Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of...
Transcript of Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of...
![Page 1: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/1.jpg)
Examples of Stochastic DifferentialEquations
Steven R. Dunbar
October 2, 2018
![Page 2: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/2.jpg)
Outline
Brownian Motions
Geometric Brownian Motions
More General Processes
Applications
![Page 3: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/3.jpg)
Standard Brownian Motion
Stochastic Differential Equation
dX = dW
Physical InterpretationMotion of small particles in a fluid buffeted by randommolecular forces.
Stochastic Process/DiffusionX (t) = W (t)
![Page 4: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/4.jpg)
Standard Brownian Motion Typical Path
0.0 0.2 0.4 0.6 0.8 1.0
−0.4
−0.2
0.0
0.2
0.4
0.6
x
Wca
retN
![Page 5: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/5.jpg)
Standard Brownian Motion
MeanE [X (t)] = 0
Variance and Standard DeviationVar [X (t)] = t, SD [X (t)] =
√t
Fokker-Planck Equation∂ρ∂t
= 12∂2ρ∂x2
Transition Probability Functionρ(x , t | y , s) = 1√
2π(t−s)e−
12(x−y)2
t−s
Typical QuestionWhat is the probability that X (t) = A before X (t) = −B?
![Page 6: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/6.jpg)
Brownian Motion with Drift
Stochastic Differential Equation
dX = r dt + dW
Physical InterpretationMotion of small particles in a convective fluid buffeted byrandom molecular forces.
Stochastic Process/DiffusionX (t) = rt +W (t)
![Page 7: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/7.jpg)
Brownian Motion with Drift Typical Path
Brownian Motion with Drift, r = 1, sigma = 1
Time
X
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
![Page 8: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/8.jpg)
Brownian Motion with Drift
MeanE [X (t)] = rt
VarianceVar [X (t)] = t, SD [X (t)] =
√t
Fokker-Planck Equation∂ρ∂t
= −r ∂ρ∂x
+ 12∂2ρ∂x2
Transition Probability Functionρ(x , t | y , s) = 1√
2π(t−s)e−
12(−rt+x−y)2
t−s
Typical QuestionWhat is the probability that X (t) = A+ s1t beforeX (t) = −B + s2t?
![Page 9: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/9.jpg)
Scaled Brownian Motion with Drift
Stochastic Differential Equation
dX = r dt +σ dW
Physical InterpretationMotion of small particles in a convective fluid buffeted bysmall (or large) random molecular forces.
Stochastic Process/DiffusionX (t) = rt + σW (t)
![Page 10: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/10.jpg)
Scaled Brownian Motion with Drift Typical Path
Brownian Motion with Drift, r = 1, sigma = 1/4
Time
X
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
![Page 11: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/11.jpg)
Scaled Brownian Motion with Drift
MeanE [X (t)] = rt
VarianceVar [X (t)] = σ2t, SD [X (t)] = σ
√t
Fokker-Planck Equation∂ρ∂t
= −r ∂ρ∂x
+ σ2
2∂2ρ∂x2
Transition Probability Functionρ(x , t | y , s) = 1√
2πσ2(t−s)e−
12(−r(t−s)+x−y)2
σ2(t−s)
Typical QuestionWhat is the probability that X (t) = A+ s1t beforeX (t) = −B + s2t?
![Page 12: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/12.jpg)
Geometric Brownian Motion (Dolean’s exponentialof Brownian motion)
Stochastic Differential Equation
dX (t) = σX (t) dW .
Physical InterpretationRelative rate of change is proportional to random effects(distributed as white noise).
Stochastic Process/DiffusionX (t) = exp(−(1/2)σ2t + σW (t))
![Page 13: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/13.jpg)
Geomtric Brownian Motion Typical Path
Geometric Brownian Motion, sigma=1
Time
X
0.0 0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
2.0
2.5
![Page 14: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/14.jpg)
Geometric Brownian Motion
MeanE [X (t)] = x0
VarianceVar [X (t)] = x2
0 [exp(σ2t)− 1]
Fokker-Planck Equation∂ρ∂t
= σ2x2
2∂2ρ∂x2 + 2σ2x ∂ρ
∂x+ σ2ρ
Transition Probability Functionρ(x , t | y , s) = 1
x√
2πσ2(t−s)e−
12 [log(x)−log(y)+σ2(t−s)/2]
2/(2σ2(t−s))
Typical QuestionFor B < 1 < A, what is the probability that X (t) = A beforeX (t) = B?
![Page 15: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/15.jpg)
General Geometric Brownian Motion
Stochastic Differential Equation
dX = rX dt +σX dW
Physical Interpretation
I Relative rate of change is proportional to base growth (ordecay) plus random effects (distributedas white noise).
I Short term growth and short term variability isproportional to the level of the process
Stochastic Process/DiffusionX (t) = x0 exp(rt − (1/2)σ2t + σW (t))
![Page 16: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/16.jpg)
General Geometric Brownian Motion Typical Path
General Geometric Brownian Motion, r = 1, sigma =1
Time
X
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.5
2.0
2.5
3.0
![Page 17: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/17.jpg)
General Geometric Brownian Motion
MeanE [X (t)] = x0 exp(rt)
VarianceVar [X (t)] = x2
0 exp(2rt)[exp(σ2t)− 1]
Fokker-Planck Equation∂ρ∂t
= σ2x2
2∂2ρ∂x2 + (2σ2 − r)x ∂
2ρ∂x2 + (σ2 − r)ρ
Transition Probability Functionρ(x , t | y , s) =
1x√
2πσ2(t−s)e−
12 [log(x)−log(y)−(r−σ2/2)(t−s)]
2/(2σ2(t−s))
Typical Question
![Page 18: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/18.jpg)
If a Geometric Brownian Motion is defined by theSDE
dX = rX dt +σX dW X (0) = x0
then the Geometric Brownian Motion is
X (t) = x0 exp((r − (1/2)σ2)t + σW (t)).
At each time the Geometric Brownian Motion has lognormaldistribution with parameters (ln(x0)+ rt − (1/2)σ2t) and σ
√t.
E [X (t)] = x0 exp(rt)
Var [X (t)] = x20 exp(2rt)[exp(σ
2t)− 1]
![Page 19: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/19.jpg)
If the primary object is the Geometric BrownianMotion
X (t) = x0 exp(rt + σW (t)).
then by Ito’s formula the SDE satisfied by this stochasticprocess is
dX = (µ+ (1/2)σ2)X (t) dt +σX (t) dW X (0) = x0.
At each time the Geometric Brownian Motion has lognormaldistribution with parameters (ln(x0) + µt) and σ
√t.
E [X (t)] = x0 exp(rt + (1/2)σ2t).
Var [X (t)] = x20 exp(2µt + σ2t)[exp(σ2t)− 1].
![Page 20: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/20.jpg)
General Ornstein-Uhlenbeck ProcessStochastic Differential Equation
dX = r(K − X ) dt +σ dW
Physical Interpretation
I Velocity (tending toward terminal velocity) of smallparticles in a resistive fluid buffeted by random molecularforces.
I Reversion to the mean K disturbed by addition of randomeffects (distributed as whilte noise)
I In economics, Vasiček model for interest rates
Stochastic Process/DiffusionX (t) = K + (x0 − K )e−rt + σ
∫ t
0 e−r(t−s) dW
![Page 21: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/21.jpg)
Ornstein-Uhlenbeck Typical Path
General Ornstein−Uhlenbeck, r=1, K=1, sigma=1
Time
X
0.0 0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
2.0
![Page 22: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/22.jpg)
General Ornstein-Uhlenbeck Process
MeanE [X (t)] = K + (x0 − K )e−rt
VarianceVar [X (t)] = σ2
2r (1− e−2rt)
Fokker-Planck Equation∂ρ∂t
= σ2
2∂2ρ∂x2 − r(K − x) ∂ρ
∂x+ rρ
Transition Probability FunctionP [X (t) = x |X (s) = y ] ∼ N(K + (x0 − K )e−rt , σ
2
2r (1− e−2rt))
Typical QuestionWhat is long-term behavior of the solution?
![Page 23: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/23.jpg)
Stochastic Verhulst Equation
Stochastic Differential Equation
dX = (r − X )X dt +σX dW
Physical InterpretationDoering, Chapter 5, page 26: Growth or decay of a populationwith birth and death rates subject to random effects that arefast relative to the deterministic time scale. Describespopulation dynamics on times scales much longer than anygeneration. Note scaling of carrying capacity into the averagegrowth rate.
![Page 24: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/24.jpg)
Stochastic Verhulst Typical Path
Stochastic Verhulst, r = 1, sigma = 0.5, X0 =1
Time
X
0 2 4 6 8 10
0.4
0.6
0.8
1.0
![Page 25: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/25.jpg)
Cox-Ingersoll-Ross Process
Stochastic Differential Equation
dX = r(K − X ) dt +σ√X dW
Physical Interpretation
I In economics, mean-reverting model for interest rates
I If Kr > 12σ
2, the process stays positive.
![Page 26: Examples of Stochastic Differential Equationssdunbar1/ProbabilityTheory/... · Examples of Stochastic Differential Equations Author: Steven R. Dunbar Subject: Stochastic Differential](https://reader033.fdocuments.in/reader033/viewer/2022052721/5f0b861a7e708231d430f080/html5/thumbnails/26.jpg)
List of Applications of SDEs (Kloeden and Platen)Many are versions of O-U or Verhulst equations:
I Population DynamicsI Protein KineticsI GeneticsI Neuronal ActivityI Option PricingI Turbulent DiffusionI Radio-AstronomyI Helicopter Rotor StabilityI Satellite Orbit StabilityI Biological Waste TreatmentI Seismology and Structural MechanicsI Fatigue CrackingI Blood Clotting DynamicsI Optical BistabilityI Josephson JunctionsI Stochastic Annealing