Ito's Lemma

Let $\mathbf{X}_t$ be a stochastic process as defined previously and $g(t,x) \in C^2([0,\infty] \times \mathfrak{R})$ is a twice differentiable function, then $\mathbf{Y}_t=g(t,\mathbf{X}_t)$ is an Ito stochastic process with $$\mathrm{d} \mathbf{Y}_t =\frac{\partial g}{\partial t}(t,\mathbf{X}_t) \mathrm{d} t + \frac{\partial g}{\partial x}(t,\mathbf{X}_t) \mathrm{d} \mathbf{X}_t + \frac{1}{2} \frac{\partial^2 g}{\partial x^2}(t,\mathbf{X}_t)~(\mathrm{d} \mathbf{X}_t)^2.$$ Integrate both sides of the above equation to get $$g(T,\mathbf{X}_T)-g(0,\mathbf{X}_0) =\int_0^T \frac{\partial g}{\partial t}(t,\mathbf{X}_t)\,\mathrm{d} t +\int_0^T \frac{\partial g}{\partial x}(t,\mathbf{X}_t)\,\mathrm{d} \mathbf{X}_t +\frac{1}{2} \int_0^T \frac{\partial^2 g}{\partial x^2}(t,\mathbf{X}_t)\,(\mathrm{d} \mathbf{X}_t)^2.$$ Substitute $\mathrm{d} \mathbf{X}_t=b(t,\mathbf{X}_t) \mathrm{d} t + v(t,\mathbf{X}_t) \mathrm{d} \mathbf{B}_t$, to get (for simplicity, we denote $b:=b(t,\mathbf{X}_t)$ and $v:=v(t,\mathbf{X}_t)$ ) $$g(T,\mathbf{X}_T)-g(0,\mathbf{X}_0) =\int_0^T \frac{\partial g}{\partial t}(t,\mathbf{X}_t)\,\mathrm{d} t +\int_0^T \frac{\partial g}{\partial x}(t,\mathbf{X}_t)\,(b \mathrm{d} t + v \mathrm{d} B_t) +\frac{1}{2} \int_0^T \frac{\partial^2 g}{\partial x^2}(t,\mathbf{X}_t)\,\left(b \mathrm{d} t + v \mathrm{d} B_t\right)^2.$$ Heuristic Use the heuristic rules to get $$g(T,\mathbf{X}_T)-g(0,\mathbf{X}_0) =\int_0^T \frac{\partial g}{\partial t}(t,\mathbf{X}_t)\,\mathrm{d} t +\int_0^T \frac{\partial g}{\partial x}(t,\mathbf{X}_t)\,(b \mathrm{d} t + v \mathrm{d} B_t) +\frac{1}{2} \int_0^T \frac{\partial^2 g}{\partial x^2}(t,\mathbf{X}_t)\, v^2 \mathrm{d} t,$$ which can be simplified by separating the stochastic (diffusion) term from the deterministic terms to get $$g(T,\mathbf{X}_T)-g(0,\mathbf{X}_0) =\int_0^T \left( \frac{\partial g}{\partial t}+ b \frac{\partial g}{\partial x} +\frac{1}{2} v^2 \frac{\partial^2 g}{\partial x^2} \right)\, \mathrm{d} t +\int_0^T v \frac{\partial g}{\partial x} \mathrm{d} B_t,$$ where the stochastic integral $$\int_0^T v \frac{\partial g}{\partial x}\,\mathrm{d} B_t$$ can be presented in a probabilistic form using Ito isometry lemma


The $n$-dimensional form of an Ito stochastic process with $m$ Brownian motions and $n$ stochastic processes is $$\mathrm{d} \mathbf{X}_{k,t}=b_k(t,\mathbf{X}_t) \mathrm{d} t + \sum_{i=1}^m \sigma_{k,i}(t,\mathbf{X}_t) \mathrm{d} B_{i,t}$$ for $1 \leq k \leq n$, and $b_i$ and $\sigma_{i,k}$ satisfy the bounded conditions $\forall~ 1\leq i \leq m$ where $$\mathbf{X}_{ t} = \left( \begin{array}{c} \mathbf{X}_{1,t}\\ \mathbf{X}_{2,t}\\ \vdots \\ \mathbf{X}_{n,t} \end{array} \right),~ b(t,\mathbf{X}_t) = \left( \begin{array}{c} b_1(t,\mathbf{X}_t)\\ b_2(t,\mathbf{X}_t)\\ \vdots \\ b_n(t,\mathbf{X}_t) \end{array} \right),~ \mathrm{d} B_t = \left( \begin{array}{c} \mathrm{d} B_{1,t}\\ \mathrm{d} B_{2,t}\\ \vdots \\ \mathrm{d} B_{m,t} \end{array} \right),$$ $$\sigma(t,\mathbf{X}_t) = \left( \begin{array}{ccc} \sigma_{1,1}(t,\mathbf{X}_t) & \cdots & \sigma_{1,m}(t,\mathbf{X}_t) \\ \vdots & \ddots & \vdots\\ \sigma_{n,1}(t,\mathbf{X}_t) & \cdots & \sigma_{n,m}(t,\mathbf{X}_t) \\ \end{array} \right).~$$ Let $g(t,x) \in C^2([0,\infty] \times \mathfrak{R})$, then the Ito process for $\mathbf{Y}_t=g(t,\mathbf{X}_t)$ using the multidimensional Ito's lemma is $$\mathrm{d} \mathbf{Y}_t= \frac{\partial g}{\partial t}(t,\mathbf{X}_t) \mathrm{d} t + \sum_{i=1}^n\frac{\partial g}{\partial x_i}(t,\mathbf{X}_t) \mathrm{d} \mathbf{X}_{i,t} + \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \frac{\partial^2 g}{\partial x_i \partial x_j}(t,\mathbf{X}_t)~\mathrm{d} \mathbf{X}_{i,t} \mathrm{d} \mathbf{X}_{j,t},$$ Exercise

1. Find the Ito process for the Asian call option $c(t,S,A)$ that is a function of $t$, $S$ and $A$.
   
2. Show that $S_{t}=S_{0} e^{(\mu - \frac{\sigma^2}{2})\mathrm{d} t + \sigma \sqrt{\mathrm{d} t} Z}$ is the solution to $\mathrm{d}S_t=\mu S_t \mathrm{d} t+ \sigma S_t \mathrm{d}\mathbf{B}_t$, where $Z \sim N(0,1)$. Hint. Apply Ito's lemma to $f(S)=\mathrm{ln~}S$.

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References

1. Kemna A.G.Z and Vorst A.C.F, (1990), A Pricing Method for Options Based on Average Asset Values, Journal of Banking and Finance, 14 (1990) P 113-129.
   
2. Oksendal B., Stochastic Differential Equations: An Introduction With Applications, 5th ed. Springer, 2000.