From Stochastic Calculus of Variations on Wiener space to Stochastic - PowerPoint PPT Presentation

From “Stochastic Calculus of Variations on Wiener space” to “Stochastic Calculus of Variations on Poisson space”. Maurizio Pratelli Department of Mathematics, University of Pisa pratelli@dm.unipi.it Brixen, July 16, 2007

Malliavin’s derivative: “Calculus of Variations approach” F ∈ L 2 (Ω) (a functional on Wiener space) Ω = C 0 (0 , T ) Cameron-Martin space CM : h ∈ CM if h ( t ) = � t h ∈ L 2 (0 , T ) 0 ˙ ˙ h ( s ) d s , and � h � CM = � ˙ h � L 2 .

Suppose that ∃ Z s ∈ L 2 (Ω × [0 , T ]) such that � T F ( ω + ǫh ) − F ( ω ) Z s ( ω )˙ lim = h ( s ) d s ǫ ε → 0 0 then F is derivable (in Malliavin’s sense) and Z s = D s F (more gen- erally D h F = � T 0 D s F ˙ h ( s ) d s ). With this definition, D is like a Fr´ echet derivative, but only along the directions in CM . Why?

Girsanov’s theorem : if � � T � T � d P ∗ h ( s )d W s − 1 h 2 ( s ) d s ˙ ˙ d P = exp = L T 2 0 0 law of � W ( . ) + h ( . ) � under P = law of W ( . ) under P ∗ (recall that on the canonical space W t ( ω ) = ω ( t )).

Introducing � T � T � � h ( s )d W s − ǫ 2 d P ǫ h 2 ( s ) d s = L ǫ ˙ ˙ d P = exp ǫ T 2 0 0 we have � F ( ω + ǫh ) − F ( ω ) � � � F ( ω ) L ǫ T − 1 I E = I E ǫ ǫ = � T L ǫ T − 1 0 ˙ since lim ǫ → 0 h ( s ) d W s ǫ

we obtain the integration by parts formula � � T � T � � � D s F ˙ ˙ I E h ( s ) d s = I E h ( s ) d W s F 0 0 h ( s ) can be replaced by H s ∈ L 2 � Ω × [0 , T ] � (˙ adapted) Intuitively : Malliavin’s calculs is the analysis of the variations of the paths along the directions supported by Girsanov’s theorem .

More generally: for k ∈ L 2 � � , define W ( k ) = � T 0 , T 0 k ( s ) d W s (Wiener’s integral) and define smooth functional F = φ � W ( k 1 ) , . . . , W ( k n ) � ( φ smooth). We obtain easily n � � W ( k 1 ) , . . . , W ( k n ) � ∂φ D s F = k i ( s ) ∂x i i =1

The operator D : S ⊂ L 2 � � → L 2 � � Ω Ω × [0 , T ] ( S space of smooth functionals) is closable (by the integration by parts formula ) (from now on we consider the closure). The adjoint operator D ∗ = δ : L 2 � � Ω × [0 , T ] is called divergence or Skorohod integral and D ∗ restricted to the adapted processes coincides with Ito’s integral . This is equivalent to the Clark-Ocone-Karatzas formula: if F is derivable � T � � � � F s F = I E[ F ] + I E d W s D s F 0

Very important is the so-called Chain rule : n � D s φ � F 1 , . . . , F n � � · · · � ∂φ D s F i = ∂x i i =1 R is derivable in the classic sense and F 1 , . . . , F n in the R n → I (if φ : I Malliavin’s sense).

Summing up: • integration by parts formula • D ∗ restricted to adapted processes coincides with Ito’s integral • Clark-Ocone-Karatzas formula • chain rule

A remark: Skorohod (anticipating) integral is not an integral (limit of Riemann’s sums). Intuitively � T � � � H s d X s = lim X t i +1 − X t i H t i 0 i Formula: if H s is adapted and F derivable � T � T � T � � δW s = F H s d W s − D s F H s d s FH s 0 0 0

Malliavin derivative in Chaos Expansion. An introductory example : alternative description on the space H 1 , 2 � � 0 , 2 π . f ∈ L 2 � 0 , 2 π � can be written � � � � | a k | 2 + | b k | 2 < + ∞ f = a 0 + a k cos kx + b k sin kx k ≥ 1 k If there is a finite number of terms � � k b k cos kx − k a k sin kx � f ′ = k

Therefore f is derivable (in weak sense) and f ′ ∈ L 2 � 0 , 2 π � if � � k 2 � | a k | 2 + | b k | 2 � � � f ′ = < + ∞ and b k cos kx − a k sin kx k k k • short and easy definition of (weak) derivative and of the space H 1 , 2 � 0 , 2 π � ; • the meaning of derivative is hidden.

Wiener Chaos Expansion � � S n = 0 < t 1 < · · · < t n < T , given f ∈ S n � � � J n ( f ) = d W t n d W t n − 1 · · · f ( t 1 , . . . , t n ) d W t 1 ]0 ,T ] ]0 ,t n ] ]0 ,t 2 ] � J n ( f ) 2 � � � � 2 � f I E = L 2 ( S n ) If C n = image of L 2 ( S n ) by J n , we have L 2 � Ω � = C 0 ⊕ C 1 ⊕ C 2 . . .

L 2 � [0 , T ] n � L 2 � [0 , T ] n � If � is the subspace of symmetric functions of � , define : � � I n ( f ) = n ! · · · f ( · · · ) d W t n · · · d W t 1 S n � I n ( f ) 2 � � � � 2 � f we have I E = n ! L 2 ([0 ,T ] n ) . F ∈ L 2 � Ω � � � can be written F = � with � n ≥ 0 n ! � f n � 2 L 2 < + ∞ n ≥ 0 I n f n

By direct calculus � � � � f n ( t 1 , . . . , t ) = n I n − 1 f n ( t 1 , . . . , t n − 1 , t ) D t I n We can define � . . . � D t F = � � n ≥ 1 n n ! � f n � 2 provided that L 2 < + ∞ n ≥ 1 n I n − 1 A similar characterization can be given for Skorohod integral .

With this approach: • concise and more elementary definitions of Malliavin’s derivative and divergence • some proof are easier, some more complicated (e.g. “chain rule” ) • the idea of derivative is hidden

A good result with this approach: Energy identity for Skorohod integral (Nualart, Pardoux, Shigekawa) �� � T � � T � T � T � 2 � � Z 2 I E = I E s d s + ( D t Z s + D s Z t ) d s d t Z s δW s 0 0 0 0 Other approaches: discretization (Ocone, Mallavin–Thalmaier), weak derivation ...

Main applications of Malliavin calculus: • Clark–Ocone–Karatzas formula (explicit characterization of the integrand) • Regularity of the law of some r.v. (solutions of S.D.E.) • Sensitivity analysis in Mathematical Finance (Monte Carlo weights for the Greek’s)

An idea of “sensitivity analysis” (Fourni´ e, Lasry, Lebuchoux, Lions, Touzi [99], and F.L.L.L. [01]): � F ζ �� � ∂ ζ F ζ � � f ′ � F ζ � ∂ ∂ζ I E = I E = f � � F ζ �� � D w ∂ ζ F ζ � � � ∂ ζ F ζ �� � F ζ � f D ∗ = I E = I E f . w D w F ζ D w F ζ � ∂ ζ F ζ � The “weight” W = D ∗ is independent of f (and not unique). w D w F ζ

In order to extend to more general situations (from diffusion models to jump–diffusion models), we need: • an integration by parts formula • chain rule .

Plain Poisson process Let P t be a Poisson process with jump times τ 1 < τ 2 < . . . ( σ i = τ i − τ i − 1 are independent exponential density) and N t = ( P t − t ) the compensated Poisson. Point of view of Chaos Expansion: Starting from � � � � J n ( f ) = d N t n d N t n − 1 · · · f ( t 1 , . . . , t n ) d N t 1 ]0 ,T ] ]0 ,t n [ ]0 ,t 2 [

A similar theory, based on chaotic representation , can be developed w.r.t. N t (Lokka, Oksendal and ...) • similar definition of derivative D c and Skorohod integral • ( D c ) ∗ coincides with ordinary stochastic integrals on predictable processes • Clark–Ocone–Karatzas formula

A serious drawback: the chain rule is not satisfied . In fact, the “chaotic” derivative satisfies the formula � � D c = F D c t G + G D c t F + D c t F D c FG t G t (Chain rule is ( morally ) equivalent to the formula D t ( FG ) = FD t G + GD t F ).

An alternative point of view: Variations on the paths (via Girsanov theorem) Given h ( t ) = � t h ∈ L 2 (0 , T ) and ˙ 0 ˙ h ( s ) d s , ˙ h uniformly bounded from below, consider a perturbed probability � T � � � � � d P ǫ d P = L ǫ ˙ 1 + ǫ ˙ T = exp − ǫ h ( s )d s h ( s )∆ P s 0 s ≤ T � h ( r ) � Let α ǫ ( t ) = � t 1 + ǫ ˙ d r (a variation on time): 0 law of P α ǫ ( . ) under P = law of P ( . ) under P ǫ

Similar definition for derivative of a Poisson functional: F ( P α ǫ . ) − F ( P . ) lim ǫ ε → 0 = � T L ǫ T − 1 0 ˙ Since lim ǫ → 0 h ( s ) d N s we obtain the integration by parts ǫ formula . Some differences with Gaussian case: only a deterministic perturba- tion is allowed, (the integration by parts formula is less immediate).

On smooth functionals of the form F = φ � � τ 1 , . . . , τ n we obtain by a direct calculus n � t φ � � � . . . � ∂φ D v = − I [0 ,τ i ] ( t ) τ 1 , . . . , τ n ∂x i i =1 Good properties : ( D v ) ∗ concides with stochastic integrals for pre- dictable processes ( Clark–Ocone–Karatzas ), the chain rule is sat- isfied.

A drawback : the analysis of divergence is more complicated (w.r.t. chaotic point of view) A serious drawback : P T is not derivable (not in the domain of the operator D v )! � P T = I [0 ,T ] ( τ i ) i ≥ 1 is not a smooth function of the jump times.

A remark : the domains of the operators D c and D v are completely different. Typical derivable functionals are: • stochastic integrals � T 0 h ( s ) d N s (or iterated stoch. int.) for the operator D c ; � � for the operator D v . • smooth functions φ τ 1 , . . . , τ n