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For a discrete-time martingale relative to the filtration $({\mathcal{F}}_n)$, each $M_n$ is ${\mathcal{F}}_n$-measurable, $\mathbb{E}|M_n|<\infty$, and $\mathbb{E}[M_n\mid {\mathcal{F}}_{n-1}]=M_{n-1}$ for $n\ge 1$. The tower property then gives the corresponding identity conditional on any earlier time.

A bounded previsible process has ${\varphi }_i$ measurable at time $i-1$ and $|{\varphi }_i|\le C$. Its martingale transform $V$ is adapted and integrable, since each increment has absolute expectation at most $C(\mathbb{E}|M_i|+\mathbb{E}|M_{i-1}|)$. Moreover,Thus $V$ is a martingale, with $V_0=0$.

For the Itô integral, begin with a bounded adapted step integrand $H_s=H_i$ on $(t_i,t_{i+1}]$. Its integral is $I={\sum }_i{H}_i(W_{t_{i+1}}-W_{t_i})$. Each summand has zero conditional mean. For $i<j$, conditioning the cross product on ${\mathcal{F}}_{t_j}$ makes its expectation zero. The conditional second moment of a Brownian increment is $t_{i+1}-t_i$, henceApproximate the predictable integrand $g(s,W_s)$ in $L^2(dsd\mathbb{P})$ by such step processes. The same identity makes their integrals Cauchy in $L^2$ and defines the limit; both its first and second moments converge. This gives the requested mean-zero result and the Itô isometry:

For the Ornstein-Uhlenbeck process, the deterministic integrating factor has no quadratic covariation with $X$. The Itô formula therefore gives $d(e^{\kappa t}{X}_t)=\kappa \theta e^{\kappa t}dt+\sigma e^{\kappa t}d{W}_t$. Integrating and dividing by $e^{\kappa t}$ yieldsThe mean-zero integral and Itô isometry giveAt $\kappa =0$ these formulas are read by continuity: $X_t=x+\sigma W_t$, with mean $x$ and variance ${\sigma }^{2}t$.

For the CIR model, let $C_0=c_0\ge 0$ and use the usual positive parameters. Before a possible visit to zero, the derivatives of $\sqrt{c}$ are $1/(2\sqrt{c})$ and $-1/(4c^{3/2})$. Applying the Itô formula gives the Square root of a CIR diffusion:The PDF's claim under $2ab={\sigma }^2$ is false: that choice leaves the nonzero drift ${\sigma }^2/(8Z_t)-a{Z}_t/2$, which is not affine in $Z_t$. The actual interior cancellation condition is $4ab={\sigma }^2$. Even then the principal square root is nonnegative and cannot be an unrestricted Gaussian OU process.

To give the intended corrected construction, let a signed process solve $d{Y}_t=-a{Y}_{t}dt/2+(\sigma /2)d{B}_t$, with $Y_0=\sqrt{c_0}$. The Itô formula givesWriting ${\tilde{W}}_t={\int }_0^t\mathrm{sgn}(Y_s)d{B}_s$, with either sign chosen at zero, produces a Brownian motion; the diffusion term is $\sigma |Y_t|d{\tilde{W}}_t$. Thus $Y^2$ is a CIR model with $ab={\sigma }^2/4$, while $Z=|Y|$ is reflected at zero. The reflection contributes local time of a semimartingale, so the interior SDE alone is not a global unrestricted OU equation. Its second moment is $c_0{e}^{-at}+{\sigma }^2(1-e^{-at})/(4a)$, agreeing with the CIR mean for that corrected parameter choice.

The requested mean under the parameters actually printed is still available without the false OU assertion. Taking expectation in the original CIR equation gives $m^{\prime }=a(b-m)$; the stochastic integral has zero mean after the usual moment/localization justification. Consequently, for the printed choice as well as general admissible parameters,