Change of measure to make things "easier"?

Question

I am familiar with the Radon-Nikodym Theorem and an R/N Derivative, but while reading a set of lecture notes on: Stochastic Calculus, Filtering, and Stochastic Control in section 1.6: "Induced measures, independence, and absolute continuity", (bottom of page 42, emphasis mine):

Absolutely continuous measures and the Radon-Nikodym theorem

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a given probability space. It is often interesting to try to find other measures on $\mathcal{F}$ with different properties. We may have gone through some trouble to construct a measure $\mathbb{P}$, but once we have such a measure, we can generate a large family of related measures using a rather simple technique. This idea will come in very handy in many situations; calculations which are difficult under one measure can often become very simple if we change to a suitably modified measure (for example, if $\{X_n\}$ is a collection of random variables with some complicated dependencies under $\mathbb{P}$, it may be advantageous to compute using a modified measure $\mathbb{Q}$ under which the $X_n$ are independent. Later on, the change of measure concept will form the basis form one of the most basic tools in our stochastic toolbox, the Girsanov theorem.

I'm having trouble conceptualizing the idea that "calculations which are difficult under one measure can often become very simple if we change to a suitably modified measure". Can someone further explain, perhaps by a couple examples, how this is the case? How can $X_n$ have complicated dependencies under $\mathbb{P}$, but simple (independent) dependencies under a $\mathbb{Q}$?

Answer 1

One example where a change of measure can make calculations simpler is the risk-neutral measure used commonly in finance.

Assume the price of a stock, $S_t$ satisfies the following SDE:

$$dS_t = \mu S_t dt + \sigma S_t dW_t$$

where $W_t$ is Brownian Motion. Using Girsanov's theorem, you can express the discounted stock price, $\tilde{S_t} = e^{-rt}S_t$ as

$$d\tilde{S_t} = \sigma \tilde{S_t}d\tilde{W_t}$$

Where $\tilde{W_t} = W_t + \frac{\mu - r}{\sigma} t$ is a martingale under a change of measure, $\mathbb{Q}$.

Under the original probability measure, $\mathbb{P}$, $\tilde{W_t}$ is not a martingale nor Brownian motion, but under the risk neutral measure, $\mathbb{Q}$, it is.

Since martingales have a lot of useful properties and are typically easier to manipulate than non-martingales, this is an example where the change-of-measure makes things easier.

Answer 2

Here are two examples from filtering and estimation theory:

1. In filtering theory, the reference measure approach makes it easier to find equations for the conditional expectations $E_P[\varphi(X_t)|\mathcal{F}^Y_t]$. The idea is to pick a reference measure $Q$ under which the state and observation processes decouple (become independent), and the observation process is some way simple. In the classical filtering problem, where the state and observation processes are diffusion process driven by independent Brownian motions and the observation noise is additive, the Girsanov theorem guarantees that we can pick a reference measure under which $Y$ becomes a Brownian motion independent of the parameters and the state $X$. We can then use the Kallianpur-Striebel formula in order to express the conditional expectations with respect to the original measure in terms of the reference measure, i.e. $$E_P[\varphi(X_t)|\mathcal{F}^Y_t]=\frac{E_Q[\tfrac{dP}{dQ}\varphi(X_t)|\mathcal{F}^Y_t]}{E_Q[\tfrac{dP}{dQ}|\mathcal{F}^Y_t]}.$$ It turns out that the expectations on the RHS are much easier to work with. This idea generalizes to a wide class of filtering problems.

2. If you have a partially observed system for which you want to estimate some parameters, you can try to find a reference measure under which the observed process is simple. To do maximum likelihood estimation, you have to restrict the measure to the observed process. For a finite number of random variables, some of which are hidden and some of which are observed, we would integrate over the hidden variables (marginalization) in order to get the probability of observed states. In the continuous-time case, this would mean that we need to integrate over all possible trajectories of the hidden process. By picking a reference measure that is independent of the parameters, we can write down a likelihood function as the Radon-Nikodym derivative of the original measure wrt. the reference measure. For the case of partially observed diffusion processes, more details can be found in Online Maximum Likelihood Estimation of the Parameters of Partially Observed Diffusion Processes.