Diffusion Processes and Linear Stochastic Equations

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Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary

Basics of simulation and statistic of dynamic Systems- Diﬀusion processes and linear stochastic equations

by Dany DJEUDEU

TU Dortmund Faculty of Statistics

November 2014 Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary

1 Introduction

2 Diffusion processes: Definition General definition Some remarks Some examples of diffusion processes

3 The Stochastic differential equation Solution of the stochastic differential equation: The integral form Existence and uniqueness of solutions to the diffusion equation

4 properties of diﬀusion processes and applications Markov property of a diﬀusion process Ito Formula Applications The Lamperti transform

5 Summary Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Introduction

differential equations are known to describe the time evolution of some phenomenon: diseases for instance It is frequently the case that economic and financial considerations will suggest that a stock price, exchange rate, interest rate, or other economic variables evolve in time according to a stochastic differential equation of the form

dXt = b(t, Xt )dt + σ(t, Xt )dBt (1)

Bt is the standard Brownian motion and b and σ are given functions of time t and the current state x. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary General definition

Deﬁnition 2.1 ([4])

A stochastic real-valued process (Xt )t≥0 is said to be a diﬀusion process if it satisﬁes the following conditions:

A)( Xt )t≥0 is a markov-process B) There exist limits:

1 b(x, t) = lim E((X (t+∆)−X (t))|X (t) = x) (2) ∆→0 ∆ 1 σ2(x, t) = lim E{(X (t + ∆) − X (t))2|X (t) = x} ∆→0 ∆ (3)

C) X (t)t≥0 is a continuous process (P(|Xt − Xs | ≥ |Xs = x) = o(t − s)) b(x, t) is called The drift (coefficient, parameter) and σ2(x, t) is the diffusion (coefficient, parameter) Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Some remarks

Non differentiability of diffusion processes The definition of a diffusion process suggests a relationship of the following form ([4]):

dXt = b(s, Xs )dt + σ(s, Xs )dWt (4) Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Some examples of diffusion processes

The standard Wiener process is a diffusion process with drift 0 and diffusion parameter 1. The class of Gaussian processes is an important class of diffusion processes ([5]) Consider the stochastic differential equation

1 2 3 3 dXt = 3Xt dt + 3Xt dWt (5)

with the initial condition X0 = 0. 3 Clearly, the process Xt ≡ 0 is a solution. But so is Xt = Wt . The problem in this example is that the coefficients 1 2 3 3 b(t) = 3Xt and σ(t) = 3Xt although continuous in x, are not smooth enough at x = 0. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Solution to the stochastic differential equation in integral form

We concentrate on the dynamic of diﬀusion processes We will be interested in solving stochastic diﬀerential equations (or SDE s) of the following form:

dXt = b(s, Xs )dt + σ(s, Xs )dWt (6)

with initial conditions X0 = Z. W : [0, ∞)xΩ → R with (t, ω) → W (t)(ω) the Brownian motion and Z a random variable with distribution µ, independent of the σ-Algebra generated by W and with a ﬁnite second moment.

b with (t, ω) → b(t, Xt ω) and σ with (t, ω) → σ(t, Xt (ω)) are deterministic measurable applications. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Solution to the stochastic differential equation in integral form

A solution X of the SDE (6) is a continuous stochastic process which satisﬁes the integral equation

Z t Z t Xt = X0 + b(s, Xs )ds + σ(s, Xs )dWs f .s. ∀t ≥ 0 (7) 0 0 R t the integral 0 b(s, Xs )ds is the usual Riemann-Integral and R t 0 σ(s, Xs )dWs the Ito-Integral. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary The stochastic differential equation: Existence and uniqueness of solutions

We consider the SDE

dXt = b(s, Xs )dt + σ(s, Xs )dWt (8)

with initial conditions X0 = Z. with b and σ deﬁned like in the previous section. We suppose that

Z T P{ sup(|b(t, x)| + σ2(t, x))dt < ∞} = 1 (9) 0 x≤R for all T , R ∈ [0, ∞) because (7) is an Ito process. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Assumptions

Assumption 3.1 (Global Lipschitz condition) There exists a constant K < +∞ such that for all x, y ∈ R and t ∈ [0, T ],

|b(t, x) − b(t, y)| + |σ(t, x) − σ(t, y)| < K|x − y|. (10)

Assumption 3.2 (Linear growth) There exists a constant C < +∞ such that for all x, y ∈ R and t ∈ [0, T ], |b(t, x)| + |σ(t, x)| < C(1 + |x|). (11) The linear growth condition controls the behavior of the solution so that Xt does not explode in a finite time. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Existence and uniqueness

Theorem 3.1 ([1]) Under the previous assumptions, the stochastic differential equation (8) has a unique, continuous, and adapted strong solution such that Z T 2 E |Xt | dt < ∞ (12) 0 Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Weak solution

Assumption 3.3 (Local assumptions)

For any x0 ∈ R , there exists a constant L(x0) > 0 and Bx0 > 0 such that: for all x ∈ R Local Lipschitz condition

|b(x) − b(x0)| + |σ(x) − σ(x0)| ≤ Lx0 |x − x0| (13)

Local linear growth

2 2 2xb(x) + σ (x) ≤ Bx0 (1 + x ) (14)

Theorem 3.2 Under the previous assumptions, the stochastic differential equation (8) has a unique weak solution [1]. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Transition probability

From the Markov property of the diffusion process, it is also possible to define the transition density from value x at time s to value y at time t by p(t, y|s, x) or, when convenient, as p(t − s, y|x). The transition density satisfies the Kolmogorov forward equation:

∂ ∂ 1 ∂2 p(t, y|s, x) = − (b(t, y)p(t, y|s, x))+ (σ2(t, x)p(t, y|s, x)) ∂t ∂y 2 ∂y 2 (15) and Kolmogorov backward equation

∂ ∂ 1 ∂2 − p(t, y|s, x) = b(t, x) p(t, y|s, x)+ σ2(t, x) p(t, y|s, x) ∂s ∂x 2 ∂x2 (16) Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Ito Formula

Useful in simulations It is the foundation of mathematical finance and stochastic calculus This formula can be seen as the stochastic version of a Taylor expansion of f (X ) stopped at the second order, where X is a diffusion process. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Ito formula

Lemma 4.1 (Ito formula) If f is a function that is two times diﬀerentiable on both t and x (f = f (t, x)), then

Z t Z t f (t, Xt ) = f (0, X0) + ft (s, Xs )ds + fx (s, Xs )dXs (17) 0 0 Z t 1 2 + fxx (s, Xs )(dXs ) 2 0 or, in differential form 1 df (t, X ) = f (t, X )dt + f (t, X )dX + f (t, X )(dX )2 (18) t t t x t t 2 xx t t ∂f (t, x) ∂f (t, x) ∂2f (t, x) f (t, x) = , f (t, x) = , f (t, x) = x ∂x t ∂t xx ∂x2 Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Proof of the Ito formula

Proof. We give here the steps of the proof. The Taylor expansion in one dimension is used, say

f (t, x) − f (s, x0) = (f (t, x) − f (s, x)) + (f (s, x) − f (s, x0))

Taylor = ft (s + α(t − s), x)(t − s) + fx (s, x0)(x − x0) 1 + f (s, x + β(x − x ))(x − x )2 2 xx 0 0 0

We ﬁrst use the partition of the intervals,

n X f (t, Xt ) − f (0, X0) = f (ti , Xti ) − f (ti−1, Xti−1 ) i=1 Then we use the Taylor expansion and the definition of the integrals Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Application of the Ito formula on the Brownian motion

Lemma 4.2

If Xt is the Brownian motion, this simpliﬁes to the following

Z t 1 Z t f (t, Wt ) = f (0, 0)+ (ft (s, Ws )+ fxx (s, Ws ))ds+ fx (s, Ws )dWs 0 2 0 (19) or, in differential form 1 df (t, W ) = (f (t, W ) + f (t, W ))dt + f (t, W )dW (20) t t t 2 xx t x t t Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Steps of the proof

(dt · dWt) and (dt)2 are of order O(dt), (dWt)2 behave like dt for the properties of the Brownian motion

(Wti − Wti−1 ) ' (ti − ti−1) Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Ito integral

Application 4.1 consider the function f (t, x) = f (x) = x2. The Ito formula applied to f (Wt) then leads to

Z t Z t 2 2 1 Wt = 0 + 2Ws dWs + 2ds 0 2 0 and therefore Z t 1 2 1 Ws dWs = Wt − t (21) 0 2 2 Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Linear stochastic differential equations

Linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution.

dXt = b1(t)Xt dt + σ1(t)Xt dWt (22) This equation is called a stochastic differential equation with multiplicative noise. We aim to solve this equation. We use the Ito formula to do so. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Linear stochastic differential equations

Theorem 4.1 Choosing f (x) = log(x) , the solution to (22) is given by

Z t Z t 1 2 Xt = X0 · exp (b1(s) − σ1(s))ds + σ1(s)dWs (23) 0 2 0 Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Application

Application 4.2 (geometric Brownian motion) It is now easy to derive the stochastic diﬀerential equation for the geometric Brownian motion. We have the Black-Scholes diﬀerential equation

dSt = rSt dt + σSt dWt (24)

b1 and σ1 are considered here constants. Consider the geometric Brownian motion

σ2 S = S exp (r − )t + σW , t > 0. (25) t 0 2 t

σ2 choosing f (t, x) == S0 exp{(r − 2 )t + σx, we easily prove with the Ito formula that the geometric Brownian motion is solution to the Black-Scholes equation (24). Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary The Lamperti transform

Theorem 4.2 Suppose we have the stochastic diﬀerential equation

dXt = b(t, Xt )dt + σ(Xt )dWt (26) where the diffusion coefficient depends only on the state variable. Such a stochastic differential equation can always be transformed into one with a unitary diffusion coefficient by applying the Lamperti transform

Z Xt 1 Yt = F (Xt ) = ds (27) z σ(s) Here z is any arbitrary value in the state space of X . Indeed, the process Yt solves the stochastic diﬀerential equation

b(t, Xt ) 1 dYt = ( − σx (Xt ))dt + dWt σ(Xt ) 2 Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Application: A trick for simulations

Application 4.3 Trick for simulation If possible, it is better to apply the Lamperti transform F to X and simulate Y = F (X ) by Euler scheme. This makes the simulation more stable and usually efficient ([6]). Then apply F −1(Y ) to obtain X on the original scale. Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary Summary

The diffusion process in one dimension General definition of a diffusion process: Solution to a stochastic differential equations Conditions for the resolution of those stochastic processes have been presented Problematic : Strictly makes the differential equation considered no sense ( the Brownian motion for example is not differentiable) Trick: Solution in the integral form, with the advantage that there is no differential term in the expression. Properties of diffusion processes: Markov property with Kolmogorov forward and backward equations Stress on the Ito form: basis of mathematical finance, Brownian motion solution to the Black-Scholes equation, Ito integral The Lamperti transform: Trick for simulations Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary

Stefano M. Iacus, Simulation and Inference for Stochastic Differential Equations. Springer- Verlag. New York. K. Webel, D. Wied (2011): Stochastische Prozesse. Eine EinführungfürStatistiker und Datenwissenschaftler,2011, XVI, 270S. 39 Abb..,ISBN 978-3-8349-2809-2 Øksendal, B. (1998): Stochastic Differential Equations. An Introduction with Applications,5th ed., Springer-Verlag, Berlin. http://kurser.math.su.se/moodle19/file.php/827/SP-Lecture- 12.pdf R. Coleman Stochastic Processes 1974 Lecture in Mathematics, Imperial College, University of London. Stefano M. Iacus 18 − 1 − 2008) Wien: Simulation and inference for SDEs,http://www.wu.ac.at/firm/talks/ws0708/sde.pdf Introduction Diffusion processes: Definition The Stochastic differential equation properties of diffusion processes and applications Summary

J. R. Movellan (2011) Tutorial on Stochastic Diﬀerential Equations, MPLab Tutorials Version 06.1.