4.3. Stochastic calculus

4.3.1. Ito’s lemma

It allows to find the differential of a time-dependent function of a stochastic process. Let \(f(t,x)\) be a twice-differentiable scalar function. Its Taylor series gives

\[ \Delta f = \dfrac{\partial f}{\partial t} \Delta t + \dfrac{\partial f}{\partial x} \Delta x + \dfrac{1}{2}\dfrac{\partial^2 f}{\partial t^2} \Delta t^2 + \dfrac{\partial^2 f}{\partial t \partial x} \Delta t \, \Delta x + \dfrac{1}{2}\dfrac{\partial^2 f}{\partial x^2} \Delta x^2 \]

If the argument \(x\) of the function \(f\) is chosen to be a random process \(X_t\) satisfying Ito drift-diffusion process,

\[d X_t = \mu_t \, dt + \sigma_t \, dW_t \ ,\]

the differential of function \(f(t,X_t)\) results from the limit of Taylor series

\[\begin{split}\begin{aligned} df & = \lim_{dt \rightarrow 0, dW_t \rightarrow 0} \left\{ \Delta f \right\} = \\ & = \lim_{dt \rightarrow 0, dW_t \rightarrow 0} \left\{ \partial_t f dt + \partial_x f d X_t + \dfrac{1}{2} \left[ \partial_{tt} f \, dt^2 + 2 \partial_{xt} \, dt dX_t + \partial_{xx} f dX_t^2 \right] \right\} = \\ & = \lim_{dt \rightarrow 0, dW_t \rightarrow 0} \left\{ \partial_t f dt + \partial_x f \left( \mu_t dt + \sigma_t dW_t \right) + \dfrac{1}{2} \left[ \partial_{tt} f \, dt^2 + 2 \partial_{xt} \, dt \left( \mu_t dt + \sigma_t dW_t \right) + \partial_{xx} f \left( \mu_t dt + \sigma_t dW_t \right)^2 \right] \right\} = \\ \end{aligned}\end{split}\]

For \(dt \rightarrow 0\), \(\left(d W_t\right)^2 = O(dt)\); keeping only terms of order lower than or equal to \(O(dt)\), the differential becomes,

\[df = \left( \partial_t f + \mu_t \partial_x f \right) dt + \sigma_t \partial_x f \, d W_t + \dfrac{\sigma_t^2}{2} \partial_{xx} f \, dW_t^2 \ .\]

Replacing \(dW_t^2\) with \(d t\) todo why?, and recalling the SDE of the Ito drift-diffusion process,

\[\begin{split}\begin{aligned} df & = \left( \partial_t f + \mu_t \partial_x f + \dfrac{\sigma_t^2}{2} \partial_{xx} f \right) dt + \sigma_t \partial_x f \, d W_t = \\ & = \left( \partial_t f + \dfrac{\sigma_t^2}{2} \partial_{xx} f \right) dt + \partial_x f \left( \mu \, dt + \sigma_t \, d W_t \right) = \\ & = \left( \partial_t f + \dfrac{\sigma_t^2}{2} \partial_{xx} f \right) dt + \partial_x f \, dX_t \ . \end{aligned}\end{split}\]

4.3.2. Ito’s calculus

Integegration w.r.t. Browinan motion produces a random variable that can be defined as

\[\int_{0}^{t} F \, dW := \lim_{n \rightarrow +\infty} \sum_{[t_{i-1},t_i] \in \pi_n} F_{t_{i-1}} \left( W_{t_i} - W_{t_{i-1}} \right) \ ,\]

being \(\pi_n\) a partition of interval \([0,t]\), and \(H\) a random proces todo with some characteristics…

Example 4.1 (Integral of a Brownian motion w.r.t. itself)

\[Y(t) = \int_{s=0}^t W_s \, dW_s = \dfrac{1}{2} W_t^2 - \dfrac{t}{2} \ .\]

The expected value for each \(t\) of the random process \(Y_t\) is zero for all \(t\), \(\mathbb{E}\left[ W_t^2 \right]= 0\), as the expected value of \(W_t^2\) is the variance of \(W_t\), and thus \(t\) by definition of the Wiener process.

Evaluation of the integral

Let \(f(t,x) = x^2\). Let’s find the differential \(df\) evaluated for \(x = W_t\) using Ito’s lemma, retaining only terms with order up to \(O(dt)\). Since \(\partial_t f \equiv 0\),

\[\begin{aligned} df & = \partial_x f|_{x=W_t} dW_t + \dfrac{1}{2} \partial_{xx} f|_{x=W_t} dW_t^2 \end{aligned}\]

and thus, replacing \(dW_t^2 = dt\),

\[\begin{aligned} d W_t^2 & = 2 W_t \, dW_t + dt \ . \end{aligned}\]

or

\[\begin{aligned} W_t \, d W_t & = d \left( \dfrac{W_t^2}{2} \right) - \dfrac{dt}{2} \ . \end{aligned}\]

Thus (todo add details if needed. A bit too much freedom in using differentials over stochastic processes here),

\[\begin{split}\begin{aligned} Y(t) & = \int_{s=0}^t W_s \, dW_s \, ds = \\ & = \int_{s=0}^t \left( \dfrac{W_s}{2} \right) \, ds - \int_{s=0}^t \dfrac{1}{2} \, ds = \\ & = \dfrac{1}{2} \left( W_t^2 - W_0^2 \right) - \dfrac{t}{2} \ . \end{aligned}\end{split}\]

4.3.3. Ito processes

4.3.3.1. Ito drift-diffusion process

An Ito drift-diffusion process is a stochastic process satisfying the stochastic differential equation (SDE)

(4.4)\[dX_t = \mu_t \, dt + \sigma_t \, dW_t \ ,\]

with \(W_t\) a Wiener process. If \(\mu_t = \mu\), \(\sigma_t = \sigma\) are constant a closed-form solution can be found using Ito’s lemma, for \(f(t,x) = x\), or by direct (stochastic) integration of the SDE (4.4), as

\[\begin{split}\begin{aligned} \int_{s=0}^t dX_s & = \int_{s=0}^t \mu \, ds + \int_{s=0}^t \sigma \, dW_s \\ \end{aligned}\end{split}\]

\[\begin{aligned} X_t - X_0 & = \mu t + \sigma \left( W_t - W_0 \right) \ , \end{aligned}\]

so that \(X_t - X_0 \sim \mathscr{N}\left( \mu t, \sigma^2 t \right)\).

Scaling of a Wiener process

Term \(\sigma W_t\) represents a scaling of a Wiener process \(W_t \sim \mathscr{N}(0, t)\) with zero expected value and variance \(t\). Multiplication by factor \(\sigma\) results in a multiplication of the expected value by \(\sigma\) and variance by \(\sigma^2\).

4.3.3.2. Geometric Brownian Motion, GBM

A geometric Brownian motion is a stochastic process satisfying the SDE

\[d X_t = \mu X_t \, dt + \sigma X_t \, dW_t \ .\]

Example 4.2 (GBM in Finance)

GBM can be used as a model of the price of an asset with constant expected return and variance of returns with normal distribution. GBM is the time-continuous counterpart of a time-discrete process, with the 1-period return \(\frac{\Delta X_n}{X_n} = \frac{X_{n+1} - X_n}{X_n}\) with expected value \(\mu\) and variance \(\sigma^2\),

\[\frac{\Delta X_n}{X_n} = \mathscr{N}\left( \mu, \sigma^2 \right) \ .\]

Let \(f(x) = \ln x\) be evaluated for \(x = X_t\). Ito’s lemma, with \(\partial_t f \equiv 0\), provides the expression of the differential

\[\begin{split}\begin{aligned} d f & = \partial_x f|_{X_t} d X_t + \dfrac{1}{2} \partial_{xx} f|_{X_t} d X_t^2 = \\ & = \partial_x f|_{X_t} \left( \mu X_t \, dt + \sigma X_t \, dW_t \right) + \dfrac{1}{2} \partial_{xx} f|_{X_t} \left( \mu X_t \, dt + \sigma X_t \, dW_t \right)^2 = \\ & = \dfrac{1}{X_t} \left( \mu X_t \, dt + \sigma X_t \, dW_t \right) - \dfrac{1}{2} \dfrac{1}{X_t^2} \sigma^2 X^2_t \, dW^2_t = \\ d \left( \ln X_t \right) & = \left( \mu - \dfrac{\sigma^2}{2} \right) \, dt + \sigma \, dW_t \ , \end{aligned}\end{split}\]

whose solution after integration reads

\[\ln X_t = \ln X_0 + \left( \mu - \dfrac{\sigma^2}{2} \right) \, t + \sigma \, W_t \ ,\]

or

\[X_t = X_0 \, e^{\left( \mu - \frac{\sigma^2}{2} \right) \, t + \sigma W_t} \ .\]

4.3.3.3. Geometric Brownian Motion with drift

A geometric Brownian motion is a stochastic process satisfying the SDE

\[d X_t = \mu X_t \, dt - C \, dt + \sigma X_t \, dW_t \ .\]

Example 4.3 (GBM with constant withdrawal in finance)

GBM with drift can be used in finance as a model to represent DCA strategy and pension withdrawal, and to show and discuss sequence risk.

The solution reads

\[X_t = X_0 e^{\left( \mu - \frac{\sigma^2}{2} \right) (t-t_0) + \sigma ( W_t - W_0)} + \int_{s=0}^{t} C e^{\left( \mu - \frac{\sigma^2}{2} \right) (t-s) + \sigma ( W_t - W_s)} \, ds \ .\]

Integration factor method for linear SDEs

Integration factor method for linear SDEs

\[d X_t = a \, dt + b \, dW_t \ ,\]

with \(a(X_t, t, W_t)\), \(b(X_t, t, W_t)\)

aims at finding an exponential factor \(e^{\alpha t + \beta W_t}\) that allows to get an integrable expression of the differential

\[d \left( e^{\alpha t + \beta W_t} X_t \right) = d \, f\left( t, W_t, X_t \right) \ .\]

Taylor expansion of this expression up to terms of order \(dt \sim dW_t^2\) reads

\[\begin{split}\begin{aligned} d \, f(t,W_t,X_t) & = \partial_t f \, dt + \partial_w f \, d W_t + \partial_x f \, \underbrace{dX_t}_{a dt + b dW_t} + \\ & + \dfrac{1}{2} \left( \underbrace{\partial_{tt} f \, dt^2}_{o(dt)} + \partial_{ww} f \, \underbrace{dW_t^2}_{dt} + \partial_{xx} f\, \underbrace{dX_t^2}_{b^2 \, dW_t^2 = b^2 \, dt} + \underbrace{2 \partial_{tw} f \, dt \, dW_t + 2 \partial_{tx} f \, dt \, dX_t}_{o(dt)} + 2 \partial_{xw} f \, \underbrace{dW_t \, dX_t}_{b dW_t^2 = b dt} \right) = \\ & = dt \left[ \partial_t f + a \partial_x f + \dfrac{1}{2} \partial_{ww} f + \dfrac{1}{2} b^2 \partial_{xx} f + 2 b \partial_{xw} f \right] + d W_t \left[ \partial_w f + b \partial_x f \right] \\ \end{aligned}\end{split}\]

Proof (with integration factor method, for linear SDEs)

GBM motion with drift and constant coefficients is governed by SDE

\[d X_t = \mu X_t \, dt - C \, dt + \sigma X_t \, dW_t \ .\]

Referring to the general expression of SDEs, coefficients \(a\), \(b\) of the GBM with drift read

\[\begin{split}\begin{aligned} a & = \mu \, X_t + C \\ b & = \sigma \, X_t \ . \end{aligned}\end{split}\]

Partial derivatives of function \(f = e^{\alpha t + \beta w} \, x\) appearing in the solution of SDEs through integration factor method read

\[\begin{split}\begin{aligned} \partial_t f & = e^{\alpha t + \beta w} \, x \, \alpha \\ \partial_w f & = e^{\alpha t + \beta w} \, x \, \beta \\ \partial_x f & = e^{\alpha t + \beta w} \\ \partial_{xx} f & = 0 \\ \partial_{wx} f & = e^{\alpha t + \beta w} \, \beta \\ \partial_{ww} f & = e^{\alpha t + \beta w} \, x \, \beta^2 \\ \end{aligned}\end{split}\]

It’s now possible to simplify the RHS of the the expression of the differential \(df\). Namely, it’s possible to choose values of \(\alpha\), \(\beta\) in order to get simpler expressions of the factors of the differentials \(dt\) and \(d W_t\)

\[\begin{split}\begin{aligned} dW_t: \quad \partial_{W_t} f & = \left.\left( \partial_w f + b \partial_x f \right)\right|_{t,W_t,X_t} = \\ & = e^{\alpha t + \beta W_t} \, \left( X_t \beta + b \right) = \\ & = e^{\alpha t + \beta W_t} \, X_t \left( \beta + \sigma \right) \\ dt : \quad \partial_{ t} f \quad & = \left.\left[ \partial_t f + a \partial_x f + \dfrac{1}{2} \partial_{ww} f + \dfrac{1}{2} b^2 \partial_{xx} f + b \partial_{xw} f \right]\right|_{t,W_t,X_t} \\ & = e^{\alpha t + \beta W_t} \left[ X_t \alpha + a + \dfrac{1}{2} X_t \beta^2 + 0 + b \beta \right] = \\ & = e^{\alpha t + \beta W_t} \left[ X_t \alpha + \mu X_t + C + \dfrac{1}{2} X_t \beta^2 + \sigma X_t \beta \right] = \\ & = e^{\alpha t + \beta W_t} \left[ X_t \left( \alpha + \mu + \dfrac{1}{2} \beta^2 + \sigma \beta \right) + C \right] \ . \end{aligned}\end{split}\]

Setting

\[\begin{split}\begin{aligned} \beta & = - \sigma \\ \alpha & = - \mu - \dfrac{1}{2} \beta^2 - \sigma \beta = - \mu + \dfrac{\sigma^2}{2} \ , \end{aligned}\end{split}\]

the differential \(d f\) becomes

\[d\left( e^{\left( -\mu + \frac{\sigma^2}{2} \right) t - \sigma W_t} \, X_t \right) = C e^{\left( -\mu + \frac{\sigma^2}{2} \right) t - \sigma W_t}\]

and integration gives

\[X_t = X_0 e^{\left( \mu - \frac{\sigma^2}{2} \right) (t-t_0) + \sigma ( W_t - W_0)} + \int_{s=0}^{t} C e^{\left( \mu - \frac{\sigma^2}{2} \right) (t-s) + \sigma ( W_t - W_s)} \, ds \ .\]