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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.