67.6. THE GENERAL STOCHASTICALLY INTEGRABLE CASE 2297

uniformly and ∫ t

0X[0,τn]φ nds→

∫ t

0φds

uniformly. Hence also Xτnn (t)→ X (t) uniformly on [0,T ] whenever ω /∈ N. Of course

|Xτnn (t)|H has the advantage of being bounded by M.

From the above,

F (T,Xτnn (T )) = F (0,X0)+

∫ T

0Ft (t,Xτn

n (t))+FX (t,Xτnn (t))X[0,τn]φ n (t)dt

+∫ T

0FX (t,Xτn

n (t))ΦndW +12

∫ T

0

(FXX (t,Xτn

n (t))X[0,τn]Φn,X[0,τn]Φn)L2(Q1/2U,H) dt

Then it is obvious that one can pass to the limit in each of the non stochastic integrals inthe above. It is necessary to consider the other one.

From the above claim, X[0,τn]Φn → Φ ◦ J−1 in L2([0,T ]×Ω;L2

(JQ1/2U,H

))and

also, from the stopping times τn, FX (t,Xτnn (t)) is bounded and converges to FX (t,X (t)) .

Hence the dominated convergence theorem applies, and letting n→ ∞, the following isobtained for a.e. ω

F (T,X (T )) = F (0,X0)+∫ T

0Ft (t,X (t))+FX (t,X (t))φ (t)dt

+∫ T

0FX (t,X (t))ΦdW +

12

∫ T

0(FXX (t,X (t))Φ,Φ)L2(Q1/2U,H) dt (67.5.13)

This is the Ito formula in case that Φ ∈ L2([0,T ]×Ω;L2

(Q1/2U,H

))and |X | is bounded

above by M.It is easy to remove this assumption on |X | . Let X be given in 67.5.12. Let τn be the

stopping timeτn ≡ inf{t > 0 : |X |> n}

Then 67.5.13 holds for the stopped process Xτn and Φ and φ replaced with ΦX[0,τn] andφX[0,τn] respectively. Then let n→ ∞ in this expression, using the continuity of X and thefact that τn→ ∞ to to recover 67.5.13 without the restriction on |X |.

67.6 The General Stochastically Integrable CaseNow suppose that Φ is only progressively measurable and stochastically integrable

P([∫ T

0∥Φ∥2

L2(Q1/2U,H) dt < ∞

])= 1.

Also φ is only progressively measurable and Bochner integrable in t. Define a stoppingtime

τ (ω) = inf{

t ≥ 0 : |X (t,ω)|H +∫ t

0||Φ||2 ds+

∫ t

0|φ |ds >C

}