Examples of Stochastic DifferentialEquations

Steven R. Dunbar

October 2, 2018

Outline

Brownian Motions

Geometric Brownian Motions

More General Processes

Applications

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)

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

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?

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)

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

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?

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)

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

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?

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))

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

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?

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))

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

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

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]

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].

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

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

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?

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.

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

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.

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