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Itô's Formula

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Theorem
StochasticDiffs

Let (Mt)t0(M_{t})_{t\ge 0} be a continuous local martingale, let (Vt)t0(V_{t})_{t\ge 0} be a continuous process of Variation (i.e. locally bounded). Let fC0(R2;R)f\in C^{0}(\mathbb{R}^{2}; \mathbb{R}) with fx,fy,2fx2\frac{ \partial f }{ \partial x },\frac{ \partial f }{ \partial y },\frac{ \partial ^{2}f }{ \partial x^{2} } C0C^{0} on R2\mathbb{R}^{2}. Then we have a.s. t0\forall t\ge 0: f(Mt,Vt)=f(M0,V0)+1[0,t]fx(M,V)dM+[0,t]fy(Ms,Vs)dVs+12[0,t]2fx2(Ms,Vs)d[M]sf(M_{t},V_{t})=f(M_{0},V_{0})+\int\limits \mathbb{1}_{[0,t]}\frac{ \partial f }{ \partial x } (M,V) \, dM +\int\limits _{[0,t]}\frac{ \partial f }{ \partial y } (M_{s},V_{s}) \, dV_{s} + \frac{1}{2}\int\limits _{[0,t]} \frac{ \partial ^{2}f }{ \partial x^{2} } (M_{s},V_{s}) \, d[M]_{s}