Constructing Picard Iterates for an SDE

From an SDE to a Sequence of Processes

For an SDE, replacing an unknown process inside its own drift and diffusion coefficients by a previously constructed process turns the equation into a sequence of explicit constructions. This is the Picard iteration method. At this stage, the goal is to define the sequence and verify that each step makes sense; defining the iterates alone does not prove that they converge or that an SDE has a solution.

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. Fix a finite horizon \(T>0\). Let the initial value \(\xi\) be \(\mathcal{F}_0\)-measurable with \(\mathbb{E}|\xi|^2<\infty\). Consider the Ito SDE \(dX_t=\mu(t,X_t)\,dt+\sigma(t,X_t)\,dB_t\), with \(X_0=\xi\). Here \(\mu\) is the drift coefficient and \(\sigma\) is the diffusion coefficient.

Assume that \(\mu\) and \(\sigma\) are Borel measurable and satisfy global Lipschitz and linear growth bounds: for some constants \(L,C<\infty\), for every \(t\in[0,T]\) and \(x,y\in\mathbb{R}\), \(|\mu(t,x)-\mu(t,y)|+|\sigma(t,x)-\sigma(t,y)|\le L|x-y|\), and \(|\mu(t,x)|^2+|\sigma(t,x)|^2\le C(1+|x|^2)\). The iterates are defined by \(X^0_t=\xi\) and, for \(n\ge0\), \(X^{n+1}_t=\xi+\int_0^t\mu(s,X^n_s)\,ds+\int_0^t\sigma(s,X^n_s)\,dB_s\).

The drift integral is an ordinary time integral. The diffusion integral is an Ito integral: its integrand must be progressively measurable and satisfy the stated \(\mathcal{L}^2\) condition. The recursion always substitutes \(X^n\) in both coefficients to define \(X^{n+1}\). Thus it specifies an explicit next step rather than an equation for an unknown process.

Why Each Step Is Well Defined

Proof. The initial process \(X^0_t=\xi\) is continuous and adapted, and \(\sup_{t\le T}\mathbb{E}|X^0_t|^2=\mathbb{E}|\xi|^2<\infty\). Suppose \(X^n\) is continuous, adapted, and \(K_n=\sup_{t\le T}\mathbb{E}|X^n_t|^2<\infty\). A continuous adapted process is progressively measurable: on each finite interval, its left-endpoint step approximations are progressively measurable and converge pointwise to it. Composing with the Borel coefficient functions shows that \(\mu(t,X^n_t)\) and \(\sigma(t,X^n_t)\) are progressively measurable. The growth bound gives \(\mathbb{E}\int_0^T|\sigma(s,X^n_s)|^2\,ds\le CT(1+K_n)<\infty\), so the diffusion integrand belongs to \(\mathcal{L}^2\). The same bound gives time-integrability of the drift in squared mean. The Ito integral has a continuous adapted version, and the time integral is continuous and adapted, so the recursion defines a continuous adapted \(X^{n+1}\).

It remains to verify its second moment. The inequality \(|a+b+c|^2\le3(|a|^2+|b|^2+|c|^2)\), Cauchy–Schwarz for the time integral, and the Ito isometry give, for \(t\le T\), \(\mathbb{E}|X^{n+1}_t|^2\le3\mathbb{E}|\xi|^2+3t\int_0^t\mathbb{E}|\mu(s,X^n_s)|^2\,ds+3\int_0^t\mathbb{E}|\sigma(s,X^n_s)|^2\,ds\). Applying the growth bound and \(t\le T\), the right side is at most \(3\mathbb{E}|\xi|^2+3C(T+1)T(1+K_n)\), a finite bound independent of \(t\). Therefore \(\sup_{t\le T}\mathbb{E}|X^{n+1}_t|^2<\infty\). Induction proves the claim for every \(n\). \(\square\)

This argument uses only a finite second-moment bound at each individual step. It does not give a bound uniform in \(n\), nor does it show that the sequence is Cauchy. Those are separate questions from whether each iterate can be constructed.

Fixed Points and the Scope of the Construction

It is useful to separate the iteration rule from the SDE itself. Define the Picard map \(\Gamma\), whenever its two integrals are defined, by \(\Gamma(Y)_t=\xi+\int_0^t\mu(s,Y_s)\,ds+\int_0^t\sigma(s,Y_s)\,dB_s\). The recursion is \(X^{n+1}=\Gamma(X^n)\). A fixed point of this map is a process \(X\) satisfying \(\Gamma(X)=X\).

Proof. By definition, a solution of the Ito SDE is a continuous adapted process satisfying, almost surely for all \(t\in[0,T]\), \(X_t=\xi+\int_0^t\mu(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dB_s\), with the indicated integrals defined. The right side is exactly \(\Gamma(X)_t\), so a solution is a fixed point. Conversely, if \(X=\Gamma(X)\) up to indistinguishability and the integrals are defined, the displayed integral identity holds almost surely for all \(t\). That identity is the integral form of the SDE, and setting \(t=0\) gives \(X_0=\xi\). Thus \(X\) is a solution. \(\square\)

There is an important limitation: constructing \(X^0,X^1,\ldots\) and proving each is well defined does not establish a fixed point. One must separately show that the sequence converges in a suitable process norm and that its limit can be passed through both integrals. Until those steps are proved, the Picard sequence is a method of generating candidate processes, not yet an existence proof.

Common Mistake: Substituting the New Iterate

In the sine-and-cosine example, the correct first step is \(X^1_t=B_t\), obtained by using \(X^0=0\) in both coefficients. A wrong calculation substitutes \(X^1\) into the diffusion while trying to define \(X^1\), writing \(X^1_t=\int_0^t\sin(X^1_s)\,ds+\int_0^t\cos(X^1_s)\,dB_s\). That is a fixed-point equation for a solution, not the first Picard iterate. The correction is to use the already known process \(X^0\) on the right: \(X^1_t=\int_0^t\sin(X^0_s)\,ds+\int_0^t\cos(X^0_s)\,dB_s=B_t\). At every stage, both coefficients use the same preceding iterate.