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 Incoming links: Initial-value integrability in normalized semimartingale decompositions

The multidimensional Itô formula for continuous semimartingales states that a

C^{2}

f (M_{t}, A_{t}) = f (M_{0}, A_{0}) + \int_{0}^{t} f_{x} (M_{s}, A_{s}) d M_{s} + \int_{0}^{t} f_{y} (M_{s}, A_{s}) d A_{s} + \frac{1}{2} \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

M_{t}^{f} = f (M_{0}, A_{0}) + \int_{0}^{t} f_{x} (M_{s}, A_{s}) d M_{s}, A_{t}^{f} = \int_{0}^{t} f_{y} (M_{s}, A_{s}) d A_{s} + \frac{1}{2} \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.

An initial-value qualification is necessary under that definition. The printed assumptions do not themselves guarantee

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

E e^{G^{2}} = \infty

. Any decomposition with

A_{0}^{f} = 0

would force the nonintegrable value

M_{0}^{f} = e^{G^{2}}

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.