Extension of calculus to [[Stochastic Processes|stochastic processes]], usually [[Wiener Process or Brownian Motion|Brownian Motion]]. While in ordinary calculus, a [[Derivative, Gradient, Jacobian and Hessian|differentiable]] [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|function]] $X(t)$ is connected to its derivative via$\mathrm{d}X(t)=f(t)\mathrm{d}t,$Itô calculus allows the function to be driven by e.g. Brownian motion, yielding an [[Itô Diffusion|Itô process]]. This is necessary to obtain a theoretical framework for studying [[Stochastic Processes|stochastic processes]] and [[SDE - Stochastic Differential Equation|SDEs]], as e.g. Brownian motion is nowhere differentiable, unbounded and continuity has to be defined as in [[Pathwise Continuity]]. >[!brainwaves] Brownian Motion >While the Itô formulation allows a wide range of stochasticity, the Brownian motion case is by far the most useful in applications. --- #### Itô Integrals For a [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|function]] $f(t)$, the classic Riemann integral is $\int_0^Tf(t)\mathrm{d}t = \lim_{n \rightarrow \infty} \sum\limits_{i=0}^{n-1}f(t^*_i)[t_{i+1}-t_i],\quad \textcolor{red}{t_i^*\in[t_i,t_{i+1}]},$which we can compute, because the [[Total Variation|variation]] of $[t_{i+1}-t_i]$ is bounded. However, the same is not true for such an increment based on Brownian motion $[W_{i+1}-W_i]$, we cannot naively extend the above (sum oscillates too wildly to take a limit). Additionally, when evaluating the integrand inside the integral, we are essentially peeking into the future, which is unproblematic for deterministic behavior. However, in the stochastic case this violates adaptation of the filtration (violates [[Martingale|martingale]], we access future information). To work around this, we can define the integral via left point evaluation as$\int_0^TH_s \, \mathrm{d}W_s = \lim_{n \rightarrow \infty} \sum\limits_{i=0}^{n-1}H_{t_i}[W_{t_{i+1}}-W_{t_i}],$which makes the evaluation of the integrand and the variation [[Statistical Independence|statistically independent]] (and a [[Martingale|martingale]] increment). This enables us to use the [[Itô Isommetry]] to obtain the limit, where the **[[Convergence of Random Variables|limit now has to be understood in a mean square sense]]** ($L^{2}$). >[!info] Definition - Itô Integral >Based on a [[Filtrations and Filtered Probability Space|filtered]] [[Probability Space|probability space]] $(\Omega, \Sigma, \mathbb{F}, \mu)$, the Itô integral constructs a [[Stochastic Processes|stochastic process]] $Y(t)$ based on an integrand $H_s$ and [[Wiener Process or Brownian Motion|Brownian motion]] $W(t)$ via$Y(t)=\int_0^t H_s \,\mathrm{d}W_s,$where we require ... >[!warning] Assumptions >- $H_s$ needs to be [[Adapted Process|adapted]] to $\mathbb{F}$ >- $H_s$ needs to be square-intergrable $(\in L^{2})$ $\mathbb{E}\Big[\int_{0}^{T}H^{2}(t)\mathrm{d}t\Big]< \infty,$which can be computed in closed form when the second moment function is known. >[!brainwaves] Intuition >- With each random noise trajectory for given $t$ boundaries, we obtain one realization of $Y$ at $t$. >- Time (generally) flows forward for [[Stochastic Processes|stochastic processes]], because each value is dependent on the previous realization, while for deterministic [[Maps and Induced Structures - Functions, Pushforwards and Pullbacks|functions]] everything is known >- The integrand of an Itô integral must be adapted to the filtration, which encodes the information about the underlying space at time $t$ via the [[Sigma-Algebra|events of the sigma algebra]] --- Interesting - Integration by Parts - Chain Rule or Itos Lemma --- #### Beyond Brownian Motion - easy if process is [[Semimartingale|semimartingale]] - ...