Reading the Structure of a Scalar SDE
A scalar stochastic differential equation can look complicated when its terms are expanded, but its dependence on the state often has a simple structure. Identifying that structure tells us which explicit-solution methods may apply later. Here the task is classification: read off the coefficients, determine whether they are affine in the state, and distinguish the homogeneous and additive-noise cases. No solution formula is needed for these tests.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\) adapted to the filtration. Fix a finite horizon \(T>0\). We interpret an SDE in integral form, with every Ito integral \(\int_0^t H_s\,dB_s\) taken for a progressively measurable integrand \(H\) in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). When the integrand depends on a solution \(X\), this requirement applies to the resulting process \(H_s\), for example \(H_s=c_sX_s+g_s\). We consider deterministic, measurable coefficient functions \(a,f,c,g\) on \([0,T]\), and equations of the form
The drift is the coefficient of \(dt\), namely \(\mu(t,x)=a_tx+f_t\). The diffusion coefficient is the coefficient of \(dB_t\), namely \(\sigma(t,x)=c_tx+g_t\). The terms \(f_t\) and \(g_t\) do not depend on the state; they are forcing terms in the drift and diffusion, respectively.
These labels describe different features. An affine equation need not be homogeneous. An additive-noise equation can still have a state-dependent drift, and it is an affine SDE only when the drift is affine as well. If \(c_t=g_t=0\), the equation has no stochastic forcing at all. All coefficient identifications below are understood pointwise in \(t\) when the functions are specified pointwise, or up to equality for almost every \(t\) when equations are identified through their time integrals.
How to Identify and Test the Coefficients
To read coefficients from an equation, first collect all terms multiplying \(dt\) and all terms multiplying \(dB_t\). Then, within each of those two coefficients, separate the term proportional to \(X_t\) from the term independent of \(X_t\). Do not assign a term to the drift or diffusion based on whether it contains \(X_t\): the differential \(dt\) or \(dB_t\) determines which coefficient it belongs to.
The affine form is not merely a convenient notation. It gives a precise test. For fixed \(t\), a function \(q(x)\) is affine if it can be written \(q(x)=mx+b\). Its slope is \(q(1)-q(0)\), and its intercept is \(q(0)\). To verify the representation for all states, one must check that \(q(x)=q(0)+x(q(1)-q(0))\) for every \(x\), not just at two selected states.
Proof. If the stated affine representations hold, substituting \(x=0\) gives \(f_t=\mu(t,0)\) and \(g_t=\sigma(t,0)\). Substituting \(x=1\) and subtracting the values at \(0\) gives \(a_t=\mu(t,1)-\mu(t,0)\) and \(c_t=\sigma(t,1)-\sigma(t,0)\). These identities prove uniqueness. Conversely, if both functions are affine in \(x\), by definition each has a representation as a slope times \(x\) plus an intercept. Calling those slopes \(a_t,c_t\) and those intercepts \(f_t,g_t\) gives the required form. The argument also includes constant functions, whose slopes are zero, and zero functions, whose slopes and intercepts are all zero. \(\square\)
Worked Examples: Extracting the Coefficients
Worked Example: Nonzero Forcing in Both Coefficients
Classify the equation \(dX_t=(2X_t+3)\,dt+(-X_t+4)\,dB_t\), and verify its coefficients by evaluating the drift and diffusion at two states.
Collect the \(dt\) and \(dB_t\) terms. The drift is \(\mu(t,x)=2x+3\), so \(a_t=2\) and \(f_t=3\). The diffusion coefficient is \(\sigma(t,x)=-x+4\), so \(c_t=-1\) and \(g_t=4\). The coefficients are constant in time.
At \(x=0\), the drift is \(3\) and the diffusion is \(4\). At \(x=1\), the drift is \(2(1)+3=5\) and the diffusion is \(-1+4=3\). The drift slope recovered from these values is \(5-3=2\), and its intercept is \(3\). The diffusion slope is \(3-4=-1\), and its intercept is \(4\). Thus the recovered coefficients agree with the equation. Since \(f_t\ne0\) and \(g_t\ne0\), it is affine but not homogeneous; since \(c_t\ne0\), its noise is not additive.
Check: substituting the recovered coefficients gives drift \(2x+3\) and diffusion \(-x+4\), exactly the original two coefficients.
Worked Example: Affine but Not Homogeneous
Classify \(dX_t=(5-2X_t)\,dt+(3X_t-6)\,dB_t\). Is its noise additive?
Reorder each coefficient into slope-times-state plus intercept. The drift is \(-2x+5\), so \(a_t=-2\) and \(f_t=5\). The diffusion is \(3x-6\), so \(c_t=3\) and \(g_t=-6\). Both coefficients are affine, but \(f_t\) and \(g_t\) are not both zero, so the equation is not homogeneous. Because \(c_t=3\ne0\), its diffusion depends on the state and the noise is not additive.
At \(x=2\), the drift evaluates to \(5-2(2)=1\), while the diffusion evaluates to \(3(2)-6=0\). The coefficient formulas give \((-2)(2)+5=1\) and \(3(2)-6=0\), confirming the identification. A zero diffusion value at this particular state does not mean the equation has no noise: at \(x=0\), for example, its diffusion coefficient is \(-6\).
Check: the slopes from the state evaluations are \(\mu(t,1)-\mu(t,0)=3-5=-2\) and \(\sigma(t,1)-\sigma(t,0)=-3-(-6)=3\), matching \(a_t\) and \(c_t\).
Worked Example: Additive Noise with Time-Dependent Coefficients
On any finite horizon \([0,T]\), classify \(dX_t=(tX_t+t^2)\,dt+e^{-t}\,dB_t\).
The drift is \(\mu(t,x)=tx+t^2\), giving \(a_t=t\) and \(f_t=t^2\). The diffusion is \(\sigma(t,x)=e^{-t}\), which has no \(x\)-dependence; hence \(c_t=0\) and \(g_t=e^{-t}\). It is an affine SDE with additive noise, but not a homogeneous one because \(f_t=t^2\) is nonzero for \(t>0\).
For a direct check at \(t=1\) and \(x=2\), the drift is \(1\cdot2+1^2=3\), and the diffusion is \(e^{-1}\). The coefficient representation gives \(a_1x+f_1=1\cdot2+1=3\) and \(c_1x+g_1=0\cdot2+e^{-1}=e^{-1}\). On \([0,T]\), these deterministic coefficients are bounded: \(0\le t\le T\), \(0\le t^2\le T^2\), and \(0<e^{-t}\le1\). In particular, if \(X\) has finite second moment integrated over \([0,T]\), then the diffusion integrand \(e^{-t}\) is in \(\mathcal{L}^2\), since \(\mathbb{E}\int_0^T e^{-2t}\,dt\le T\).
Check: the diffusion evaluated at \(x=0\) and \(x=1\) is \(e^{-t}\) in both cases, so its slope is \(e^{-t}-e^{-t}=0\), as required for additive noise.
Worked Example: Detecting a Non-Affine Drift
Decide whether \(dX_t=(X_t^2+1)\,dt+(X_t+1)\,dB_t\) is an affine SDE.
The diffusion is affine: \(\sigma(x)=x+1\), with \(c_t=1\) and \(g_t=1\). The drift candidate is \(\mu(x)=x^2+1\). Its values at \(x=0,1,2\) are \(1,2,5\), respectively. If it were affine, its slope from the first two values would be \(2-1=1\), and its intercept would be \(1\). That proposed affine function would have value \(1(2)+1=3\) at \(x=2\), whereas the actual drift is \(2^2+1=5\). The drift is therefore not affine, so the SDE is not affine, even though its diffusion is affine.
Check: the first differences of the drift values are \(2-1=1\) and \(5-2=3\). An affine function has equal first differences over equal state increments; these are unequal, confirming the failure of the affine test.
Why the Classification Helps
Affine coefficients also make the standard Lipschitz and growth checks direct. These checks are useful when deciding whether the existence-and-uniqueness results for SDEs apply. The proposition below gives the specific bounds; it does not provide an explicit solution.
Proof. Let \(A=\sup_{t\in[0,T]}|a_t|\), \(F=\sup_{t\in[0,T]}|f_t|\), \(C_0=\sup_{t\in[0,T]}|c_t|\), and \(G=\sup_{t\in[0,T]}|g_t|\); these are finite by the boundedness assumption. For any \(x,y\), subtracting the drift values gives \(\mu(t,x)-\mu(t,y)=a_t(x-y)\), so its absolute value is at most \(A|x-y|\). Subtracting the diffusion values gives \(\sigma(t,x)-\sigma(t,y)=c_t(x-y)\), so its absolute value is at most \(C_0|x-y|\). Adding yields the Lipschitz bound with \(L=A+C_0\). For growth, the triangle inequality gives \(|\mu(t,x)|+|\sigma(t,x)|\le(A+C_0)|x|+F+G\). Taking \(C=A+C_0+F+G\) bounds this by \(C(1+|x|)\), because both \(|x|\) and \(1\) are nonnegative. This proves the claims, including cases where any or all of the coefficient bounds are zero. \(\square\)
The affine structure therefore identifies not only special labels such as homogeneous or additive-noise, but also a simple route to verifying global coefficient bounds. Under the usual square-integrability assumptions on the initial condition and the solution, these bounds ensure the drift and diffusion terms have the integrability required in the SDE formulation. The classification itself does not assert a particular explicit solution or probability law.
Common Pitfall: Reversing Slope and Intercept
A frequent error is to read the constant term as the state coefficient. For the diffusion \(-x+4\), a wrong identification would be \(c_t=4\) and \(g_t=-1\), producing \(4x-1\). At \(x=0\), this incorrect expression gives \(-1\), while the original diffusion gives \(4\); at \(x=1\), it gives \(3\), while the original gives \(3\). Matching at \(x=1\) alone is not enough. The correction is to use the intercept at zero and the difference between values at one and zero: \(g_t=\sigma(t,0)=4\) and \(c_t=\sigma(t,1)-\sigma(t,0)=3-4=-1\). The reconstructed coefficient \(-x+4\) then matches at both states and, by direct algebra, for every \(x\).
Another classification error is to call an equation homogeneous merely because its coefficients contain terms proportional to \(X_t\). Homogeneity requires that there be no state-independent forcing in either coefficient: both \(f_t\) and \(g_t\) must vanish. Likewise, additive noise requires the diffusion slope \(c_t\) to vanish; a state-independent drift does not make the noise additive if the diffusion still depends on \(X_t\).
Check Your Understanding
Use the coefficient tests above. When classifying, specify whether each claim holds for all times or only almost every time under the assumptions given.
- For \(dX_t=(4X_t-2)\,dt+7X_t\,dB_t\), identify \(a_t,f_t,c_t,g_t\). Is the equation homogeneous? Does it have additive noise?
- For \(dX_t=(tX_t)\,dt+3\,dB_t\), identify the four coefficient functions and classify the equation.
- Give a direct state-value test showing that the drift \(x^2-2\) is not affine.
- Suppose an affine diffusion has \(\sigma(t,0)=5\) and \(\sigma(t,1)=2\). Recover \(c_t\) and \(g_t\), and determine whether its noise is additive.
- Explain why finding matching affine values at \(x=0\) and \(x=1\) does not by itself prove that an arbitrary function of \(x\) is affine.