Reading a Scalar SDE as a Drift-Diffusion Model
A scalar stochastic differential equation describes the local change of a real-valued process using a time increment and a Brownian increment. Its two coefficients have distinct roles: the drift specifies the \(dt\) contribution, while the diffusion coefficient multiplies the random increment \(dB_t\). Either coefficient may depend on time, the current state, or both. Formulating the equation means identifying those dependencies and writing the corresponding integral equation; it does not, by itself, assert that a solution exists or is unique.
Throughout, work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a Brownian motion \(B_t\) relative to the filtration. For a fixed horizon \(T>0\), an Ito integral \(\int_0^t H_s\,dB_s\), \(0\le t\le T\), is taken with a progressively measurable integrand \(H\) satisfying \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\); this class is denoted by \(\mathcal{L}^2\). Drift integrals below are ordinary time integrals. Whenever an equation contains a state-dependent Ito integrand, it is understood that a candidate process is adapted and has the integrability required for that integral. The examples formulate equations; they do not establish existence or uniqueness.
The symbol \(dB_t\) marks the Brownian-driven term, whereas \(dt\) marks accumulated drift. In Ito multiplication notation, \(dB_t\,dB_t=dt\), \(dB_t\,dt=0\), and \(dt\,dt=0\). In particular, replacing an Ito equation with an ordinary chain-rule calculation can lose a correction term. A Stratonovich equation uses \(\circ dB_t\) instead: for a sufficiently regular scalar diffusion coefficient, \(dX_t=b(t,X_t)\,dt+\sigma(t,X_t)\circ dB_t\) has Ito drift \(b(t,X_t)+\tfrac12\sigma(t,X_t)\partial_x\sigma(t,X_t)\). The examples here use Ito notation throughout.
Additive and State-Dependent Noise
Additive noise means that the diffusion coefficient does not depend on the state. It may still depend on time. For example, \(\sigma(t,X_t)=g(t)\) gives random increments whose coefficient is set by the clock rather than by the current value of \(X_t\). By contrast, state-dependent multiplicative noise has a diffusion coefficient that changes with \(X_t\), such as \(\sigma(t,X_t)=cX_t\). A coefficient proportional to the state is one common multiplicative model, but the general state-dependent case need not be linear.
Worked Example: Mean-Reverting Drift with Additive Noise
Consider a process pulled toward level \(m=2\) at rate \(\kappa=3\), with time-varying additive noise \(g(t)=1+t\). Formulate its Ito SDE, identify the coefficients, and write its integral form from \(X_0=x_0\).
The restoring drift is \(\kappa(m-X_t)\). Substituting the given parameters gives \(3(2-X_t)=6-3X_t\). The diffusion is \(g(t)=1+t\), which has no state dependence. Thus \(\mu(t,x)=6-3x\) and \(\sigma(t,x)=1+t\), and the equation is
Its integral form is \(X_t=x_0+\int_0^t(6-3X_s)\,ds+\int_0^t(1+s)\,dB_s\). The second integrand is deterministic and continuous, so it belongs to \(\mathcal{L}^2\) on every finite interval. The first integral is defined whenever the candidate process has integrable drift on that interval. As a check on the coefficient reading, at state \(x=2\) the drift is \(6-3(2)=0\), while the diffusion remains \(1+t\); being at the mean-reversion level cancels the drift but does not remove the noise.
Worked Example: State-Dependent Multiplicative Noise
Consider \(dX_t=0.08X_t\,dt+0.3X_t\,dB_t\), with deterministic \(X_0=x_0\). Identify the two coefficients and evaluate them at state \(x=4\).
Directly matching the \(dt\) and \(dB_t\) terms gives \(\mu(t,x)=0.08x\) and \(\sigma(t,x)=0.3x\). At \(x=4\), the drift coefficient is \(0.08(4)=0.32\), and the diffusion coefficient is \(0.3(4)=1.2\). The integral equation is \(X_t=x_0+\int_0^t0.08X_s\,ds+\int_0^t0.3X_s\,dB_s\), provided the candidate \(X\) is adapted and \(\mathbb{E}\int_0^T(0.3X_s)^2\,ds<\infty\) on the time horizon considered.
Check: at \(x=0\), both coefficients evaluate to zero: \(0.08(0)=0\) and \(0.3(0)=0\). This is a statement about the coefficient functions, not a claim about existence or uniqueness of a process solving the equation.
In a model description, “noise proportional to the current value” identifies a state-dependent diffusion coefficient, not an additional drift term. Similarly, saying that the expected local change is proportional to the current value describes a drift specification. Keeping those statements separate prevents a common source of incorrect equations.
Deterministic Time-Dependent Coefficients
A coefficient can vary with time even when it does not depend on \(X_t\). For deterministic \(a(t)\) and \(g(t)\), the equation \(dX_t=a(t)\,dt+g(t)\,dB_t\) has time-dependent drift and additive, time-dependent diffusion. The drift contribution accumulated from \(0\) to \(t\) is \(\int_0^t a(s)\,ds\); the random contribution is the Ito integral of \(g\).
Proof. Set \(I_t=\int_0^t g(s)\,dB_s\). For deterministic step functions, the integral is a sum of constants times Brownian increments. Each increment has mean zero, and distinct increments are independent, so the sum has mean zero and variance equal to the sum of the squared constants times the lengths of their intervals. Approximate \(g\) in \(L^2([0,t])\) by deterministic step functions \(g_n\). By the Ito isometry, \(\int_0^t g_n(s)\,dB_s\) converges to \(I_t\) in \(L^2(\mathbb{P})\). This convergence implies convergence in \(L^1(\mathbb{P})\), so the zero means pass to the limit. The isometry also gives \(\mathbb{E}[I_t^2]=\int_0^t g(s)^2\,ds\), and therefore \(\operatorname{Var}(I_t)=\int_0^t g(s)^2\,ds\). The other part of \(X_t\), \(x_0+\int_0^t a(s)\,ds\), is deterministic. Adding it shifts the expectation and leaves the variance unchanged, proving both formulas. \(\square\)
Worked Example: A Time-Varying Drift and Diffusion
Let \(X_0=x_0\) and consider \(dX_t=(2t-1)\,dt+\tfrac12e^{-t}\,dB_t\). Identify the coefficients and compute the accumulated drift and variance of the stochastic contribution at \(t=2\).
The drift coefficient is \(\mu(t,x)=2t-1\), which depends on time but not on the state. The diffusion coefficient is \(\sigma(t,x)=\tfrac12e^{-t}\), also deterministic and time-dependent. The integral form is
At \(t=2\), the drift contribution is \(\int_0^2(2s-1)\,ds=[s^2-s]_0^2=(4-2)-(0-0)=2\). By the proposition, the variance of the stochastic contribution is \(\int_0^2(\tfrac12e^{-s})^2\,ds=\tfrac14\int_0^2e^{-2s}\,ds=\tfrac14[-\tfrac12e^{-2s}]_0^2=\tfrac18(1-e^{-4})\). The expectation is therefore \(x_0+2\), and the variance is \(\tfrac18(1-e^{-4})\).
Check: the deterministic drift integral has the right derivative, since \(\frac{d}{dt}(t^2-t)=2t-1\), and it is zero at \(t=0\). The variance formula uses the square of the diffusion coefficient, \((\tfrac12e^{-s})^2=\tfrac14e^{-2s}\), not the coefficient itself.
How a Time-Dependent Rescaling Changes the Coefficients
Changing the units or scale of a state variable changes its coefficient functions. The following result gives the transformation directly. It also illustrates how Ito's formula justifies a new drift-diffusion formulation. Assume the original equation has Ito form and all integrals involved satisfy the stated integrability conditions.
Proof. Apply Ito's formula to \(f(t,x)=\alpha(t)x+\beta(t)\). Its partial derivatives are \(f_t(t,x)=\alpha'(t)x+\beta'(t)\), \(f_x(t,x)=\alpha(t)\), and \(f_{xx}(t,x)=0\). For an Ito process with drift \(\mu\) and diffusion \(\sigma\), the formula is \(df(t,X_t)=[f_t+f_x\mu+\tfrac12 f_{xx}\sigma^2]\,dt+f_x\sigma\,dB_t\), with all derivatives and coefficients evaluated at \((t,X_t)\). Substitution gives the drift \(\alpha'(t)X_t+\beta'(t)+\alpha(t)\mu(t,X_t)+\tfrac12(0)\sigma(t,X_t)^2\) and diffusion \(\alpha(t)\sigma(t,X_t)\). Since \(Y_t=f(t,X_t)\), these are the asserted coefficients. If \(\alpha(t)\ne0\), solving \(Y_t=\alpha(t)X_t+\beta(t)\) for \(X_t\) gives the stated substitution. \(\square\)
Worked Example: Rescaling a Mean-Reverting Model
Suppose \(dX_t=\kappa(m-X_t)\,dt+\sigma\,dB_t\), and define \(Y_t=e^{\kappa t}(X_t-m)\), where \(m,\kappa,\sigma\) are constants. Find the SDE for \(Y_t\).
Here \(\alpha(t)=e^{\kappa t}\), \(\beta(t)=-me^{\kappa t}\), and \(\alpha'(t)=\kappa e^{\kappa t}\), \(\beta'(t)=-m\kappa e^{\kappa t}\). The theorem gives the drift as \(\kappa e^{\kappa t}X_t-m\kappa e^{\kappa t}+e^{\kappa t}\kappa(m-X_t)\). Expanding the last term gives \(\kappa e^{\kappa t}X_t-m\kappa e^{\kappa t}+m\kappa e^{\kappa t}-\kappa e^{\kappa t}X_t=0\). The diffusion is \(\sigma e^{\kappa t}\). Thus \(dY_t=\sigma e^{\kappa t}\,dB_t\).
Check: substituting \(Y_t=e^{\kappa t}(X_t-m)\) into the drift calculation cancels the \(X_t\) terms and the \(m\) terms separately. The transformed diffusion retains the time-varying factor \(e^{\kappa t}\); the transformation removes the drift, not the Brownian noise.
Common Formulation Pitfalls
A frequent error is to place a state-dependent noise term in the drift. For the multiplicative example, writing \(dX_t=(0.08+0.3)X_t\,dt\) is wrong: it treats \(0.3X_t\,dB_t\) as though it were a time-integrated contribution. The correct equation has drift \(0.08X_t\) multiplying \(dt\) and diffusion \(0.3X_t\) multiplying \(dB_t\). The two terms have different meanings and cannot be combined as coefficients of one increment.
Another error is to infer solution properties from a coefficient reading. Identifying \(\mu\) and \(\sigma\) does not prove a solution exists, establish uniqueness, or specify whether a solution is strong (built on the given stochastic basis) or weak (with a probability space and Brownian motion allowed to vary). Those are separate questions. Likewise, the equations here are under \(\mathbb{P}\); changing to a risk-neutral measure \(\mathbb{Q}\) would require stating that measure and how the drift changes. Coefficients alone do not perform a change of measure.
| Model feature | Coefficient pattern | Interpretation |
|---|---|---|
| Additive noise | \(\sigma(t,x)=g(t)\) | Diffusion does not depend on the state |
| Multiplicative noise | \(\sigma(t,x)=c(t)x\) | Diffusion scales with the current state |
| Time-varying drift | \(\mu(t,x)=a(t)\) | Drift depends on time but not the state |
| General scalar model | \(\mu(t,x),\sigma(t,x)\) | Each coefficient may depend on time and state |
Check Your Understanding
For each question, keep the \(dt\) and \(dB_t\) terms separate and state any integrability condition needed for an Ito integral.
- In \(dX_t=(1-t)X_t\,dt+2\,dB_t\), which coefficient is state-dependent, and is the noise additive or multiplicative?
- Write the integral form of \(dX_t=a(t)\,dt+g(t)\,dB_t\), and state the condition on \(g\) over \([0,T]\).
- For \(dX_t=2X_t\,dt+X_t\,dB_t\), evaluate the drift and diffusion coefficients at \(X_t=-3\).
- If \(Y_t=3X_t+1\) and \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\), identify the drift and diffusion of \(Y_t\).
- Explain why a model statement that noise is proportional to the state affects the diffusion coefficient rather than adding a term to the drift.