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Multidimensional Itô Formula

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Theorem
StochasticDiffs

Let (Mt1)t0,,(Mtm)t0(M_{t}^{1})_{t\ge 0},\dots,(M_{t}^{m})_{t\ge 0} be continuous local martingales. Let (Vt1)t0,,(Vtn)t0(V_{t}^{1})_{t\ge 0},\dots,(V_{t}^{n})_{t\ge 0} be continuous adapted processes of Variation, let Zt=(Mt1,,Mtm,Vt1,,Vtn)t0Z_{t}=(M_{t}^{1},\dots,M_{t}^{m},V_{t}^{1},\dots,V_{t}^{n})\quad\forall t\ge 0Let f:Rm×RnRf:\mathbb{R}^{m}\times \mathbb{R}^{n}\to \mathbb{R} be C0C^{0} s.t. fxi,2fxixj,1i,m\frac{ \partial f }{ \partial x^{i} },\frac{ \partial ^{2}f }{ \partial x^{i}x^{j} },1\le i,\le m, fyk,1kn\frac{ \partial f }{ \partial y^{k} },1\le k\le n exist and are C0C^{0}. Then, a.s. t0\forall t\ge 0 f(Zt)f(Z0)=i=1m1[0,t]fxi(Z)dMi+k=1n[0,t]fyk(Zs)dVs+12i=1mj=1m[0,t]2fxixj(Zs)d[Mi,Mj]sf(Z_{t})-f(Z_{0})=\sum_{i=1}^{m}\int\limits \mathbb{1}_{[0,t]}\frac{ \partial f }{ \partial x^{i} } (Z) \, dM^{i}+\sum_{k=1}^{n}\int\limits _{[0,t]}\frac{ \partial f }{ \partial y^{k} } (Z_{s}) \, dV_{s}+ \frac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{m}\int\limits _{[0,t]}\frac{ \partial ^{2}f }{ \partial x^{i}x^{j} } (Z_{s}) \, d[M^{i},M^{j}]_{s}