# Lecture 24: Stochastic Calculus Source: https://www.youtube.com/watch?v=5cruqmIF6l0 Recap page: https://rapidrecap.app/video/5cruqmIF6l0 Generated: 2025-12-03T16:12:37.286+00:00 --- ## Quick Overview Stochastic calculus extends ordinary calculus to model asset dynamics using stochastic processes, particularly Brownian motion, introducing the necessity of the Ito integral defined through nonanticipatory variables and foundational properties like the Ito isometry for convergence. **Key Points:** - Stochastic calculus models asset dynamics as integrals with respect to Brownian motion, accommodating randomness in increments. - The increment for Brownian motion with drift (dX = mu dt + sigma dBt) follows a normal distribution with mean mu delta t and variance sigma squared delta t. - The infinitesimal squared value of the Brownian motion increment has an expectation of order delta t, specifically equal to "sigma squared delta t plus little o of delta t". - The Ito integral is formally defined by considering a probability process specified by a sample space (omega), a sigma field, and a probability measure, using the filtration Ft which represents information known up to time t. - The Ito integral of B dBs is shown to be $\frac{1}{2} B_T^2 - \frac{1}{2} T$, which differs from the real calculus result of $\frac{1}{2} x^2$ due to the term dependent on the variation of the process. - Ito's formula for a function $f$ of $B_t$ with a continuous second derivative states $df(B_t) = f'(B_t)dB_t + \frac{1}{2} f''(B_t)dt$, contrasting with the standard Taylor expansion by including the $\frac{1}{2} f'' dt$ term. - The ruin problem for Brownian motion with drift uses martingale theory, defining a martingale $M_t = h(X_t)$ where $h$ must satisfy the PDE $\frac{\partial h}{\partial t} = -\frac{1}{2} \sigma^2 \frac{\partial^2 h}{\partial x^2}$ to find the probability of stopping at A before -B. **Context:** This lecture, delivered by Peter Kempthorne, introduces stochastic calculus as an essential extension of ordinary calculus for quantitative finance, specifically focusing on modeling asset values using stochastic processes driven by Brownian motion. The discussion moves from reviewing Brownian motion with drift to formally defining the Ito integral, establishing its properties, and deriving Ito's formula for single and multiple variables, culminating in an application to the ruin problem via martingale theory. ## Detailed Analysis Stochastic calculus extends real calculus to systems dependent on stochastic processes, like Brownian motion, which characterizes asset dynamics; the increment $\Delta X$ for Brownian motion with drift is normally distributed with mean $\mu \Delta t$ and variance $\sigma^2 \Delta t$, and its squared increment approaches $\sigma^2 \Delta t$ as $\Delta t \to 0$. The formal definition of the Ito integral requires understanding probability foundations, including the sample space $\Omega$, the sigma field, the probability measure, and the filtration $F_t$, which captures information known up to time $t$. The Ito integral is defined through limits of sums involving nonanticipatory variables ($a_i$ measurable in $F_{t_i}$), leading to the Ito isometry, which relates the variance of the integral to the $L^2$ norm of the integrand, ensuring convergence. A key result is that integrating $B_s dB_s$ yields $\frac{1}{2} B_T^2 - \frac{1}{2} T$, demonstrating the non-equivalence to standard calculus. Ito's formula generalizes this, stating $df(B_t) = f'(B_t)dB_t + \frac{1}{2} f''(B_t)dt$; applying this in reverse allows integration, for example, $\int f(B_s)dB_s = F(B_T) - \frac{1}{2} \int f'(B_s)ds$, where $F' = f$. For functions of two variables, $f(t, B_t)$, Ito's formula connects martingale properties to partial differential equations (PDEs): if the integrand of the time derivative term is zero ($\frac{\partial f}{\partial t} + \frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2} = 0$), the resulting process is a martingale, as demonstrated with the exponential function leading to a scaled lognormal variable whose expectation is 1. Finally, the ruin problem probability is solved by finding a function $h(X_t)$ satisfying the corresponding PDE, yielding an explicit solution based on martingale theory. ### Brownian Motion Fundamentals - Brownian motion with drift $X$ is defined by $dX = \mu dt + \sigma dB_t$ - Key properties include independent increments and variance proportional to interval length - The conditional distribution of $X_{t+\Delta t}$ given $X_t$ is normal. ### Definition of Ito Integral - Ito integrals require defining a probability process via a triple (sample space, sigma field, probability model) and the filtration $F_t$ (information up to time t) - Integrands must be nonanticipatory variables measurable in $F_{t_i}$ - The expectation of $\int_0^T dBs$ is 0 and variance is $T \sigma^2$. ### Ito Isometry and Convergence - The variance of an Ito integral is characterized as the squared $L^2$ norm of the integrand $f$ - This norm allows characterizing limits and proving the existence of the limiting Ito integral as time increments approach zero - Convergence of the discretized sum of increments of Brownian motion squared converges to $T$. ### Ito's Formula for Single Variable - For $f$ with continuous second derivative, $df(B_t) = f'(B_t)dB_t + \frac{1}{2} f''(B_t)dt$ - This allows reversing integration: $\int_0^T f(B_s)dB_s = F(B_T) - \frac{1}{2} \int_0^T f'(B_s)ds$ where $F'=f$ - $\int B_s dB_s = \frac{1}{2} B_T^2 - \frac{1}{2} T$. ### Ito's Formula for Multiple Variables - For $f(t, x)$ that is $C^{1,2}$, the change is $df(t, B_t) = \frac{\partial f}{\partial t}dt + \frac{\partial f}{\partial x}dB_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2}dt$ - If the function is a martingale, $\frac{\partial f}{\partial t} = -\frac{1}{2} \sigma^2 \frac{\partial^2 f}{\partial x^2}$ - $e^{\sigma B_t - \frac{1}{2} \sigma^2 t}$ is explicitly shown to be a martingale with expected value 1. ### Application to Ruin Problem - The probability of stopping at $A$ before $-B$ for $X_t$ (Brownian motion with drift) is found by defining a martingale $M_t = h(X_t)$ - $h$ must satisfy the PDE $\frac{\partial h}{\partial t} = -\frac{1}{2} \sigma^2 \frac{\partial^2 h}{\partial x^2}$ - Solving this differential equation for $h$ provides the explicit solution for the stopping probability.