Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete

Lecture on Stochastic Differential Equations

Erik Lindström

FMS161/MASM18 Financial Statistics

Erik Lindström Lecture on Stochastic Differential Equations

Motivation

I Continuous time models are more ’interpretable’ thandiscrete time models, at least if you have a background inscience or engineering.

I It is often argued that continuous time models need fewerparameters compared to discrete time models, as theparameters often can be given an interpretation.

I Consistent with option valuation due to path wiseproperties.

I Integration between time scales (e.g. irregularly sampleddata)

I Heteroscedasticity is easily integrated into the models.


Motivation







Motivation







Motivation







Motivation







ODEs in physics

Physics is often modelled as (a system of) ordinary differentialequations

dXdt

(t) = µ(X (t)) (1)

Similar models are found in financeBond dB

dt (t) = rB(t)

Stock dSdt (t) = (µ + “noise′′(t))S(t)

CAPM dSdt (t) = (r + βσ + σ“noise′′(t))S(t)


ODEs in physics


dXdt

(t) = µ(X (t)) (1)


dt (t) = rB(t)




ODEs in physics


dXdt

(t) = µ(X (t)) (1)


dt (t) = rB(t)




Noise processes

The noise process should ideally be the time derivative of whitenoise. Examples of continuous time processes (see Chapter7.5)

I Brownian motion W (t)I Poisson process N(t)

I Compound Poisson process S(t) = ∑N(t)n=1 Yn

I Lévy process L(t)


Noise processes






Noise processes






Noise processes






Noise processes






Wiener process aka Standard Brownian Motion

A processes satisfying the following conditions is a StandardBrownian Motion

I X (0) = 0 with probability 1.I The increments W (t)−W (u), W (s)−W (0) with

t > u ≥ s > 0 are independent.I The increment W (t)−W (s)∼ N(0, t−s)

I The process has continuous trajectories.


Time derivative of the Wiener process

Study the object

ξh =W (t + h)−W (t)

h(2)

(Think dW (t)/dt = limh→0 ξh). ComputeI E[ξh]

I Var[ξh]

The limit does not converge in mean square sense!


Time derivative of the Wiener process

Study the object

ξh =W (t + h)−W (t)

h(2)

(Think dW (t)/dt = limh→0 ξh). ComputeI E[ξh]

I Var[ξh]

The limit does not converge in mean square sense!


Re-interpreting ODEs

In physics,dXdt

(t) = µ(X (t)) (3)

really meansdX (t) = µ(X (t))dt (4)

or actually ∫ t

0dX (s) = X (t)−X (0) =

∫ t

0µ(X (s))ds (5)

NOTE: No derivatives needed!



In physics,dXdt

(t) = µ(X (t)) (3)


or actually ∫ t

0dX (s) = X (t)−X (0) =

∫ t

0µ(X (s))ds (5)




In physics,dXdt

(t) = µ(X (t)) (3)


or actually ∫ t

0dX (s) = X (t)−X (0) =

∫ t

0µ(X (s))ds (5)



Stochastic differential equations

InterpretdXdt

=(µ(X (t)) + “noise′′(t)

)(6)

as

X (t)−X (0)≈∫ t

0

(µ(X (s)) + “noise′′(s)

)ds (7)

The mathematically correct approach is to define StochasticDifferential Equations as

X (t)−X (0) =∫

µ(X (s))ds +∫

σ(X (s))dW (s) (8)


Stochastic differential equations

InterpretdXdt

=(µ(X (t)) + “noise′′(t)

)(6)

as

X (t)−X (0)≈∫ t

0

(µ(X (s)) + “noise′′(s)

)ds (7)

The mathematically correct approach is to define StochasticDifferential Equations as

X (t)−X (0) =∫

µ(X (s))ds +∫

σ(X (s))dW (s) (8)


Integrals

The ∫µ(X (s))ds (9)

integral is an ordinary Riemann integral,whereas the∫σ(X (s))dW (s) (10)

integral is an Ito integral.


Integrals

The ∫µ(X (s))ds (9)

integral is an ordinary Riemann integral,whereas the∫σ(X (s))dW (s) (10)

integral is an Ito integral.


The Ito integral

The Ito integral is defined (for a piece-wise constant integrandσ(s,ω)) as

b∫a

σ(s,ω)dW (s) =n−1

∑k=0

σ(tk ,ω)(W (tk+1)−W (tk )). (11)

General functions are approximated by piece-wise constantfunctions, while letting the discretization tend to zero. The limitis computed in L2(P) sense.


The Ito integral

The Ito integral is defined (for a piece-wise constant integrandσ(s,ω)) as

b∫a

σ(s,ω)dW (s) =n−1

∑k=0

σ(tk ,ω)(W (tk+1)−W (tk )). (11)

General functions are approximated by piece-wise constantfunctions, while letting the discretization tend to zero. The limitis computed in L2(P) sense.


Properties

Stochastic integrals are martingales.

Definition: A stochastic process {X (t), t ≥ 0} is called amartingale with respect to a filtration {F (t)}t≥0 if

I X (t) is F (t)-measurable for all tI E [|X (t)|] < ∞ for all t , andI E [X (t)|F (s)] = X (s) for all s ≤ t .

Proof:

E[X (t)|F (s)] = E[X (s) + (X (t)−X (t)|F (s)] (12)

= X (s) + E[∫

σ(u,ω)dW (u)|F (s)] (13)

= X (s) + E[E[n−1

∑k=0

σ(tk ,ω)(W (tk+1)−W (tk ))F (tk )]|F (s)] = X (s)

(14)


Properties

Stochastic integrals are martingales.

Definition: A stochastic process {X (t), t ≥ 0} is called amartingale with respect to a filtration {F (t)}t≥0 if

I X (t) is F (t)-measurable for all tI E [|X (t)|] < ∞ for all t , andI E [X (t)|F (s)] = X (s) for all s ≤ t .

Proof:

E[X (t)|F (s)] = E[X (s) + (X (t)−X (t)|F (s)] (12)

= X (s) + E[∫

σ(u,ω)dW (u)|F (s)] (13)

= X (s) + E[E[n−1

∑k=0

σ(tk ,ω)(W (tk+1)−W (tk ))F (tk )]|F (s)] = X (s)

(14)


Other properties (Theorem 7.1)

I Stochastic integrals are linear operatorsI The unconditional expectation of a stochastic integral is

zeroI Stochastic integrals are measurable wrt the Filtration of the

driving Brownian motionI The Ito isometry is useful when computing the covariance

E

[(∫σ(s)dW (s)

)2]

=∫

E[σ

2(s)]

ds (15)


Solving SDEs

Generally rather difficult... Use the definitions if possible.The Ito formula states the if

dX (t) = µ(X (t))dt + σ(X (t))dW (t) (16)

Y (t) = F (t ,X (t)) ∈ C1,2 (17)

Then the Ito formula applies

dY (t) =

(Ft + µFX +

12

σσT FXX

)dt + σFX dW (t) (18)

where the dependence on X (t) is suppressed andFt = ∂F/∂ t ,FX = ∂F/∂X , . . .“Proof”: Essentially Taylor expansions, and using that X andhence Y is continuous.


Solving SDEs


dX (t) = µ(X (t))dt + σ(X (t))dW (t) (16)

Y (t) = F (t ,X (t)) ∈ C1,2 (17)


dY (t) =

(Ft + µFX +

12

σσT FXX

)dt + σFX dW (t) (18)



Solving SDEs


dX (t) = µ(X (t))dt + σ(X (t))dW (t) (16)

Y (t) = F (t ,X (t)) ∈ C1,2 (17)


dY (t) =

(Ft + µFX +

12

σσT FXX

)dt + σFX dW (t) (18)



Solving SDEs


dX (t) = µ(X (t))dt + σ(X (t))dW (t) (16)

Y (t) = F (t ,X (t)) ∈ C1,2 (17)


dY (t) =

(Ft + µFX +

12

σσT FXX

)dt + σFX dW (t) (18)



Solving SDEs


dX (t) = µ(X (t))dt + σ(X (t))dW (t) (16)

Y (t) = F (t ,X (t)) ∈ C1,2 (17)


dY (t) =

(Ft + µFX +

12

σσT FXX

)dt + σFX dW (t) (18)



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Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete