Stochastic Analogs

Stochastic models #

Since the differential equation involves randomness, we can only say something about the probability distributions of the solutions

simple population growth model #

$$ \begin{align*} \frac{dN}{dt}=a(t)N(t)\\ N(0)=N_{0}\\ a(t)=r(t)+\text{noise} \end{align*} $$

Electrical circuit #

$$ \begin{align*} LQ’’+RQ’ + \frac{1}{C}Q(t)=F(t)\\ F(t)=G(t)+ \text{noise} \end{align*} $$

Filter #

Kalman-Bucy filter gives a procedure for estimating the state of a system which satisfies a noisy linear differential equation, based on a series of noisy observations

Like observing Q(s) until a time $s\le t$, call it $Z(s)$ and finding the best estimation for Q(t)

Optimal stopping problem #

$X_{t}$: price of asset

$$ \frac{dX_{t}}{dt}=rX_{t}+\alpha X_{t}*\text{noise} $$

If we know the value of $X_{s}$ until present time t, then what is the stopping strategy to maximize expected profit when inflation is taken into account

Stochastic Control #

Risky investment #

$$ \frac{dp_{1}}{dt}=(a+\alpha*\text{noise})p_{1} $$

safe investment #

$$ \frac{dp_{2}}{dt}=bp_{2} $$

portfolio #

portfolio $u_{t}\in[0,1]$ such that a person places $u_{t}X_{t}$ in risky investment and $(1-u_{t})X_{t}$ in safe investment

Find the investment $u_{t}0<t<T;$ which maximizes expected utility $$\max_{u_{t}}(E(U(X_{T}^{u})))$$

Mathematical Finance #

At t=0, the person is offered the right to buy 1 unit from risky asset at a price K. How much should he be willing to pay?

Solved by Fischer Black and Myron Scholes