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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.
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.
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.
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.
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.
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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!
Erik Lindström Lecture on Stochastic Differential Equations
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!
Erik Lindström Lecture on Stochastic Differential Equations
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!
Erik Lindström Lecture on Stochastic Differential Equations
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!
Erik Lindström Lecture on Stochastic Differential Equations
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!
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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)
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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.
Erik Lindström Lecture on Stochastic Differential Equations
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