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Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Large Deviations in Dynamical SystemsPart 1: A market model with Markovian switching
Terry Lynch1
(joint work with J. Appleby1, X. Mao2 and H. Wu1)
1 Dublin City University, Ireland.
2 Strathclyde University, Glasgow, U.K.
Stochastic Evolutionary ProblemsTheory, Modelling and Numerics
University of ChesterOct 31st - Nov 2nd 2007
Supported by the Irish Research Council for Science, Engineering and Technology
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large fluctuations of a stochasticdifferential equation with Markovian switching, concentratingon processes which obey the law of the iterated logarithm:
lim supt→∞
|X (t)|√2t log log t
= c a.s.
The results are applied to financial market models which aresubject to random regime shifts (bearish to bullish) and tochanges in market sentiment.
We show that the security model exhibits the same long-rungrowth and deviation properties as conventional geometricBrownian motion.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large fluctuations of a stochasticdifferential equation with Markovian switching, concentratingon processes which obey the law of the iterated logarithm:
lim supt→∞
|X (t)|√2t log log t
= c a.s.
The results are applied to financial market models which aresubject to random regime shifts (bearish to bullish) and tochanges in market sentiment.
We show that the security model exhibits the same long-rungrowth and deviation properties as conventional geometricBrownian motion.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large fluctuations of a stochasticdifferential equation with Markovian switching, concentratingon processes which obey the law of the iterated logarithm:
lim supt→∞
|X (t)|√2t log log t
= c a.s.
The results are applied to financial market models which aresubject to random regime shifts (bearish to bullish) and tochanges in market sentiment.
We show that the security model exhibits the same long-rungrowth and deviation properties as conventional geometricBrownian motion.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Classical geometric Brownian motion
Characterised as the unique solution of the linear S.D.E
dS∗(t) = µS∗(t) dt + σS∗(t) dB(t)
where µ is the instantaneous mean rate of growth of the price,σ its instantaneous volatility and S∗(0) > 0.
S∗ grows exponentially according to
limt→∞
log S∗(t)
t= µ− 1
2σ2, a.s. (1)
The maximum size of the large deviations from this growthtrend obey the law of the iterated logarithm
lim supt→∞
| log S∗(t)− (µ− 12σ2)t|
√2t log log t
= σ, a.s. (2)
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Classical geometric Brownian motion
Characterised as the unique solution of the linear S.D.E
dS∗(t) = µS∗(t) dt + σS∗(t) dB(t)
where µ is the instantaneous mean rate of growth of the price,σ its instantaneous volatility and S∗(0) > 0.
S∗ grows exponentially according to
limt→∞
log S∗(t)
t= µ− 1
2σ2, a.s. (1)
The maximum size of the large deviations from this growthtrend obey the law of the iterated logarithm
lim supt→∞
| log S∗(t)− (µ− 12σ2)t|
√2t log log t
= σ, a.s. (2)
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Classical geometric Brownian motion
Characterised as the unique solution of the linear S.D.E
dS∗(t) = µS∗(t) dt + σS∗(t) dB(t)
where µ is the instantaneous mean rate of growth of the price,σ its instantaneous volatility and S∗(0) > 0.
S∗ grows exponentially according to
limt→∞
log S∗(t)
t= µ− 1
2σ2, a.s. (1)
The maximum size of the large deviations from this growthtrend obey the law of the iterated logarithm
lim supt→∞
| log S∗(t)− (µ− 12σ2)t|
√2t log log t
= σ, a.s. (2)
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
S.D.E with Markovian switching
We study the stochastic differential equation with Markovianswitching
dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t) (3)
where
X (0) = x0,
f , g : R× S× [0,∞) → R are continuous functions obeyinglocal Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with finitestate space S and is independent of the Brownian motion B.
Under these conditions, there exists a unique continuous andadapted process which satisfies (3).
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
S.D.E with Markovian switching
We study the stochastic differential equation with Markovianswitching
dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t) (3)
where
X (0) = x0,
f , g : R× S× [0,∞) → R are continuous functions obeyinglocal Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with finitestate space S and is independent of the Brownian motion B.
Under these conditions, there exists a unique continuous andadapted process which satisfies (3).
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
S.D.E with Markovian switching
We study the stochastic differential equation with Markovianswitching
dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t) (3)
where
X (0) = x0,
f , g : R× S× [0,∞) → R are continuous functions obeyinglocal Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with finitestate space S and is independent of the Brownian motion B.
Under these conditions, there exists a unique continuous andadapted process which satisfies (3).
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1
Let X be the unique adapted continuous solution satisfying
dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t).
If there exist ρ > 0 and real numbers K1 and K2 such that∀ (x , y , t) ∈ R× S× [0,∞)
xf (x , y , t) ≤ ρ, 0 < K2 ≤ g2(x , y , t) ≤ K1
then X satisfies
lim supt→∞
|X (t)|√2t log log t
≤√
K1, a.s.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1
Let X be the unique adapted continuous solution satisfying
dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t).
If there exist ρ > 0 and real numbers K1 and K2 such that∀ (x , y , t) ∈ R× S× [0,∞)
xf (x , y , t) ≤ ρ, 0 < K2 ≤ g2(x , y , t) ≤ K1
then X satisfies
lim supt→∞
|X (t)|√2t log log t
≤√
K1, a.s.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1
Let X be the unique adapted continuous solution satisfying
dX (t) = f (X (t),Y (t), t) dt + g(X (t),Y (t), t) dB(t).
If there exist ρ > 0 and real numbers K1 and K2 such that∀ (x , y , t) ∈ R× S× [0,∞)
xf (x , y , t) ≤ ρ, 0 < K2 ≤ g2(x , y , t) ≤ K1
then X satisfies
lim supt→∞
|X (t)|√2t log log t
≤√
K1, a.s.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1 continued
If, moreover, there exists an L ∈ R such that
inf(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
g2(x , y , t)=: L > −1
2
Then X satisfies
lim supt→∞
|X (t)|√2t log log t
≥√
K2, a.s.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Main results
Theorem 1 continued
If, moreover, there exists an L ∈ R such that
inf(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
g2(x , y , t)=: L > −1
2
Then X satisfies
lim supt→∞
|X (t)|√2t log log t
≥√
K2, a.s.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under anappropriate change of time and scale, are recurrent andstationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined bymeans of a classical theorem of Motoo.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under anappropriate change of time and scale, are recurrent andstationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined bymeans of a classical theorem of Motoo.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under anappropriate change of time and scale, are recurrent andstationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined bymeans of a classical theorem of Motoo.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline of proof
The proofs rely on
time-change and stochastic comparison arguments,
constructing upper and lower bounds on |X | which, under anappropriate change of time and scale, are recurrent andstationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined bymeans of a classical theorem of Motoo.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Security Price Model
The large deviation results are now applied to a security pricemodel, where the security price S obeys
dS(t) = µS(t)dt + S(t)dX (t), t ≥ 0
Here we consider the special case
dX (t) = f (X (t),Y (t), t) dt + γ(Y (t)) dB(t), t ≥ 0
with X (0) = 0 and γ : S → R \ {0}.Under the conditions
sup(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
γ2(y)≤ ρ and
inf(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
γ2(y)> −1
2,
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Security Price Model
The large deviation results are now applied to a security pricemodel, where the security price S obeys
dS(t) = µS(t)dt + S(t)dX (t), t ≥ 0
Here we consider the special case
dX (t) = f (X (t),Y (t), t) dt + γ(Y (t)) dB(t), t ≥ 0
with X (0) = 0 and γ : S → R \ {0}.Under the conditions
sup(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
γ2(y)≤ ρ and
inf(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
γ2(y)> −1
2,
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Security Price Model
The large deviation results are now applied to a security pricemodel, where the security price S obeys
dS(t) = µS(t)dt + S(t)dX (t), t ≥ 0
Here we consider the special case
dX (t) = f (X (t),Y (t), t) dt + γ(Y (t)) dB(t), t ≥ 0
with X (0) = 0 and γ : S → R \ {0}.Under the conditions
sup(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
γ2(y)≤ ρ and
inf(x ,y ,t)∈R×S×[0,∞)
xf (x , y , t)
γ2(y)> −1
2,
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Fluctuations in the Markovian switching model
Theorem 2
Let S be the unique continuous adapted process governed by
dS(t) = µS(t)dt + S(t)dX (t), t ≥ 0
with S(0) = s0 > 0, where X is defined on the previous slide.Then:
1
limt→∞
log S(t)
t= µ− 1
2σ2∗, a.s.
2
lim supt→∞
| log S(t)− (µt − 12
∫ t0 γ2(Y (s))ds)|
√2t log log t
= σ∗
where σ2∗ =
∑j∈S γ2(j)πj , and π is the stationary probability
distribution of Y .
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Fluctuations in the Markovian switching model
Theorem 2
Let S be the unique continuous adapted process governed by
dS(t) = µS(t)dt + S(t)dX (t), t ≥ 0
with S(0) = s0 > 0, where X is defined on the previous slide.Then:
1
limt→∞
log S(t)
t= µ− 1
2σ2∗, a.s.
2
lim supt→∞
| log S(t)− (µt − 12
∫ t0 γ2(Y (s))ds)|
√2t log log t
= σ∗
where σ2∗ =
∑j∈S γ2(j)πj , and π is the stationary probability
distribution of Y .
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Fluctuations in the Markovian switching model
Theorem 2
Let S be the unique continuous adapted process governed by
dS(t) = µS(t)dt + S(t)dX (t), t ≥ 0
with S(0) = s0 > 0, where X is defined on the previous slide.Then:
1
limt→∞
log S(t)
t= µ− 1
2σ2∗, a.s.
2
lim supt→∞
| log S(t)− (µt − 12
∫ t0 γ2(Y (s))ds)|
√2t log log t
= σ∗
where σ2∗ =
∑j∈S γ2(j)πj , and π is the stationary probability
distribution of Y .
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (whichintroduces regime shifts) and the X -dependent drift term f(which introduces inefficiency), we have shown that the newmarket model obeys the same asymptotic properties ofstandard geometric Brownian motion models.
However, the analysis is now more complicated because theincrements are neither stationary nor Gaussian, and it is notpossible to obtain an explicit solution.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (whichintroduces regime shifts) and the X -dependent drift term f(which introduces inefficiency), we have shown that the newmarket model obeys the same asymptotic properties ofstandard geometric Brownian motion models.
However, the analysis is now more complicated because theincrements are neither stationary nor Gaussian, and it is notpossible to obtain an explicit solution.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
1 Introduction
2 Mathematical Results
3 Application to Financial Market Models
4 Extensions and future work
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions and future work
We can also get upper and lower bounds on both positive andnegative large fluctuations under the assumption that the driftis integrable.
A result can be proven about the large fluctuations of theincremental returns when the diffusion coefficient depends onX ,Y and t.
Can be extended to the case in which the diffusion coefficientin X depends not only on the Markovian switching term butalso on a delay term, once that diffusion coefficient remainsbounded.
We hope in the future to extend the model to the case ofunbounded noise.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions and future work
We can also get upper and lower bounds on both positive andnegative large fluctuations under the assumption that the driftis integrable.
A result can be proven about the large fluctuations of theincremental returns when the diffusion coefficient depends onX ,Y and t.
Can be extended to the case in which the diffusion coefficientin X depends not only on the Markovian switching term butalso on a delay term, once that diffusion coefficient remainsbounded.
We hope in the future to extend the model to the case ofunbounded noise.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions and future work
We can also get upper and lower bounds on both positive andnegative large fluctuations under the assumption that the driftis integrable.
A result can be proven about the large fluctuations of theincremental returns when the diffusion coefficient depends onX ,Y and t.
Can be extended to the case in which the diffusion coefficientin X depends not only on the Markovian switching term butalso on a delay term, once that diffusion coefficient remainsbounded.
We hope in the future to extend the model to the case ofunbounded noise.
Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Extensions and future work
We can also get upper and lower bounds on both positive andnegative large fluctuations under the assumption that the driftis integrable.
A result can be proven about the large fluctuations of theincremental returns when the diffusion coefficient depends onX ,Y and t.
Can be extended to the case in which the diffusion coefficientin X depends not only on the Markovian switching term butalso on a delay term, once that diffusion coefficient remainsbounded.
We hope in the future to extend the model to the case ofunbounded noise.