= Solution
The multidimensional <Itô formula> for continuous <semimartingales> states that a $C^2$ transformation has first-order stochastic and finite-variation integrals plus one-half its Hessian contracted with the <quadratic covariations>. A continuous <finite-variation process> has zero <quadratic variation> and zero quadratic covariation with a continuous local martingale. Thus in the present case
$$
f(M_t,A_t)=f(M_0,A_0)+\int_0^t f_x(M_s,A_s)\,dM_s+\int_0^t f_y(M_s,A_s)\,dA_s+\frac12\int_0^t f_{xx}(M_s,A_s)\,d[M]_s.
$$
The first integrand is continuous adapted, hence <predictable>, and locally bounded by stopping the two coordinate paths on exiting compact sets. The standard construction of the <Itô integral> then makes this integral a zero-starting <continuous local martingale>. The remaining integrals are pathwise <Lebesgue-Stieltjes integrals>. On each compact time interval their integrands are bounded along the continuous paths, so their total variations are bounded by a constant times $V_t(A)+[M]_t$, which is finite. Therefore the normalized terms are
$$
\boxed{M_t^f=f(M_0,A_0)+\int_0^t f_x(M_s,A_s)\,dM_s,\qquad A_t^f=\int_0^t f_y(M_s,A_s)\,dA_s+\frac12\int_0^t f_{xx}(M_s,A_s)\,d[M]_s.}
$$
The process $A^f$ is continuous adapted of finite variation and $A_0^f=0$. When $f(M_0,A_0)$ is integrable, $M^f$ is a local martingale in the definition of part (a). Uniqueness follows from part (a): the difference of two normalized decompositions is both a continuous local martingale and a continuous finite-variation process starting at zero, hence vanishes.
\b[An initial-value qualification is necessary under that definition.] The printed assumptions do not themselves guarantee $\mathbb E|f(M_0,A_0)|<\infty$. Take $M=0$, $A_t=G$ with $G$ a standard <normal random variable> known at time zero, and $f(x,a)=e^{a^2}$. All path assumptions hold, but $\mathbb E e^{G^2}=\infty$. Any decomposition with $A_0^f=0$ would force the nonintegrable value $M_0^f=e^{G^2}$, contradicting the local-martingale definition. Without an integrability hypothesis, the always-valid <continuous semimartingale decomposition> is $X=X_0+N+A^f$, with $N$ the zero-starting integral above. Alternatively one may use the convention that a local martingale permits an arbitrary finite initial value and localizes its centered increments. With deterministic initial data, as in part (c), there is no issue. This is <initial-value integrability in normalized semimartingale decompositions>.
Back to article page
