CNH3C3 Persamaan Diferensial Parsial (The art of Modeling PDEs) DR. PUTU HARRY GUNAWAN

CNH3C3 Persamaan Diferensial Parsial (The art of Modeling PDEs) DR. PUTU HARRY GUNAWAN

Partial Differential Equations Content 1. Part II: Derivation of PDE in Brownian Motion

PART II DERIVATION OF PDE IN BROWNIAN MOTION

Copyright SOME PARTS OF MATERIAL IN THIS SLIDE ARE OBTAINED FROM HTTPS://EN.WIKIPEDIA.ORG/WIKI/BROW NIAN_MOTION) HTTPS://YOUTUBE.COM

Brownian motion Brownian motion or pedesis (from Ancient Greek: πήδησις /pέːdεːsis/ "leaping") is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fastmoving atoms or molecules in the gas or liquid.

Examples This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions. This is a simulation of the Brownian motion of 5 particles (yellow) that collide with a large set of 800 particles. The yellow particles leave 5 blue trails of random motion and one of them has a red velocity vector.

Motivations In this course, two applications of Brownian motion will be given in: 1. Chemistry: Colloid motion 2. Finance : Geometry of Brownian motion

Chemistry: Colloid motion https://www.youtube.com/watch?v=xeailm2zuvc

Chemistry: Colloid motion Colloidal particles in a sol are continuously bombarded by the molecules of the dispersion medium on all sides. The impacts are however not equal in every direction. As a result, the sol particles show random or zig-zag movements. This random or zig-zag motion of the colloidal particles in a sol is called Brownian motion or Brownian movement. Or see (Courtesy) https://www.youtube.com/watch?v=yax0gs00f Tw

Finance : Geometry of Brownian motion A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. [1] It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black Scholes model. Two sample paths of Geometric Brownian motion, with different parameters. The blue line has larger drift, the green line has larger variance.

Finance : Geometry of Brownian motion Intuitively, we may think of a Brownian motion as a limiting case of some random walk as its time increment goes to zero. The upper graph depicts a realization of a random walk. The lower graph depicts a similar realization of a Brownian motion.

Goals In this course, we will not talk about the applications of Brownian motion in deeply. Next, we will try to elaborate the derivation of Brownian motion equations from physics to PDEs

Mathematical formulation Consider a particle on arbitrary position in one-dimensional space p(x, t) t = 0 p(x 0, 0) x 0 x

Mathematical formulation Then a particle try randomly to move p(x, t) t = Δt Particle moves to the right with probability α p(x 1, Δt) p(x 0, 0) x 0 x 1 x

Mathematical formulation Then a particle try randomly to move p(x, t) t = Δt Particle moves to the right with probability α p(x 1, Δt) p(x 0, 0) Δx 1 x 0 x 1 x

Mathematical formulation Then a particle try randomly to move p(x, t) p(x 2, 2Δt) t = 2Δt Particle moves to the right with probability α p(x 1, Δt) p(x 0, 0) Δx 1 Particle moves to the left with probability β x 2 x 0 x 1 x

Mathematical formulation Then a particle try randomly to move p(x, t) p(x 2, 2Δt) t = 2Δt Particle moves to the right with probability α p(x 1, Δt) p(x 0, 0) Δx 2 Δx 1 Particle moves to the left with probability β x 2 x 0 x 1 x

Mathematical formulation Then a particle try randomly to move p(x, t) t = 2Δt p(x 2, 2Δt) p(x 1, Δt) Δx 2 Or particle does not move, thus the probability is 1 α β p(x 0, 0) Δx 1 x 2 x 0 x 1 x

Mathematical formulation Then a particle try randomly to move p(x, t) p(x 2, 2Δt) p(x 1, Δt) p(x 0, 0) t = 2Δt Δx 2 Δx 1 Combining the previous rules in the interval x Δx, x + Δx at time t + Δt, p x, t + Δt = α p x + Δx, t + β p x Δx, t + p x, t 1 α β x 2 x 0 x 1 x

Mathematical formulation p x, t + Δt = α p x + Δx, t + β p x Δx, t + p x, t 1 α β For now we shall consider the case where leftward and rightward movements are equally likely, i.e. α = β.

Mathematical formulation p x, t + Δt = α p x + Δx, t + β p x Δx, t + p x, t 1 α β For now we shall consider the case where leftward and rightward movements are equally likely, i.e. α = β. p x, t + Δt = α p x + Δx, t + α p x Δx, t + p x, t 1 2α

Mathematical formulation p x, t + Δt = p x, t + α p x + Δx, t + α p x Δx, t 2α p x, t p x, t + Δt p x, t = α p x + Δx, t + α p x Δx, t 2α p x, t Dividing by Δt p x, t + Δt p x, t Δt = α Δt p x + Δx, t + p x Δx, t 2 p x, t

Mathematical formulation p x, t + Δt p x, t Δt = α(δx2 ) Δt p x + Δx, t + p x Δx, t 2 p x, t Δx 2 p x, t + Δt p x, t Δt = D p x Δx, t 2 p x, t + p x + Δx, t Δx 2 with D = α(δx2 ) Δt

Remember The derivative of a function f at a point x is defined by the limit df x dx = f x = lim h 0 f x + h f x h f x f x + h f x h

Remember The higher derivative of a function f at a point x is defined by the limit d 2 f x dx 2 = f x = lim h 0 f x + h 2f x + f(x h) h 2 f x f x + h 2f x + f x h h 2

Mathematical formulation Taking limit Δt 0 and Δx 0 p x, t + Δt p x, t lim Δt 0 Δt = D lim Δx 0 p x + Δx, t 2 p x, t + p x Δx, t Δx 2

Mathematical formulation Taking limit Δt 0 and Δx 0 p x, t + Δt p x, t lim Δt 0 Δt = D lim Δx 0 p x + Δx, t 2 p x, t + p x Δx, t Δx 2 Then we obtain, with D = α(δx2 ) Δt p x, t t = D 2 p x, t x 2

Mathematical formulation This equation is called the random walk model of diffusion p x, t t = D 2 p x, t x 2 with D = α(δx2 ) Δt Extend the idea into the three dimensions p x, y, z, t t = D p(x, y, z, t)

Application: Brownian Motion in Stochastic differential equation Let the stochastic proses X be the solution of the stochastic differential equation dx t = 2k db t X 0 = 0 where B is the Wiener process. Then the probability density function of X is given at any time t by 1 2πkt exp x2 4kt which is the solution of the initial value problem u t x, t ku xx x, t = 0, x, t R (0, + ) u(x, 0) = δ(x)

Summary PART II The particle motion has been elaborated The derivation of random walk model in PDE has been explained in detail U t D 2 U x 2 = 0 D = α Δx2 Δt

End of Presentation