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 - Find the Ito process for the Asian call option \(c(t,S,A)\) that is a function of \(t\), \(S\) and \(A\).
- 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 \(f(S)=\mathrm{ln~}S\).
References
- 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.
- Oksendal B.,
Stochastic Differential Equations: An Introduction With Applications
, 5th ed. Springer, 2000.
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