# Change of Measure and Girsanov theory

[1] J. Michael Steele - Stochastic Calculus and Financial Applications
[2] [Girsanov theorem for multifractional Brownian processes (tandfonline.com)](https://www.tandfonline.com/doi/full/10.1080/17442508.2022.2027948)
## Introduction

Can a stochastic process $X_t$ with drift $\mu$ be viewed as another stochastic process $Y_t$ without drift? This is related to the fact that almost any question about Brownian motion with drift may be paraphresed as an equivalent but slightly modified question about standard Brownian motion $B_t$.

In this note we will be focussing on this sort of inquiry condensed into a few theorems which is known as **Girsanov's theory**. This theory is crucial in **Mathematical Finance** which provides the foundation for **risk-neutral** pricing of financial instruments like **Options** and **Derivatives**. For example, it is applied to the celebrated **Black-Scholes** model to find a probability measure which transfoms the *current value* of the stock price $S_t$ into a very interesting mathematical object called a **martigale**. Almost everything about **Option and Derivative Pricing** theory is depedent on such mathematical objects together with **semi/sub/super-martingales**. But here we will not dive deeper into the mathematical intricacies of such objects except for stating their important properties.

To motivate the question asked on the first sentence of this note we will begin with a simple simulation technique called ***importance sampling***. The idea of importance sampling is then extended in a natural way to random processes and in short order this extension leads to the first Girsanov theorem. We will illustrate the effectiveness of this theorem with a few examples at end one of them being the derivation of the *elegant* **Levy-Bachelier** formula for the density of [first hitting time](https://en.wikipedia.org/wiki/First-hitting-time_model) of the Brownian motion to a slope. First hitting time is also known as the **first passage time** which is more familair to physicists.

Later we will discuss a set of similar theorems of several flavors which will be established by several different methods. Finally we will point out their applications in latest scientific research one of them being **stochastic thermodynamics** By the end of this note, the development of the Girsanov theory will leave us with new perspective on *continuous random processes*.

> A continous stochastic process is best viewed as a random variable with a value that can associated with a point chosen from path space $C[0,T]$.

## Importance Sampling
The purpose of all sorts of sampling techniques is to device an *effective* and *accurate* route to calculate the expectation of a random variable. Here we consider the calculation of $E[f(X)]$ where $f$ is a known function and $X$ is random variable with zero mean and unit variance. A direct but naive way to calculate $E[f(X)]$ is via direct simulation where (in the simplest ca\tag{1}\label{1}se) one tries to estimate $E[f(X)]$ by an approximation of the form
$$
\begin{equation}
\begin{aligned}
E[f(X)] \approx \frac{1}{n}\sum_{i=1}^{n}f(X_{i})
\end{aligned}
\end{equation}\label{1}\tag{1}
$$

$$
\begin{equation}
\begin{aligned}\Pr(y_{t+1}|c_t)
&= \sum_{j-0}^K \Pr(y_{t+1}, \tau_{t+1} = j |c_t) \\
&= \sum_{j-0}^K \Pr(y_{t+1} | \tau_{t+1} = j, c_t) \cdot \Pr(\tau_{t+1} = j | c_t)\\
\end{aligned}
\end{equation}\tag{2}\label{2}
$$