Stochastic Calculus

Sum of Centered Gaussians

Consider the sum of two independent random variables XX and YY , where each variable is normally distributed with zero mean. Denote the variance of XX as σX2\sigma^{2}_{X} and the variance of YY as σY2\sigma^{2}_{Y} . XN(0,σX2);YN(0,σY2)X \sim \mathcal{N}(0, \sigma^{2}_{X}) \quad ; \quad Y \sim \mathcal{N}(0, \sigma^{2}_{Y}) Recall that the sum of XX and YY is distributed with variance σX2+σY2\sigma^{2}_{X} + \sigma^{2}_{Y} .

Z=X+Y    ZN(0,σX2+σY2)\begin{equation} \label{eqn-1}\tag{1} Z = X + Y \implies Z \sim \mathcal{N}(0, \sigma^{2}_{X} + \sigma^{2}_{Y}) \end{equation}

A Continuous Random Walk

Imagine a continuous-time random walk W(t)W(t) for which observed differences between W(ti)W(t_{i}) and W(tj)W(t_j) are always normally distributed with zero mean:

W(tj)W(ti)N(0,σij2)\begin{equation} \label{eqn-2}\tag{2} W(t_j) - W(t_i) \sim \mathcal{N}(0, \sigma^{2}_{ij}) \end{equation}

Assume also that the random walk is uncorrelated with itself when comparing non-overlapping time intervals.

Consistency between Eq. (2) and Eq. (1), i.e.,  ti<tj<tk,σij2+σjk2=σik2\forall ~ t_{i} < t_{j} < t_{k}, \quad \sigma^{2}_{ij} + \sigma^{2}_{jk} = \sigma^{2}_{ik} dictates that σij2tjti\sigma^{2}_{ij} \propto |t_j - t_i| Let us choose to define WW such that the constant of proportionality is 1, identifying this random walk as a Wiener process. W(tj)W(ti)N(0,tjti)W(t_{j}) - W(t_{i}) \sim \mathcal{N}\big(0, |t_{j} - t_{i}|\big) For positive, infinitesimal differences in time, we may write

dWN(0,dt)\begin{equation} \label{eqn-3}\tag{3} {\rm d}W \sim \mathcal{N}(0, {\rm d} t) \end{equation}

Quadratic Variation

If we consider infinitesimal differences in time and recall that, for any random variable ZZ Var[Z]E[Z2]E[Z]2{\rm Var}[Z] \equiv \mathbb{E}[Z^{2}] - \mathbb{E}[Z]^{2} For a Wiener process, this implies Var[dW]=E[(dW)2]{\rm Var}[{\rm d}W] = \mathbb{E}[({\rm d}W)^{2}] We may therefore rewrite Eq. (3) in terms of a quadratic variation E[(dW)2]=dt\mathbb{E}[({\rm d}W)^{2}] = {\rm d}t This observation has an important consequence: First-order approximations in of a process X(t)X(t) depending on W(t)W(t) must account for the quadratic variation in WW , where, in expectation,

dX(t)dXdtdt+dXdWdW(t)+12d2XdW2(dW(t))2=(dXdt+12d2XdW2)dt+dXdWdW(t)\begin{equation} \label{eqn-4}\tag{4} \begin{aligned} {\rm d}X(t) &\approx {\frac{{\rm d} X}{{\rm d} t}}{\rm d}t + {\frac{{\rm d} X}{{\rm d} W}}{\rm d}W(t) + \frac{1}{2} {\frac{{\rm d}^{2} X}{{\rm d} W^{2}}} ({\rm d}W(t))^{2} \\ &= \bigg( {\frac{{\rm d} X}{{\rm d} t}} + \frac{1}{2} {\frac{{\rm d}^{2} X}{{\rm d} W^{2}}} \bigg) {\rm d}t + {\frac{{\rm d} X}{{\rm d} W}}{\rm d}W(t) \end{aligned} \end{equation}

Drift-Diffusion Processes

An Itô drift-diffusion process may be represented in terms of differentials in tt and WW as dX(t)=μX(t)dt+σX(t)dW(t){\rm d}X(t) = \mu_{X}(t){\rm d}t + \sigma_{X}(t) {\rm d}W(t) where μX(t)\mu_{X}(t) and σX(t)\sigma_{X}(t) are deterministic (e.g., as functions of the process up to time tt ) and W(t)W(t) is a Wiener process as described above. Note that μX(t)\mu_{X}(t) and σ2(t)\sigma^{2}(t) are the mean and variance, respectively, in this example for dXdt\frac{{\rm d}X}{{\rm d} t} , not for XX !

Itô’s Lemma

Twice-differentiable functions applied to drift-diffusion stochastic processes also define drift-diffusion stochastic processes. For example, given a twice-differentiable function f(t,x)f(t, x) where the second argument is given by a stochastic process XX , dX(t)=μX(t)dt+σX(t)dW(t){\rm d}X(t) = \mu_{X}(t){\rm d}t + \sigma_{X}(t) {\rm d}W(t) it follows that df(X)=μf(t)dt+σf(t)dW(t){\rm d}f(X) = \mu_{f}(t){\rm d}t + \sigma_{f}(t) {\rm d}W(t) where, applying Eq. (4) and the observation that d2fdW2=d2fdX2(dXdW)2+dfdXd2XdW2\frac{{\rm d}^{2} f}{{\rm d} W^{2}} = \frac{{\rm d}^{2} f}{{\rm d} X^{2}} \bigg(\frac{{\rm d} X}{{\rm d} W}\bigg)^{2} + \frac{{\rm d} f}{{\rm d} X} \frac{{\rm d}^{2} X}{{\rm d} W^{2}} yields Itô’s Lemma: μf=ft+fxμX+122fx2σX2;σf=fxσX(t)\mu_{f} = {\frac{\partial f}{\partial t}} + {\frac{\partial f}{\partial x}} \mu_{X} + {\frac{1}{2} \frac{\partial^{2} f}{\partial x^{2}}} \sigma^2_{X} \quad ; \quad \sigma_{f} = {\frac{\partial f}{\partial x}} \sigma_{X}(t)

Geometric Brownian Motion

Consider a stochastic process XX for which the proportional growth dXX\frac{{\rm d}X}{X} is an affine transformation of a Wiener process, i.e., dX(t)=X(t)(μdt+σdW(t)){\rm d}X(t) = X(t) \bigg( \mu{\rm d}t + \sigma {\rm d}W(t) \bigg) for constant μ\mu and σ\sigma .

Recognizing that this is a transformation of the standard drift-diffusion process by the function f=logXf = \log X , Itô’s Lemma provides the standard solution.

Nonetheless, we may also directly equate this equation with Eq. (4): dX(t)=X(t)(μdt+σdW(t))=(dXdt+12d2XdW2)dt+dXdWdW(t){\rm d}X(t) = X(t) \bigg( \mu{\rm d}t + \sigma {\rm d}W(t) \bigg) = \bigg( {\frac{{\rm d} X}{{\rm d} t}} + \frac{1}{2} {\frac{{\rm d}^{2} X}{{\rm d} W^{2}}} \bigg) {\rm d}t + {\frac{{\rm d} X}{{\rm d} W}}{\rm d}W(t) from which we obtain the separable equations X(t)μ=dXdt+12d2XdW2;σX(t)=dXdWX(t) \mu = {\frac{{\rm d} X}{{\rm d} t}} + \frac{1}{2} {\frac{{\rm d}^{2} X}{{\rm d} W^{2}}} \quad ; \quad \sigma X(t) = {\frac{{\rm d} X}{{\rm d} W}} From the second equation, we note that XeσW and therefore X(t)μ=dXdt+12σ2X(t)X \propto e^{\sigma W} \quad \text{ and therefore } \quad X(t) \mu = {\frac{{\rm d} X}{{\rm d} t}} + \frac{1}{2} \sigma^{2} X(t) Recognizing an exponential as the solution class again, we arrive at the unique solution X(t)=X(0)e(μ12σ2)t+σW(t)X(t) = X(0) e^{\Big(\mu - \frac{1}{2}\sigma^{2}\Big) t + \sigma W(t)}