ito form; applying an Ito scheme without that conversion generally simulates a different equation.

For an Ito step, \(\Delta B_n\) has conditional mean zero given \(\mathcal{F}_{t_n}\), and variance \(h_n\). When the coefficient is square-integrable, independence gives \(\mathbb{E}[\sigma(t_n,X_n^{\mathrm{EM}})\Delta B_n\mid\mathcal{F}_{t_n}]=0\). This conditional statement about one step does not assert that the whole Euler path is a martingale: the drift contribution is present. Nor does writing a stochastic integral by itself establish a true martingale property; the relevant integrability conditions must be checked. Strong convergence concerns approximation of paths on the same Brownian probability space, whereas weak convergence concerns expectations of test functions of the approximation. The recursion alone establishes neither convergence rate; those are separate error questions.

A frequent scaling error is to use \(h_n Z_n\) for the random term. For the constant-coefficient first step of the first example, \(h_n Z_n=\tfrac14\cdot2=\tfrac12\) would be the Brownian increment only if \(Z_n=2\), but it would have variance \(h_n^2\), not \(h_n\), for a standard normal draw. The correct draw is \(\sqrt{h_n}Z_n\), whose variance is \(h_n\). With \(h_n=\tfrac14\) and \(Z_n=1\), the correct increment is \(\tfrac12\), while the incorrect one is \(\tfrac14\). Multiplying by diffusion coefficient 3 gives respective noise contributions \(\tfrac32\) and \(\tfrac34\), a factor-of-two error.

A second common error is to evaluate coefficients at the newly computed state. In the second worked example, the first update must use \(\mu(0,2)=-1\) and \(\sigma(0,2)=1\), not the coefficients at a future value. Euler–Maruyama is explicit precisely because the current approximation \(X_n^{\mathrm{EM}}\) determines both coefficients before the next update is calculated. Keep the drift scaling \(h_n\), the noise scaling \(\Delta B_n\), and the left-endpoint coefficient evaluations distinct.