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7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Intro Brownian motion Integration SDE Applications
Stochastic Integration and Stochastic DifferentialEquations: a gentle introduction
Oleg MakhninNew Mexico Tech
Dept. of Mathematics
October 26, 2007
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
2/28
Intro Brownian motion Integration SDE Applications
Intro: why Stochastic?
Brownian Motion/ Wiener process
Stochastic IntegrationStochastic DEs
Applications
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Intro Brownian motion Integration SDE Applications
Intro
Deterministic ODE X(t) = a(t,X(t)) t> 0X(0) = X0
However, noise is usually presentX(t) = a(t,X(t)) + b(t,X(t))(t), t> 0X(0) = X0
Natural requirements for noise :
- (t) is random with mean 0- (t) is independent of(s), t= s White noise- is continuous
Strictly speaking, does not exist!
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Intro Brownian motion Integration SDE Applications
Brownian motion
T
c
T
T
c
$$$
Intuitive: Random walk, DiffusionStocks: you cannot predict the future behavior based on pastperformance
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Intro Brownian motion Integration SDE Applications
Brownian motion
0 200 400 600 800 1000
60
40
20
0
20
40
60
Several paths of Brownian Motion and LIL bounds
t
Bt
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
S
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Intro Brownian motion Integration SDE Applications
2D Brownian motion
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
I B i i I i SDE A li i
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
7/28
Intro Brownian motion Integration SDE Applications
Brownian motion: Axioms
BM is continuous (but not smooth, see later!), W(0) = 0
Normality:W(t+t)W(t) Normal(mean = 0, variance 2 = t)
3 2 1 0 1 2 3
0
.0
0.
1
0.
2
0.
3
0.
4
2
=
t
Normal because sum of many small, independent increments.
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
I t B i ti I t ti SDE A li ti
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
8/28
Intro Brownian motion Integration SDE Applications
Brownian motion: Axioms
Independence:
0.00 0.02 0.04 0.06 0.08 0.10
0
.4
0.3
0.2
0.1
0.0
t
W(t)
0.00
0
.4
0.3
0.2
0.1
0.0
W(t)
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Intro Brownian motion Integration SDE Applications
Fractal nature
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5
0 5 10 15 20 25
0.0 0.2 0.4
0 1 2
0 5 10
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
10/28
Intro Brownian motion Integration SDE Applications
Integration
Need
T
0 h(t) dW(t)
How would you define it?- For which functions h?- Answer is random (since W(t) is)
Riemann integral
T
0h(t) dt
ni=1
h(ti )ti
Stieltjes integral T
0
h(t) dF(t)
n
i=1
h(ti )[F(ti+1)
F(ti)]
where F has finite variation
- when F exists, then
T
0h(t) dF(t) =
T
0h(t)F(t) dt
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
11/28
Intro Brownian motion Integration SDE Applications
Stochastic Integration
Try the same:
T
0h(t) dW(t)
ni=1
h(ti )[W(ti+1)W(ti)]
does this still work?
No, W does not have finite variation
let maxti 0, limit in what sense? Depends on ti
Iton
i=1
h(ti)[W(ti+1)W(ti)]Stratonovichn
i=1
h
ti + ti+1
2
[W(ti+1)W(ti)]
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
12/28
Intro Brownian motion Integration SDE Applications
Stochastic Integration: examples
T
0dW(t)
W(ti+1)W(ti) = W(T) of course (recall W(0) = 0)
T
0t dW(t)
ti[W(ti+1)W(ti)] =? to be continued
so far, Ito and Stratonovich integrals agree.However,
T
0W(t) dW(t)
W(ti)[W(ti+1)W(ti)]
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
13/28
g pp
Stochastic Integration:T
0 W(t) dW(t)
note,2
W(ti)[W(ti+1)W(ti)] = W2(ti+1)W
2(ti) (W(ti+1)W(ti))
2
W2(T) T(brown part by Law of Large numbers)
Hence,
T
0W(t) dW(t) =
W2(T)
2 T
2
Wish it were easier?
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
14/28
Stochastic Differential Equations
X(t) =
t
0b(s,X(s)) dW(s), t> 0
dX(s) = b(s,X(s)) dW(s) - Stochastic differentialFull form
dX(s) = a(s,X(s)) ds + b(s,X(s)) dW(s),X(0) = X0 (can be random)
(note that dW/dt does not exist)
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
15/28
Ito 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) = gtdt+gxdX(t) + gxxb2(t,X(t))
2dt - extra term
heuristic: follows from Taylor series assuming (dW(t))2
= dt
Oleg Makhnin New Mexico Tech Dept. of Mathematics
Stochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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for example,
d(W2(s)) = W2(s + ds)W2(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
Hence,
t
0 W(s) dW(s) =
W2(t)
2 t
2
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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dg(t,X(t)) = gtdt+ gxdX(t) + gxxb2
2 dt
Exercise:
d(W3(s)) =?
= 3W2(s) dW(s) + 3W(s)ds
therefore,t
0W2(s) dW(s) = W
3
(t)3
t0
W(s) ds
d(sW(s)) =?
= W(s)ds + s dW(s),
therefore
t
0s dW(s) = tW(t)
t
0W(s) ds
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Solution
What does the solution look like?
Note that for Ito integrals, we always have
Et
0
h(s) dW(s) = 0 Expected value
E
t
0h(s) dW(s)
2=
t
0E[h2(s)] ds Variance
(can see using the Riemann sums).For example, the result of
t
0s dW(s) is a Normal random
variable, with mean 0 and variance = t3/3
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Langevin equation
More realistic Brownian motion: resistance/friction
dX(t) = X(t)dt + dW(t)
X(t) = particle velocityThen, can obtain using Ito formula, integrating factor et
X(t) = etX0 + t
0
e(ts) dW(s)
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
20/28
Stock price
dX(t) = rX(t) dt + X(t) dW(t)
can express d log(X(t)) and get
X(t) = exp
(r 22
)t + W(t)
Note that for both equations, expected value E[X(t)] coincideswith the solution of deterministic equation, here
d X(t) = r X(t) dt
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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A nonlinear equation
Consider a logistic equation for population dynamics: non-linear
dX(t) = rX(t)(1
X(t)) dt + X(t) dW(t)
Here, deterministic solution of
d X(t) = r X(t)(1 X(t)) dt
is different from the mean of stochastic solutions
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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A nonlinear equation
dX(
t) =
rX(
t)(1
X(
t))
dt+
X(
t)
dW(
t)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
t
X(t)
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
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Math Finance
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
7/29/2019 Stochastic Integration and Stochastic Differential Equations a Gentle Introduction
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Math Finance
- Optimal control of investments
- Black-Scholes formula for option pricing
- Based ondX(t) = rX(t) dt + X(t) dW(t)
and beyond
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
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Other applications
Boundary value problems: e.g. solving a Dirichlet problemf = 0 (harmonic) in the region U Rnf = g on the boundary U, given g
a stochastic solution is
f(x) = Ex[g(point of first exit from U)],
Ex refers to Brownian motion started from x.
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
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Other applications
Hydrology: porous media flow
div(V) = 0 incompressible flowV =
K
P Darcys law
V = velocity, P = pressure fieldK = conductivity is stochastic
Filtering: determine the position of a stochastic process from noisyobservation history. Kalman filter (discrete), Kalman-Bucy
filter (continuous)
Predator-prey models, Chemical reactions, Reservoir models ... ...
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
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QUESTIONS?
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction
Intro Brownian motion Integration SDE Applications
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THANK YOU!
Oleg Makhnin New Mexico Tech Dept. of MathematicsStochastic Integration and Stochastic Differential Equations: a gentle introduction