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Lecture on Stochastic Differential Equations
Erik Lindström
FMS161/MASM18 Financial Statistics
Erik Lindström Lecture on Stochastic Differential Equations
![Page 2: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/2.jpg)
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
![Page 3: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/3.jpg)
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
![Page 4: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/4.jpg)
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
![Page 5: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/5.jpg)
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
![Page 6: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/6.jpg)
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
![Page 7: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/7.jpg)
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
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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
![Page 9: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/9.jpg)
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
![Page 10: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/10.jpg)
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
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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
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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
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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
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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
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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.
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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!
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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!
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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!
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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
![Page 20: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/20.jpg)
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
![Page 21: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/21.jpg)
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
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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
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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.
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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
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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
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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
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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
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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)
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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)
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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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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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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
![Page 33: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/33.jpg)
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
![Page 34: Lecture on Stochastic Differential EquationsErik Lindström Lecture on Stochastic Differential Equations Motivation I Continuous time models are more ’interpretable’ than discrete](https://reader033.vdocuments.net/reader033/viewer/2022060314/5f0b861a7e708231d430f07e/html5/thumbnails/34.jpg)
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