Stochastic Differential Equations SIMBA, Barcelona. David Ba nos - PowerPoint PPT Presentation

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equations SIMBA, Barcelona. David Ba˜ nos April 7th, 2014. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Table of contents 1 Quick review on ODE’s 2 Brownian motion 3 Densities of the solution SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Ordinary Differential Equation Let f : [ t 0 , T ] × R d → R d and x : [0 , T ] → R d x ( t 0 ) = x 0 ∈ R d . dx ( t ) = f ( t , x ( t )) dt , t 0 � t � T , (1) If a solution to the Cauchy problem (1) exists we can write � t x ( t ) = x 0 + f ( s , x ( s )) ds , t 0 � t � T . (2) t 0 SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Ordinary Differential Equation Let f : [ t 0 , T ] × R d → R d and x : [0 , T ] → R d x ( t 0 ) = x 0 ∈ R d . dx ( t ) = f ( t , x ( t )) dt , t 0 � t � T , (1) If a solution to the Cauchy problem (1) exists we can write � t x ( t ) = x 0 + f ( s , x ( s )) ds , t 0 � t � T . (2) t 0 Example If f ( s , x ( s )) = x ( s ) then (1) has a closed form solution given by x ( t ) = x 0 e t . SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Existence and Uniqueness of Solutions Picard-Lindel¨ of theorem (suff. cond. for local existence and uniqueness) Peano’s theorem (suff. cond. for existence) Carath´ eodory’s theorem (weaker version of Peano’s theorem) Okamura’s theorem (nec. an suff. conditions for uniqueness) ... SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Brownian motion In 1827, while examining grains of pollen of the plant Clarkia pulchella suspended in water under a microscope, Brown observed small particles ejected from the pollen grains , executing a continuous jittery motion . He then observed the same motion in particles of inorganic matter, enabling him to rule out the hypothesis that the effect was life-related. Although Brown did not provide a theory to explain the motion, and Jan Ingenhousz already had reported a similar effect using charcoal particles, in German and French publications of 1784 and 1785, the phenomenon is now known as Brownian motion . SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Brownian motion and SDE’s Let (Ω , F , P ) be a probability space. A stochastic process B : [0 , T ] × Ω → R d defined on (Ω , F , P ) is said to be a standard Brownian motion (moviment Browni` a) or a Wiener process if 1 B 0 = 0, P − a . s . 2 Given two times s , t ∈ [0 , T ], s < t , the law of B t + s − B s is the same as B t . 3 The increments B t − B s and B v − B u are independent for all u < v , s < t . 4 B t ∼ N (0 , t ) for all t ∈ [0 , T ]. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Brownian motion and SDE’s Let (Ω , F , P ) be a probability space. A stochastic process B : [0 , T ] × Ω → R d defined on (Ω , F , P ) is said to be a standard Brownian motion (moviment Browni` a) or a Wiener process if 1 B 0 = 0, P − a . s . 2 Given two times s , t ∈ [0 , T ], s < t , the law of B t + s − B s is the same as B t . 3 The increments B t − B s and B v − B u are independent for all u < v , s < t . 4 B t ∼ N (0 , t ) for all t ∈ [0 , T ]. In addition, we can choose a version such that B t is almost surely continuous. The existence of a stochastic process defined as above is not immediate (Kolmogorov’s existence theorem). SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Brownian motion To have intuition working with { B t ( ω ) , t ∈ [0 , T ] , ω ∈ Ω } we present four sample paths of a standard Brownian motion. Figure: Four realizations of a standard Brownian motion on the interval [0 , 1]. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation A (ordinary) stochastic differential equation with additive noise is an equation of the form: dX t = b ( t , X t ) dt + σ dB t , t ∈ [0 , T ] , (3) X 0 = x ∈ R d where σ is a parameter often called volatility . SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation A (ordinary) stochastic differential equation with additive noise is an equation of the form: dX t = b ( t , X t ) dt + σ dB t , t ∈ [0 , T ] , (3) X 0 = x ∈ R d where σ is a parameter often called volatility . Then if a solution to (4) exists we write � t X t = x + b ( s , X s ) ds + σ B t . 0 SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation A (ordinary) stochastic differential equation with additive noise is an equation of the form: dX t = b ( t , X t ) dt + σ dB t , t ∈ [0 , T ] , (3) X 0 = x ∈ R d where σ is a parameter often called volatility . Then if a solution to (4) exists we write � t X t = x + b ( s , X s ) ds + σ B t . 0 Observe that for each ω ∈ Ω � t X t ( ω ) = x + b ( s , X s ( ω )) ds + σ B t ( ω ) . 0 SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation A (ordinary) stochastic differential equation is an equation of the form: dX t = b ( t , X t ) dt + σ ( t , X t ) dB t , t ∈ [0 , T ] , (4) X 0 = x ∈ R d where b : [0 , T ] × R d → R d is a measurable function and σ : [0 , T ] × R d → R d × m is a suitable function. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation A (ordinary) stochastic differential equation is an equation of the form: dX t = b ( t , X t ) dt + σ ( t , X t ) dB t , t ∈ [0 , T ] , (4) X 0 = x ∈ R d where b : [0 , T ] × R d → R d is a measurable function and σ : [0 , T ] × R d → R d × m is a suitable function. Then if a solution to (4) exists we write � t � t X t = x + b ( s , X s ) ds + σ ( s , X s ) dB s . 0 0 SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation Figure: Two samples of Brownian motion with drift at different starting points. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic Differential Equation Figure: Three samples fo a geometric Brownian motion with µ = 0 . 05 and σ = 0 . 02. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Path properties of the Brownian motion The function the function t �→ B t ( ω ) has the following properties: Takes both strictly positive and strictly negative numbers on (0 , ε ) for every ε > 0. It is continuous everywhere but differentiable nowhere. It has infinite variation. Finite quadratic variation. The set of zeros is a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2. H¨ older-continuous paths of index α < 1 / 2. Hausdorff dimension 1.5. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Stochastic integral Let B t ( ω ), t ∈ [0 , T ], ω ∈ Ω be a standard Brownian motion. Consider a process X t satisfying some conditions. Then � T � X t dB t := lim X t i ( B t i +1 − B t i ) (Itˆ o integral) n →∞ 0 [ t i , t i +1 ] ∈ π n � T � X t dB t := lim X ti + ti +1 ( B t i +1 − B t i ) (Stratonovich integral) n →∞ 0 2 [ t i , t i +1 ] ∈ π n The convergence above is in probability (in fact, in L 2 (Ω)). SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations

Quick review on ODE’s Brownian motion Densities of the solution Back to ODE theory Consider the Cauchy problem � dx ( t ) = b ( t , x ( t )) dt , t ∈ [0 , T ] , x ( t 0 ) = x 0 ∈ R . Theorem (Picard-Lindel¨ of) If b is continuous in t and Lipschitz continuous in x then there exists a unique (local) strong solution to the Cauchy problem above. SIMBA, Barcelona. David Ba˜ nos Stochastic Differential Equations