The document discusses a stochastic differential equation (SDE) with Markovian switching that is used to model financial market behavior subject to random regime shifts. Key results shown include: 1) The SDE model exhibits the same long-run growth and deviation properties as classical geometric Brownian motion, obeying the law of the iterated logarithm. 2) Mathematical analysis is presented of the SDE with Markovian switching. 3) The results are applied to financial market models to account for random regime shifts between bullish and bearish states and changes in market sentiment.

1. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Large Deviations in Dynamical Systems
Part 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 Problems
Theory, Modelling and Numerics
University of Chester
Oct 31st - Nov 2nd 2007
Supported by the Irish Research Council for Science, Engineering and Technology

2. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
Introduction
1
Mathematical Results
2
Application to Financial Market Models
3
Extensions and future work
4

3. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
Introduction
1
Mathematical Results
2
Application to Financial Market Models
3
Extensions and future work
4

4. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large fluctuations of a stochastic
differential equation with Markovian switching, concentrating
on processes which obey the law of the iterated logarithm:
|X (t)|
lim sup √ =c a.s.
2t log log t
t→∞
The results are applied to financial market models which are
subject to random regime shifts (bearish to bullish) and to
changes in market sentiment.
We show that the security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.

5. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large fluctuations of a stochastic
differential equation with Markovian switching, concentrating
on processes which obey the law of the iterated logarithm:
|X (t)|
lim sup √ =c a.s.
2t log log t
t→∞
The results are applied to financial market models which are
subject to random regime shifts (bearish to bullish) and to
changes in market sentiment.
We show that the security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.

6. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Introduction
We consider the size of the large fluctuations of a stochastic
differential equation with Markovian switching, concentrating
on processes which obey the law of the iterated logarithm:
|X (t)|
lim sup √ =c a.s.
2t log log t
t→∞
The results are applied to financial market models which are
subject to random regime shifts (bearish to bullish) and to
changes in market sentiment.
We show that the security model exhibits the same long-run
growth and deviation properties as conventional geometric
Brownian motion.

7. 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
log S∗ (t) 1
= µ − σ2,
lim a.s. (1)
t 2
t→∞
The maximum size of the large deviations from this growth
trend obey the law of the iterated logarithm
| log S∗ (t) − (µ − 1 σ 2 )t|
2
√
lim sup = σ, a.s. (2)
2t log log t
t→∞

8. 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
log S∗ (t) 1
= µ − σ2,
lim a.s. (1)
t 2
t→∞
The maximum size of the large deviations from this growth
trend obey the law of the iterated logarithm
| log S∗ (t) − (µ − 1 σ 2 )t|
2
√
lim sup = σ, a.s. (2)
2t log log t
t→∞

9. 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
log S∗ (t) 1
= µ − σ2,
lim a.s. (1)
t 2
t→∞
The maximum size of the large deviations from this growth
trend obey the law of the iterated logarithm
| log S∗ (t) − (µ − 1 σ 2 )t|
2
√
lim sup = σ, a.s. (2)
2t log log t
t→∞

10. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
Introduction
1
Mathematical Results
2
Application to Financial Market Models
3
Extensions and future work
4

11. 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 Markovian
switching
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 obeying
local Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with finite
state space S and is independent of the Brownian motion B.
Under these conditions, there exists a unique continuous and
adapted process which satisfies (3).

12. 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 Markovian
switching
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 obeying
local Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with finite
state space S and is independent of the Brownian motion B.
Under these conditions, there exists a unique continuous and
adapted process which satisfies (3).

13. 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 Markovian
switching
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 obeying
local Lipschitz continuity and linear growth conditions,
Y is an irreducible continuous-time Markov chain with finite
state space S and is independent of the Brownian motion B.
Under these conditions, there exists a unique continuous and
adapted process which satisfies (3).

14. 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, ∞)
0 < K2 ≤ g 2 (x, y , t) ≤ K1
xf (x, y , t) ≤ ρ,
then X satisfies
|X (t)|
lim sup √ ≤ K1 , a.s.
2t log log t
t→∞

15. 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, ∞)
0 < K2 ≤ g 2 (x, y , t) ≤ K1
xf (x, y , t) ≤ ρ,
then X satisfies
|X (t)|
lim sup √ ≤ K1 , a.s.
2t log log t
t→∞

16. 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, ∞)
0 < K2 ≤ g 2 (x, y , t) ≤ K1
xf (x, y , t) ≤ ρ,
then X satisfies
|X (t)|
lim sup √ ≤ K1 , a.s.
2t log log t
t→∞

17. 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
xf (x, y , t) 1
=: L > −
inf 2 (x, y , t)
(x,y ,t)∈R×S×[0,∞) g 2
Then X satisfies
|X (t)|
lim sup √ ≥ K2 , a.s.
2t log log t
t→∞

18. 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
xf (x, y , t) 1
=: L > −
inf 2 (x, y , t)
(x,y ,t)∈R×S×[0,∞) g 2
Then X satisfies
|X (t)|
lim sup √ ≥ K2 , a.s.
2t log log t
t→∞

19. 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 an
appropriate change of time and scale, are recurrent and
stationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.

20. 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 an
appropriate change of time and scale, are recurrent and
stationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.

21. 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 an
appropriate change of time and scale, are recurrent and
stationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.

22. 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 an
appropriate change of time and scale, are recurrent and
stationary processes whose dynamics are not determined by Y .
The large deviations of these processes are determined by
means of a classical theorem of Motoo.

23. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
Introduction
1
Mathematical Results
2
Application to Financial Market Models
3
Extensions and future work
4

24. 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 price
model, where the security price S obeys
t≥0
dS(t) = µS(t)dt + S(t)dX (t),
Here we consider the special case
t≥0
dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t),
with X (0) = 0 and γ : S → R {0}.
Under the conditions
xf (x, y , t)
≤ ρ and
sup
γ 2 (y )
(x,y ,t)∈R×S×[0,∞)
xf (x, y , t) 1
>− ,
inf 2 (y )
γ 2
(x,y ,t)∈R×S×[0,∞)

25. 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 price
model, where the security price S obeys
t≥0
dS(t) = µS(t)dt + S(t)dX (t),
Here we consider the special case
t≥0
dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t),
with X (0) = 0 and γ : S → R {0}.
Under the conditions
xf (x, y , t)
≤ ρ and
sup
γ 2 (y )
(x,y ,t)∈R×S×[0,∞)
xf (x, y , t) 1
>− ,
inf 2 (y )
γ 2
(x,y ,t)∈R×S×[0,∞)

26. 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 price
model, where the security price S obeys
t≥0
dS(t) = µS(t)dt + S(t)dX (t),
Here we consider the special case
t≥0
dX (t) = f (X (t), Y (t), t) dt + γ(Y (t)) dB(t),
with X (0) = 0 and γ : S → R {0}.
Under the conditions
xf (x, y , t)
≤ ρ and
sup
γ 2 (y )
(x,y ,t)∈R×S×[0,∞)
xf (x, y , t) 1
>− ,
inf 2 (y )
γ 2
(x,y ,t)∈R×S×[0,∞)

27. 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
t≥0
dS(t) = µS(t)dt + S(t)dX (t),
with S(0) = s0 > 0, where X is defined on the previous slide.
Then:
1
log S(t) 12
= µ − σ∗ ,
lim a.s.
t 2
t→∞
2
t
| log S(t) − (µt − 1 0 γ 2 (Y (s))ds)|
2
√
lim sup = σ∗
2t log log t
t→∞
where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability
2
distribution of Y .

28. 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
t≥0
dS(t) = µS(t)dt + S(t)dX (t),
with S(0) = s0 > 0, where X is defined on the previous slide.
Then:
1
log S(t) 12
= µ − σ∗ ,
lim a.s.
t 2
t→∞
2
t
| log S(t) − (µt − 1 0 γ 2 (Y (s))ds)|
2
√
lim sup = σ∗
2t log log t
t→∞
where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability
2
distribution of Y .

29. 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
t≥0
dS(t) = µS(t)dt + S(t)dX (t),
with S(0) = s0 > 0, where X is defined on the previous slide.
Then:
1
log S(t) 12
= µ − σ∗ ,
lim a.s.
t 2
t→∞
2
t
| log S(t) − (µt − 1 0 γ 2 (Y (s))ds)|
2
√
lim sup = σ∗
2t log log t
t→∞
where σ∗ = j∈S γ 2 (j)πj , and π is the stationary probability
2
distribution of Y .

30. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (which
introduces regime shifts) and the X -dependent drift term f
(which introduces inefficiency), we have shown that the new
market model obeys the same asymptotic properties of
standard geometric Brownian motion models.
However, the analysis is now more complicated because the
increments are neither stationary nor Gaussian, and it is not
possible to obtain an explicit solution.

31. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Comments
Despite the presence of the Markov process Y (which
introduces regime shifts) and the X -dependent drift term f
(which introduces inefficiency), we have shown that the new
market model obeys the same asymptotic properties of
standard geometric Brownian motion models.
However, the analysis is now more complicated because the
increments are neither stationary nor Gaussian, and it is not
possible to obtain an explicit solution.

32. Introduction Mathematical Results Application to Financial Market Models Extensions and future work
Outline
Introduction
1
Mathematical Results
2
Application to Financial Market Models
3
Extensions and future work
4

33. 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 and
negative large fluctuations under the assumption that the drift
is integrable.
A result can be proven about the large fluctuations of the
incremental returns when the diffusion coefficient depends on
X , Y and t.
Can be extended to the case in which the diffusion coefficient
in X depends not only on the Markovian switching term but
also on a delay term, once that diffusion coefficient remains
bounded.
We hope in the future to extend the model to the case of
unbounded noise.

34. 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 and
negative large fluctuations under the assumption that the drift
is integrable.
A result can be proven about the large fluctuations of the
incremental returns when the diffusion coefficient depends on
X , Y and t.
Can be extended to the case in which the diffusion coefficient
in X depends not only on the Markovian switching term but
also on a delay term, once that diffusion coefficient remains
bounded.
We hope in the future to extend the model to the case of
unbounded noise.

35. 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 and
negative large fluctuations under the assumption that the drift
is integrable.
A result can be proven about the large fluctuations of the
incremental returns when the diffusion coefficient depends on
X , Y and t.
Can be extended to the case in which the diffusion coefficient
in X depends not only on the Markovian switching term but
also on a delay term, once that diffusion coefficient remains
bounded.
We hope in the future to extend the model to the case of
unbounded noise.

36. 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 and
negative large fluctuations under the assumption that the drift
is integrable.
A result can be proven about the large fluctuations of the
incremental returns when the diffusion coefficient depends on
X , Y and t.
Can be extended to the case in which the diffusion coefficient
in X depends not only on the Markovian switching term but
also on a delay term, once that diffusion coefficient remains
bounded.
We hope in the future to extend the model to the case of
unbounded noise.