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So let S = (St )0≤t≤T be an Rd -valued continuous semimartingale with canonical decomposition S = S0 +M +A. The processes M = (Mt )0≤t≤T and A = (At )0≤t≤T are both Rd -valued, continuous and null at 0. Moreover, M is a local P -martingale and A is adapted and of finite variation. The bracket process M of M is the adapted, continuous, d × d-matrix-valued process with components M ij = M i , M j for i, j = 1, . . , d; it exists because M is continuous, hence locally square-integrable. e. d Ait = t d j λu d M j =1 0 ij u = j =1 0 t j λu d M i , M j u for i = 1, .

2. A nonnegative process X is a local Q-martingale if and only if Y Q X is a local P-martingale. Proof Start by assuming that X is a local Q-martingale. Since Q, Xτ n ∧τ = X0 for all n ∈ N and all stopping times τ , where (τn )n∈N is a localizing sequence, (τn )n∈N can be assumed to also localize Q. Then, since Xτ n ∧τ ∈ Fτ n for all n ∈ N and all stopping times τ , and since Qn := Q|Fτ n is countably additive with dQn /( dP|Fτ n ) = YτQn , it follows that Y0Q X0 = X0 = Q, Xτ n ∧τ = E[YτQn Xτ n ∧τ ] = E[E[YτQn | Fτ n ∧τ ]Xτ n ∧τ ] = E[YτQn ∧τ Xτ n ∧τ ] for all n ∈ N and all stopping times τ .

D}. · 3. There exists a d-dimensional, predictable process ρ such that A = 0 (ct ρt ) dGt , T as well as 0 ρt , ct ρt dGt < ∞. 32 C. Kardaras Proof We prove (1) ⇒ (3), (3) ⇒ (2), and (2) ⇒ (1) below. (1) ⇒ (3). We shall show that if statement (3) of Theorem 4 is not valid, then {XT | X ∈ X (1)} is not bounded in probability. In view of Proposition 1, (1) ⇒ (3) will be established. Suppose that one cannot find a predictable d-dimensional process ρ such that · A = 0 (ct ρt ) dGt . In that case, linear algebra combined with a simple measurable selection argument gives the existence of some bounded predictable process T · θ such that (a) 0 θt dGt = 0, (b) 0 θt , dAt is a nondecreasing process, and (c) T P[ 0 θt , dAt > 0] > 0.