Answer to Question #168135 in Financial Math for kenny dug

Question

Let A be a Brownian motion. A stochastic process P has both a Newtonian term based on dt, where t is time, and a Brownian term based on dA_t , the infinitesimal increment of A.

A stochastic process P can be defined as a continuous process (P_t , t ≥ 0) such that P_t can be written in differential form

dP_t / P_t = Q_t dt + C_t dA_t

Explain the meaning of C_t and Q_t invoking probability density functions?

Answer 1

The given equation is-

$\dfrac{dP_t}{P_t} = Q_t dt + C_t dA_t$

 $p_t$ gives the probability density function, or pdf. 

Here $C_t$ denotes the commulative distribution function. cdf. $q_t$ gives the quantile function, which is the inverse of the cdf. The quantile function is used, for example, when constructing confidence intervals, to find the endpoints of an interval which contains 90 (or 95%, or 99%...) of the mass of the distribution: