Quadratic Variation in SDEs.Let X t be a stochastic process satisfying: X t = X 0 + ∫ 0 t μ ( s , ω ) d s + ∫ 0 t ν ( s , ω ) d B s Shorthand: d X t = μ t d t + ν t d B t where the last integral is a Brownian motion integral, and where μ ( t , ω ) , ν ( t , ω ) are F t adapted L 2 functions. (In class we so far only did the case where μ and ν are deterministic).Let f(t,x) be a twice differentiable deterministic function.The usual presentation of Ito's lemma is: d f ( t , X t ) = ( ∂ f ∂ t + μ t ∂ f ∂ x + 1 2 ∂ 2 f ∂ x 2 ν t 2 ) d t + ∂ f ∂ x ν t d B t The professor offered us this shorthand: d f ( t , X t ) = ∂ f ∂ t d t + ∂ f ∂ x d X t + 1 2 ∂ 2 f ∂ 2 x ( d X t ) 2 He explained that the notation ( d X t ) 2 is to be interpreted as the taking the quadratic variation whenever the algebra would suggest you multiply the differentials. For example, d B t d B t = d ⟨ B t , B t ⟩ T = d ( T ) a.s. = d t (this last step has its own answer on s.e.) Why is the formal multiplication of (stochastic) differentials interpreted as the quadratic variation?

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Quadratic Variation in SDEs.Let       X    t   be a stochastic process satisfying:            X      t        =          X      0        +          ∫      0      t        μ    (    s    ,    ω    )              d        s    +          ∫      0      t        ν    (    s    ,    ω    )              d              B      s      Shorthand:       d        X    t    =      μ    t          d    t  +      ν    t        d        B    t  where the last integral is a Brownian motion integral, and where   μ  (  t  ,  ω  )  ,  ν  (  t  ,  ω  ) are             F        t   adapted       L    2   functions. (In class we so far only did the case where   μ and   ν are deterministic).Let f(t,x) be a twice differentiable deterministic function.The usual presentation of Ito&#039;s lemma is:            d        f    (    t    ,          X      t        )    =          (                                    ∂            f                                ∂            t                          +                  μ          t                                      ∂            f                                ∂            x                          +                  1          2                                                    ∂              2                        f                                ∂                          x              2                                                ν          t          2                    )              d        t    +                  ∂        f                    ∂        x                    ν      t                  d              B      t      The professor offered us this shorthand:            d        f    (    t    ,          X      t        )    =                  ∂        f                    ∂        t                        d        t    +                  ∂        f                    ∂        x                        d              X      t        +          1      2                                ∂          2                f                              ∂          2                x                  (          d              X      t              )      2      He explained that the notation   (      d        X    t        )    2   is to be interpreted as the taking the quadratic variation whenever the algebra would suggest you multiply the differentials. For example,       d        B    t        d        B    t    =      d    ⟨      B    t    ,      B    t        ⟩    T    =      d    (  T  )    a.s.  =      d    t (this last step has its own answer on s.e.) Why is the formal multiplication of (stochastic) differentials interpreted as the quadratic variation?

Steinherrjm · Accepted Answer

Step 1Well, to my modest opinion, we only have the slightly boring answer: Because it works. Via the ito isommetry one can show, that for any bounded, predictable       f    1    ,      f    2   and semimartingales X, Y one has:  ⟨      ∫    0    ⋅        f    s    1    d      X    s    ,      ∫    0    ⋅        f    s    2    d      Y    s        ⟩    t    =      ∫    0    t        f    s    1        f    s    2    d  ⟨  X  ,  Y      ⟩    s  Step 2which is basically the justification of doing it. Furthermore, one can even make it rigorous with some algebraic structures.

Quadratic Variation in SDEs. Let Xt be a stochastic process satisfying: X_t=X_0+int_0^t mu (s, omega)ds+int_0^t ν(s,omega)dB_s

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