20171117 Notes: Stochastic Process, Parameter Estimation, PDE

Stochastic process

  • Brownian Motion (BM): $Z_t \sim \mathcal{N}(0,t)$;
  • Arithmetic Brownian Motion (ABM): $d \mu = \alpha dt + \sigma dZ_t$
  • Geometric Brownian Motion (GBM): $dY_t = (\alpha + \frac{1}{2} \sigma^2) dt + \sigma dZ_t$
  • Ornstein-Uhlenbeck process (O-U Process): $du = \alpha(\mu-u)dt + \sigma dZ_t$, where $\alpha$ is speed of reversion and $\mu$ is long-term mean (A.K.A steady state mean). The distribution of O-U Process is $u \sim \mathcal{N} (\mu_{\mathcal{N}}, \sigma_{\mathcal{N}})$, whereAs $t \rightarrow \infty$:
    • if $\alpha > 0$,Converge.
    • if $\alpha = 0$,Diverge.
    • if $\alpha < 0$, Diverge.

OU Process

  • Cox-Ingersoll-Ross (CIR) process: $dX_t = \alpha (\mu-X_t)dt+\sigma \sqrt{X_t} dZ_t$
  • Stochastic Differential Equation for general 1-Spatial dimension Itô drift-diffusion process: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dZ_t$
  • Ito’s lemma: $Y$ is Itô drift-diffusion process, $f: \mathbb{R}^2 \to \mathbb{R}$ is smooth, then$(dt)^2 = dt dW_t$; $(dW_t)^2 = dt$

Parameter Estimation

  • Using Maximum Likelihood Estimation (MLE):
    • Exact MLE: Using known density function $f(x_i|\theta)$;
    • Inexact MLE:
  • Fokker Planck Equation:

Partial Differential Equation

  • General linear second order Partial Differential Equation (PDE)23:
    • Elliptic: if $B^2-AC < 0$. Simple example is 2-D Laplace Equation: $u_{xx} + u_{yy}= 0$ (where $X = x$, $Y = y$, $A = 1$, $B = 0$, $C = 1$), which can describe the electrostatic field.
    • Parabolic: if $B^2-AC = 0$. Simple example is 1-D Heat Equation: $u_t - K u_{xx} = 0$ (where $X = x$, $Y = t$, $A = -K$, $B=C=0$), which can describe heat transportation process
      • Fokker Planck Equation:
      • Black Scholes Eqaution:
    • Hyperbolic: if $B^2-AC > 0$. Simple example is 1-D Wave Equation: $a^2 u_{xx} - u_{tt} = 0$ (where $X = x$, $Y = t$, $A = a^2$, $B = 0$, $C = -1$), which can describe light wave.
  • MATLAB function pdepe can solve initial-boundary value problems for parabolic-elliptic PDEs in 1-D.

Stochastic Optimal Control Process

  • Stochastic Control problem concept:

    • Agent
    • State: $X_t$, SDE:
    • Control: $u(X_t,t)$
    • Objective Function: $F(u, t)$
  • Using Brand Management on Twitter as an example:

    • Agent: Some firm, such as AT&T Co.
    • State: Sentiment Score of customers on Twitter;
    • Control: Effort AT&T Co. pay to increase the sentiment;
    • Objective Function: Return;
  • If we want to minimize value function:

    where $\gamma$ is discount factor.

  • Taylor expansion on $\eqref{1}$:

  • $\eqref{2}$ into $\eqref{1}$:
  • Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. The solution of the HJB equation is the value function which gives the minimum cost for a given dynamical system with an associated cost function.
    Expectation on $\eqref{3}$:
  • Guess Objective Function (such as the return of the firm) follow the form of $F(u, t) = \beta X - C u^2$. Drift term of State $dX_t$ follow the form: $f(u, t) = k_1 u \sqrt{b-X} - k_2 X$; Volatility term of State $dX_t$ follow the form: $g(u, t) = k_3$. Thus replace $f$ and $g$ in $\eqref{4}$ HJB equation:
  • Optimal Control: Partial derivative of the term inside $Max_{u(X, t)}$ in RHS w.r.t. $u$:
  • Guesswhere $\lambda_1 > 0$, $V_X = \lambda_1$, $V_{XX} = V_t = 0$. Substitute $\eqref{5}$ and $\eqref{6}$ into $\eqref{4*}$:
  • (X) part of $\eqref{7}$:
  • (Constant) part of $\eqref{7}$:
  • Solve $\eqref{8-1}$ and $\eqref{8-2}$ ($\lambda_1 > 0$):
  • Substitute $\eqref{9-1}$ and $\eqref{9-2}$ in $\eqref{5}$:
  • Substitute $\eqref{10}$ in $\eqref{0}$, note that $f(u, t) = k_1 u \sqrt{b-X} - k_2 X$, $g(u, t) = k_3$:Define:Then $dX = \alpha_{HJB} (\mu_{HJB} - X) dt + \sigma_{HJB} dZ$, which is O-U process!
  • According to O-U Process, state $X \sim \mathcal{N} (\mu_{\mathcal{N}}, \sigma_{\mathcal{N}})$, where converge with $t \to \infty$ when $\eqref{12-1}$ $\alpha_{HJB} > 0$.

Read More

  • Princeton-Economics 521, Advanced Macroeconomics I:

Lecture 4: Hamilton-Jacobi-Bellman Equations, Stochastic Differential Equations

Lecture 5: Stochastic HJB Equations, Kolmogorov Forward Equations

1. Han, C., & Phillips, P. C. (2013). First difference maximum likelihood and dynamic panel estimation. Journal of Econometrics, 175(1), 35-45.
2. Partial Differential Equations in Finance
3. Stanford MATH220a: 4 Classification of Second-Order Equations
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