Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27
Intro: why Stochastic? Brownian Motion/ Wiener process Stochastic Integration Stochastic DE s Applications
Intro Deterministic ODE { X (t) = a(t, X (t)) t > X () = X However, noise is usually present { X (t) = a(t, X (t)) + b(t, X (t)) ω(t), t > X () = X Natural requirements for noise ω: - ω(t) is random with mean - ω(t) is independent of ω(s), t s White noise - ω is continuous Strictly speaking, ω does not exist!
Brownian motion Intuitive: Random walk, Diffusion Stocks: you cannot predict the future behavior based on past performance
Brownian motion Several paths of Brownian Motion and LIL bounds Bt 6 4 2 2 4 6 2 4 6 8 1 t
2D Brownian motion
Brownian motion: Axioms BM is continuous (but not smooth, see later!), W() = Normality: W (t + t) W (t) Normal(mean =, variance σ 2 = t) yc..1.2.3.4 2σ 3 2 1 1 2 3 xc σ = t Normal because sum of many small, independent increments.
Brownian motion: Axioms Independence: W(t).4.3.2.1. W(t).4.3.2.1...2.4.6.8.1. t
Fractal nature..2.4.6.8 1...2.4 1 2 3 4 5 1 2 5 1 15 2 25 5 1
Integration Need T h(t) dw (t) How would you define it? - For which functions h? - Answer is random (since W (t) is) T n Riemann integral h(t) dt h(ti ) t i Stieltjes integral T where F has finite variation - when F exists, then i=1 h(t) df (t) T n h(ti )[F (t i+1 ) F (t i )] i=1 h(t) df (t) = T h(t)f (t) dt
Stochastic Integration Try the same: T h(t) dw (t) n h(ti )[W (t i+1 ) W (t i )] i=1 does this still work? No, W does not have finite variation let max t i, limit in what sense? Depends on ti Itô Stratonovich n n ( ) ti + t i+1 h(t i )[W (t i+1 ) W (t i )] h [W (t i+1 ) W (t i )] 2 i=1 i=1
Stochastic Integration: examples T T T dw (t) W (t i+1 ) W (t i ) = W (T ) of course (recall W () = ) t dw (t) t i [W (t i+1 ) W (t i )] =? to be continued so far, Itô and Stratonovich integrals agree. However, W (t) dw (t) W (t i )[W (t i+1 ) W (t i )]
Stochastic Integration: T W (t) dw (t) note, 2 W (t i )[W (t i+1 ) W (t i )] = W 2 (t i+1 ) W 2 (t i ) (W (t i+1 ) W (t i )) 2 W 2 (T ) T (brown part by Law of Large numbers) Hence, T Wish it were easier? W (t) dw (t) = W 2 (T ) 2 T 2
Stochastic Differential Equations X (t) = t b(s, X (s)) dw (s), t > dx (s) = b(s, X (s)) dw (s) - Stochastic differential Full form { dx (s) = a(s, X (s)) ds + b(s, X (s)) dw (s), X () = X (can be random) (note that dw /dt does not exist)
Itô Formula Let dx (t) = a(t, X (t)) dt + b(t, X (t)) dw (t) then, the chain rule is dg(t, X (t))(t) = g t dt+g x dx (t) + g xx b 2 (t, X (t)) 2 dt - extra term heuristic: follows from Taylor series assuming (dw (t)) 2 = dt
for example, d(w 2 (s)) = W 2 (s + ds) W 2 (s) = [W (s + ds) + W (s)][w (s + ds) W (s)] = [W (s + ds) W (s)][w (s + ds) W (s)] +2W (s)[w (s + ds) W (s)] = 2W (s) dw (s) + ds t Hence, W (s) dw (s) = W 2 (t) t 2 2
dg(t, X (t)) = g t dt + g x dx (t) + g xx b 2 2 dt Exercise: d(w 3 (s)) =? = 3W 2 (s) dw (s) + 3W (s)ds therefore, t d(sw (s)) =? = W (s)ds + s dw (s), therefore t W 2 (s) dw (s) = W 3 (t) 3 s dw (s) = tw (t) t t W (s) ds W (s) ds
Solution What does the solution look like? Note that for Itô integrals, we always have E t h(s) dw (s) = Expected value [ t 2 E h(s) dw (s)] = t (can see using the Riemann sums). For example, the result of t variable, with mean and variance = t 3 /3 E[h 2 (s)] ds Variance s dw (s) is a Normal random
Langevin equation More realistic Brownian motion: resistance/friction dx (t) = βx (t)dt + σ dw (t) X (t) = particle velocity Then, can obtain using Itô formula, integrating factor e βt t X (t) = e βt X + e β(t s) dw (s)
Stock price can express d log(x (t)) and get dx (t) = rx (t) dt + σx (t) dw (t) ] X (t) = exp [(r σ2 )t + σw (t) 2 Note that for both equations, expected value E[X (t)] coincides with the solution of deterministic equation, here dx (t) = rx (t) dt
A nonlinear equation Consider a logistic equation for population dynamics: non-linear dx (t) = rx (t)(1 X (t)) dt + σx (t) dw (t) Here, deterministic solution of dx (t) = rx (t)(1 X (t)) dt is different from the mean of stochastic solutions
A nonlinear equation dx (t) = rx (t)(1 X (t)) dt + σx (t) dw (t) X(t)..5 1. 1.5..2.4.6.8 1. t
Math Finance
Math Finance - Optimal control of investments - Black-Scholes formula for option pricing - Based on dx (t) = rx (t) dt + σx (t) dw (t) and beyond
Other applications Boundary value problems: e.g. solving a Dirichlet problem f = (harmonic) in the region U R n f = g on the boundary U, given g a stochastic solution is f (x) = E x [g(point of first exit from U)], E x refers to Brownian motion started from x.
Other applications Hydrology: porous media flow { div(v ) = incompressible flow V = K P Darcy s law V = velocity, P = pressure field K = conductivity is stochastic Filtering: determine the position of a stochastic process from noisy observation history. Kalman filter (discrete), Kalman-Bucy filter (continuous) Predator-prey models, Chemical reactions, Reservoir models......
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